Continue reading "Hilbert C*-modules"
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is called a pre-Hilbert C*-module over $\mathcal{A}$. The completion of this module with respect to the norm
$$||x||^2 = ||\langle x, x \rangle||^{1/2}$$
is called a right Hilbert C*-module over $\mathcal{A}$ or just a Hilbert $\mathcal{A}$-module.
By convention the above sesquilinear form is linear in the second variable and conjugate linear in the first, this is not what I’m used to, but you can’t argue with conventions, so let’s stick with this for now.
Here are some norm-inner product identities which will come in handy. They can all be found in the first chapters of [1] and [2].
Property (2) resembles Cauchy-Schwartz inequality, and property (4) is also very similar to the Hilbert space setting.
Let $\mathcal{M}$ and $\mathcal{N}$ be $\mathcal{A}$-modules. A morphism $f$ between $\mathcal{M}$ and $\mathcal{N}$ is a continuous $\mathcal{A}$-module homomorphism, that is, a continuous $\mathbb{C}$-linear map $f: \mathcal{M}\to \mathcal{N}$ satisfying
$$f(xa) = f(x) a \qquad \text{for all } x\in \mathcal{M} \text{ and for all } a \in \mathcal{A}.$$
The collection of all such morphisms from $\mathcal{M}$ to itself will be denoted $Hom_\mathcal{A}(\mathcal{M})$ and is a Banach algebra with respect to the “usual” operator norm, i.e.
$$||f|| = \sup\{ ||f(x)|| ~: ~ x\in \mathcal{M}, ||x|| = 1 \}.$$
This follows from the fact that $Hom_\mathcal{A}(\mathcal{M})$ is a closed subalgebra of the algebra of bounded $\mathbb{C}$-linear maps on $\mathcal{M}$, know from Banach space theory to be complete. We donete by $\mathcal{L}(\mathcal{M})$ the collection of maps $f\in Hom_\mathcal{A}(\mathcal{M})$ for which there exists a map $s \in Hom_\mathcal{A}(\mathcal{M})$ such that
$$\langle f(x), y \rangle = \langle x, s(y) \rangle \qquad \text{for all } x, y \in \mathcal{M}.$$
The map $s$ is uniquely determined and will be denoted $f^\star$, imitating the adjoint operator for Hilbert spaces. $\mathcal{L}(\mathcal{M})$ is easily seen to be a closed subalgebra of $Hom_\mathcal{A}(\mathcal{M})$, hence a Banach algebra in its own right. Elements of $\mathcal{L}(\mathcal{M})$ are called adjointable operators. Remarkably the collection $\mathcal{L}(\mathcal{M})$ can also be characterized simply as the set of all maps $f$ on $\mathcal{M}$ for which there exists an adjoint map $f^\star$ such that
$$\langle f(x), y \rangle = \langle x, f^\star(y) \rangle$$
the point being that continuity and $\mathcal{A}$-linearity of both $f$ and $f^*$, and uniqueness of $f^*$ are all implied (see Lemma 2.1.1 [2]).
$\mathcal{L}(\mathcal{M})$ is actually a C*-algebra since the involution $f\mapsto f^*$ is an isometry, and
\[
\begin{align}
|| f^*f || & := \sup\{ ||f^*f x|| ~ : ~ ||x|| \leq 1 \} \\ & = \sup\{ ||\langle f^*f x, y \rangle || ~ : ~ ||x||, ||y|| \leq 1 \} \\
& \geq \sup \{ ||\langle f^*fx, x \rangle|| ~ :~ ||x|| \leq 1 \} \\ & = \sup \{ ||\langle fx, fx \rangle||~ :~ ||x|| \leq 1\} = ||f||^2
\end{align}
\]
where the second equality follows from Proposition 1 (4).
Another interesting subalgebra of $Hom_\mathcal{A}(\mathcal{M})$, denoted $\mathcal{K}(\mathcal{M})$, is the closed linear span of the collection of all operators of the form
$$ \theta_{x,y}: \mathcal{M} \to \mathcal{M} \qquad \text{given by } \theta_{x,y}(z) = x\langle y, z \rangle$$
where $x, y$ are fixed elements in $\mathcal{M}$. Elements of $\mathcal{K}(\mathcal{M})$ are called compact operators, since they mirror the characterization of compact operators on Hilbert spaces as the closure of the span of the rank-1 operators, and as such if $\mathcal{A} = \mathbb{C}$, they are precisely the usual compact operators. Keep in mind though that these operators need not in general be compact operators when viewed as maps between the underlying Banach space (see [2] page 10 for a counterexample).Since $\theta_{x,y}^* = \theta_{y,x}$ one has that $\mathcal{K}(\mathcal{M}) \subset \mathcal{L}(\mathcal{M})$, but equality does not hold in general. Recently it was shown by E. Troitsky that for an adjointable map $T:\mathcal{M} \to \mathcal{N}$ with $\mathcal{N}$ countably generated, the two notions of compactness actually coincide.
Let’s check out some of the common operations that can be done with the Hilbert C*-modules. Given a countable collection of Hilbert $\mathcal{A}$-modules $\mathcal{M}_i$, their direct sum is defined to be the the set
$$\bigoplus \mathcal{M}_i := \left\{ (x_i) \subset \prod \mathcal{M}_i ~~ : ~~ \sum \langle x_i, x_i \rangle \text{ is (norm) convergent in } \mathcal{A} \right\}$$
with inner produce
$$\langle (x_i), (y_i) \rangle := \sum \langle x_i, y_i \rangle.$$
It follows easily from Proposition 1 (2) that the inner product is well defined on $\bigoplus \mathcal{M}_i$, and the proof of completeness with respect to the induced norm is similar to the usual proof of completeness of $l^\infty$ (see for instance [1] Example 1.3.5).
The tensor product is not as straight forward. Let $\mathcal{M}, \mathcal{N}$ be Hilbert C*-modules over $\mathcal{A}$ and $\mathcal{B}$ respectively. The natural starting point is the tensor product $\mathcal{M} \otimes_{alg} \mathcal{N}$ of $\mathcal{M}$ and $\mathcal{N}$ (treated as linear spaces over $\mathbb{C}$). We are going to define an inner product and a $\mathcal{A} \otimes_{min} \mathcal{B}$-module structure on $ \mathcal{M} \otimes_{alg} ,\mathcal{N},$ where $\mathcal{A} \otimes_{min} \mathcal{B}$ is the minimal tensor product of the C*-algebras $\mathcal{A}$ and $\mathcal{B}$, henceforth simply denoted $\mathcal{A} \otimes \mathcal{B}$. This is commonly done in the following two ways
In both cases the action of $\mathcal{A} \otimes \mathcal{B}$ is defined in the natural way, by its action on decomposable tensors; $m\otimes n \cdot a \otimes b := ma \otimes nb$. There is a lot to check here, which I will not be covering in this post, as it would get way to bloated, but it is all covered well in both [2] and [1] or any introductory book to Hilbert C*-modules.
For projective and injective limits of Hilbert C*-modules over over fixed C*-algebras, one needs to be a little careful as to the type of directed system and inverse systems we take the limit over, since the category of Hilbert C*-modules over a fixed C*-algebra seems not to be complete. See this post for more on projective and injective limits. Here is one example where an injective (direct) limit does exists:
If $(\mathcal{M}_i, T_{ij})$ is a directed system of Hilbert $\mathcal{A}$-modules, with $T_{ij}:\mathcal{M}_i \mapsto \mathcal{M}_j$ (whenever $i \prec j$) are $\mathcal{A}$-linear module homomorphisms which preserve the inner product (but not necessarily adjointable) , we can define an inner product on the algebraic direct limit of the modules;
$$\bigsqcup \mathcal{M}_i/\sim $$
(recall that $m_i \sim m_j$ if there exist a $k$ such that $i, j \prec k$, and $T_{ik}(m_i) = T_{jk}(m_j)$). One can do this by picking arbitrary representatives, that is $\langle [m_i], [m_j] \rangle := \langle T_{ik}m_i, T_{jk} m_j \rangle$, since the module maps in the directed system preserves inner products. This turns out to be a Hilbert C*-module over $\mathcal{A}$ with the desired universal properties as is shown in Proposition 1.3 of this article.
Every Hilbert C*-module has the structure of a Banach space over $\mathbb{C}$ (if we neglect the module structure) and morphisms of Hilbert C*-modules as defined above are continuous linear maps of the underlying Banach spaces. Hence many of the results that hold for Banach spaces carry over to the setting of Hilbert C*-modules. Off the top of my head, among these are the usual suspects from Banach space theory; the open mapping theorem, the closed graph theorem and the uniform boundedness principle. Additionally, as we have seen, the collection of adjointable operators forms a C*-algebra, on which we may use the spectral theory and functional calculus as usual without any hassle.
Here is a list of things that do carry over, and some known pathologies for Hilbert C*-modules. The list is by no means exhaustive, but hopefully it gets you quickly up to speed. In what follows $H$ will be a Hilbert space over $\mathbb{C}$ and $\mathcal{M}$ will be a Hilbert $\mathcal{A}$-module. The inner products of both $H$ and $\mathcal{M}$ will be denoted by $\langle \cdot, \cdot \rangle$, as this is unlikely to cause serious confusion.
Riesz representation theorem and sesquilinear forms. On Hilbert spaces the Riesz representation theorem asserts that every bounded linear functional $\phi$ on $H$ is of the form $x \mapsto \langle x, y \rangle$ for some uniquely determined $y\in H$, and that $||\phi|| = ||y||.$
For Hilbert C*-modules we have a similar statement, namely that the C*-algebra $\mathcal{K}(\mathcal{M}, \mathcal{A})$, where $\mathcal{M}$ is an $\mathcal{A}$-module, consists entirely of functions of the form $ f_x(y) = \langle x, y \rangle$, and it’s easy to check uniqueness and that $||f_x|| = ||x||$.
On Hilbert spaces there is is also a Riesz representation theorem for continuous sesquilinear forms, which says that there is a 1-1 correspondence between contunuous (or bounded) sesquilinear forms and bounded operators on a Hilbert space. The correspondence is give by
$$[\cdot, \cdot ] \mapsto T\qquad \text{where} \qquad [x, y ] = \langle Tx, y \rangle.$$
It would be nice to find some similar statement (or counterexamples) for Hilbert C*-modules.
Polarization Identity: The identity $$\langle x, y \rangle = \frac{1}{4} \sum_{n=0}^3 i^n \langle x + i^ny, x + i^ny \rangle$$ holds for any sesquilinear form, so it clearly also holds for inner products on Hilbert C*-modules.
Polar decomposition The polar decomposition from Hilbert space theory, decomposing operators on Hilbert spaces into a product of a partial isometry and a positive operator does not in general work on Hilbert C*-modules. That is, if $T$ is an adjointable operator on a Hilbert $\mathcal{A}$-module $\mathcal{M}$, then it does not follow that the partial isometry $U$ in the polar decomposition $T = U|T|$ of $T$ is adjointable. The operator $|T|$ is found using functional calculus on the C*-subalgebra of $\mathcal{L}(\mathcal{M})$ generated by $T$ and $T^*$, so this is an adjointable module morphism, but recall that the partial isometry $U$ is contained in the von Neumann algebra generated by $T$, which may not be contained in the C*-algebra of adjointable operators. There are situations where the decomposition works though. One such situation is when both $T$ and $T^*$ have dense ranges (see Proposition 3.8 [2]).
Orthogonal complementing subspace: The most striking pathology of Hilbert C*-modules is the non-existence of an orthogonal complementing subspace. On Hilbert spaces any closed subspace has a natural complementing subspace, namely the orthogonal complement. This does not hold in the setting of Hilbert C*-modules. That is, if $K \subset \mathcal{M}$ is a closed submodule of a Hilbert C*-module $\mathcal{M}$, then $$K^\perp = \{ x\in \mathcal{M} ~:~ \langle x, y\rangle = 0 ~\text{for all } y\in K \}$$ is indeed also a closed submodule of $\mathcal{M}$, but $K$ and $K^\perp$ need not be complementary, that is $K\oplus K^\perp$ may not be isomorphic to $\mathcal{M}$. In fact, there may be no complementary submodule at all. However, if either of the following conditions holds;
then $K \oplus K^\perp \simeq \mathcal{M}$ (section 2 of [1] and Th 1.4.6 [2]).
Existence of a minimal distance : On Hilbert spaces, for any closed subspace $K\subset H$ and any $x\in H\backslash K$ there exists an element $y \in K $ such that $$||x – y|| = \inf_{z\in K} || x – z|| \tag{$(\star)$}.$$
This does not hold for Hilbert C*-modules in general. Try to find a counterexample, or look at Exercise 3 below.
Dual space: The dual space of a Hilbert space $H$ has a natural Hilbert space structure, with respect to the inner product $\langle \phi_x, \phi_y \rangle = \langle x, y \rangle$ where $\phi_x, \phi_y \in H^\star$ are the functionals corresponding to $x$ and $y$ in $H$ by Riesz. The Riesz representation theorem is an (antilinear) isomorphism of these Hilbert spaces, hence all Hilbert spaces are isomorphic to their bidual. Such spaces are called reflexive. For Hilbert C*-modules the dual space of a $\mathcal{A}$-module $\mathcal{M}$, denoted $\mathcal{M}^*$, is defined as
$$M^* : = Hom_\mathcal{A}(\mathcal{M}, \mathcal{A})$$
and is a complete banach module over $\mathcal{A}$ with respect to the usual operator. The action of $\mathbb{C}$ and $\mathcal{A}$ on $\mathcal{M}^*$ is given by $(\lambda \cdot f)(x) = \overline{\lambda} f(x)$ and $(f \cdot a)(x) = a^*f(x)$ respectively, which makes the natural inclusion of $\mathcal{M}$ into $\mathcal{M}^\star$ given by
$$x \mapsto \langle x, \cdot \rangle =: \hat{x}$$
an imbedding of Banach spaces. When it is surjective, the module $\mathcal{M}$ is called self-dual. Two important property of self-dual Hilbert C*-modules are
When the C*-algebra $\mathcal{A}$ is a vN-algebra the dual module $\mathcal{M}^*$ can be given a Hilbert $\mathcal{A}$-module structure with an inner product for which $\langle f, \hat{x} \rangle := f(x)$ for all $f,\hat{x}\in M^*$ (see Theorem 3.2.1 of [1]).
Bidual and Reflexivity
Luckily the bidual $\mathcal{M}^{**} = Hom_\mathcal{A}(\mathcal{M}^*, \mathcal{A})$ of a Hilbert C*-module always admits a Hilbert C*-module structure, determined by the inner product
$$\langle F, G \rangle = F(\tilde{G})$$
where $F, G\in \mathcal{M}^{**}$ and $\tilde{G} \in \mathcal{M}^*$ is given by $\tilde{G}(x) := G(\hat{x})$ (see “dual space” above for the notation). The norm induced by this inner product is the same as the usual operator norm (Theorem 4.1.4 [1]).
Existence of a basis and dimensionality : This may be silly, but for correctness let’s assert that C*-algebras are not division rings, which makes it difficult to talk about dimensions and basis in general. There are non-trivial examples where a basis does exist, like $H_\mathcal{A}$, where $\mathcal{A}$ is (say) a finite dimensional C*-algebra with basis $\{ a_j\}_{j=1}^n$. The $\mathcal{A}$-module $H_\mathcal{A}$ has basis $\{(e_i, a_j) ~ :~ j= 1, …, n,~ i = 1, … \}$. One often has to make due with generating sets rather than a basis. Many results rely on the existence of a countable generating set, and as a sidenote, a useful characterization of countably generated $\mathcal{A}$ modules $\mathcal{M}$ is the following; $\mathcal{M}$ is countably generated if and only if $\mathcal{K}(\mathcal{M})$ is $\sigma$-unital (if and only if $\mathcal{K}(\mathcal{M})$ posesses a strictly positive element).
Spectral theory and functional calculus: As previously mentioned, the theory carries over to the adjointable operators without issues, since $\mathcal{L}(\mathcal{M})$ is a C*-algebra.
Characterization of Positive elements: Just as in the case of operators on (complex) Hilbert spaces, a map T is a positive element of the C*-algebra $\mathcal{L}(\mathcal{N})$ if and only if
$$\langle T x, x \rangle \geq 0 \qquad \text{for all } x\in \mathcal{M}.$$ (see [1] Prop. 2.1.3).
Unitary operators: An operator $u: \mathcal{M} \to \mathcal{N}$ is unitary if and only if it is surjective isometric and $\mathcal{A}$-linear (Th. 2.3.5 [2])
Boundedness Let T be any map on a Hilbert $\mathcal{A}$-module $\mathcal{M}$. Then $T\in Hom_\mathcal{A}(\mathcal{M})$ if and only if there exists a $K \geq 0$ such that $\langle Tx, Tx \rangle \leq K \langle x, x \rangle$ (see Theorem 2.1.4 of [1]).
Invertibility For Hilbert spaces an operator $T: H \to H$ is invertible if and only if $Im(T)$ is dense in $H$ and $T$ is bounded away from zero, that is, there exists a real constant $k>0 $ such that $||Tx|| \geq k||x||$ for all non-zero x.
For Hilbert C*-modules a similar statement holds, at least for self adjoint operators. Let $t^* = t\in \mathcal{L}(\mathcal{M})$ be self adjoint, then $t$ is invertible if and only if there exists a $k>0 $ such that $||tx|| \geq k||x||$ for all $x\in \mathcal{M}$. (See Lem. 3.1 [2])
Here are some common and useful examples to keep in mind when working with Hilbert C*-modules. First of which, any C*-algebra $\mathcal{A}$, (or any right ideal of $\mathcal{A}$) can be treated as a Hilbert $\mathcal{A}$-module with respect to the inner product
$$\langle x, y \rangle = x^*y.$$
(note again the linearity in the second term!). I will denote this module by $\mathcal{M}_\mathcal{A}$.
The standard Hilbert C*-module over $\mathcal{A}$ is defined to be the Hilbert $\mathcal{A}$-module
$$H_\mathcal{A} := \bigoplus^\infty \mathcal{M}_\mathcal{A},$$
that is, the set of all sequences $(a_i)\subset \mathcal{A}$ such that $\sum a_i^*a_i$ is norm convergent, with inner product $$\langle (a_i), (b_i) \rangle = \sum a_i^*b_i.$$
Next, the continuous section of bundles of Hilbert subspaces of a common Hilbert space $H$, over a compact Hausdorff space $X$, can be endowed with a Hilbert $C(X)$-module structure by pointwise operations, that is if $\xi$ is a continuous section, and $f\in C(X)$, then define the $C(X)$-module action by $(f\cdot \xi)(x) = f(x)\xi(x)$, and if $\nu$ is another continuous section, define the $C(X)$-valued inner product $(\langle \xi, \nu \rangle)(x) = \langle \xi(x), \nu(x) \rangle$ which is clearly a continuous function, being the composition of continuous functions. One can check this inner product satisfies the conditions of Definition 1.
The following identities are worth memorizing as they are often tacitly employed in the literature. Let $\mathcal{M}$ and $\mathcal{E}$ be Hilbert $\mathcal{A}$-modules, and $\mathcal{N}$ be a Hilbert $\mathcal{B}$ module, then
Note how the first and fifth equality together implies that $\mathcal{L}(\mathcal{A})$ is isomorphic to the multiplier algebra of $\mathcal{A}$ and that $\mathcal{A}$ imbeds into $\mathcal{L}(\mathcal{A})$ as the C*-subalgebra of compact operators.
In general we only have an imbedding $\mathcal{L}(\mathcal{M}) \otimes \mathcal{L} (\mathcal{N}) \rightarrow \mathcal{L}(\mathcal{M} \otimes \mathcal{N})$ given in the most natural way as the unique lift of the bilinear map from $\mathcal{L}(\mathcal{M}) \times \mathcal{L}(\mathcal{N}) \to \mathcal{L}(\mathcal{M} \otimes \mathcal{N}),$ determined by sending $(s, t) \mapsto s\otimes t,$ where $(s\otimes t)(x\otimes y) = sx\otimes ty,$ to the space $\mathcal{L}(\mathcal{M}) \otimes \mathcal{L} (\mathcal{N}).$
Here are some useful facts which I have left as exercises, most of which are taken from the cited references where they pop-up in various proofs and remarks.
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]]>Continue reading "Constructing new C*-algebras – Universal C*-algebras (1)"
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]]>This series of posts introduces the notion of universal C*-algebras. In this post I introduce the general construction of a universal C*-algebra, and a simple method for computing the the universal C*-algebra of a family of unitaries and (sufficiently nice) relations. In subsequent posts I hope to cover more general constructions with possibly non-unitary operators and more subtle relations.
Like most universal object, one should think of a universal C*-algebra generated by a family of operators and relations as the “largest” C*-algebra containing operators satisfying said relations and containing no “excess junk”. Formally, if it exists, the universal C*-algebra generated by a family $\{u_i\}_{I}$ of operators (or indeterminates) and relations $\mathcal{R}$, is a C*-algebra with the following universal property:
The surjectivity of $\phi$ captures the idea of being the “largest” C*-algebra with the give properties, while the uniqueness of the map $\phi$ in the above definition captures the idea of having no “excess junk”. If it exists, it is clearly unique up to isomorphism (do you see why?)
One can also define the universal C*-algebra as the initial object in the category of all C*-algebras generated by some elements $\{ v_i \}$ satisfying relations $\mathcal{R}$ with morphisms the C*-morphisms which map generators to generators, though it’s not immediately clear how this characterization would be useful.
Let’s show how one would construct such a C*-algebra. If $\{v_i \}$ is a set of indeterminates and $\mathcal{R}$ a collection of relations on this set, we may form the free *-algebra generated by $(\{ v_i \} | \mathcal{R})$ as the quotient of the free *-algebra of all (non-commuting complex) polynomials in $v_i$ and $v_i^*$, modulo the ideal generated by the relations $\mathcal{R}$. Denote this algebra by $F(\{ v_i \} | \mathcal{R})$. The completion of $F(\{ v_i \} | \mathcal{R})$ with respect to the C*-seminorm
$$ ||a|| = \sup \{ ||\pi(a)|| ~ | ~ \pi \text{ is a *-representation of } F(\{ v_i \} | \mathcal{R}) \}$$
is the universal C*-algebra given by the indeterminates $v_i$ and relations $\mathcal{R}$. We will denote this algebra by $C^*(\{ v_i\}, \mathcal{R})$. The above construction is nonsensical if one does not specify an upper bound on the norms on $\pi(v_i)$ for all generators $v_i$, and the lack of such an upper is easily seen to be equivalent to the non-existence of a universal C*-algebra.
If we now specialize to the case where the indeterminates $\{ u_i \}_{I}$ are all known to be unitaries and the relations $\mathcal{R}$ can be expressed by means of products alone, the universal C*-algebra can be constructed in a particularly simple way,
Here are some of the many examples of the use of universal construction of C*-algebras, chose at random.
Among the most common examples are the Cuntz algebras $\mathcal{O}_n$ which are the algebras given by $n$ isometries $u_1,…, u_n$ satisfying $\sum u_iu_i^* = I.$
Next consider the the C*-algebra generated by unitaries satisfying the relations
$$UV = e^{2\pi i\theta} V U.$$
if $\theta = 0$ then, employing the above proposition, we have $C^*(\{ u, v | uv= vu \}) = C^*(\mathbb{Z}^2) \simeq C(\mathbb{T}^2)$, where the last inequality follows from duality theory, which says that $C^*(G) = C(G^*)$ for any locally compact abelian group (here $G^*$ denotes the (Pontryagin) dual group). If $\theta$ is irrational we get the socalled irrational rotation algebra or noncommutative torus $\mathcal{A}_\theta$.
As a final example let $u, w$ be unitaries satisfying the relations $$ uw = w^p u.$$ for some positive integer $p$. Applying the above proposition we thus have that
$$C^*(\{ u, w | \mathcal{R}\}) = C^*(\mathbb{Z}(\frac{1}{p}) \rtimes_\alpha \mathbb{Z} ) \simeq C^*(\mathbb{Z}(\frac{1}{p}))\rtimes_\alpha \mathbb{Z}$$
where $\mathbb{Z}(\frac{1}{p})$ are the p-adic rationals, and $\mathbb{Z}$ acts by $z \cdot n = p^n z.$ The last identification is a black magic from the theory of cross-product C*-algebras, which was introduced in this post, but unfortunately I didn’t prove this particular identification (the proof is rather straightforward, so I will probably add it in the near future)
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]]>Continue reading "Constructing new C*-algebras – Crossed product."
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]]>The next proposition and proof, in addition to its intrinsic value, will come in handy later, when we define two norms on a dense subset of the crossed-product C*-algebra.
One can check these are indeed well defined representations, and that $\lambda^H_g$ is unitary, since it acts on $\xi \in l^2(G, H)$ by permuting the indices $G$, hence $||\lambda_g^T\xi||^2 = \sum_{h\in G }||\xi(g^{-1}h)||^2 = \sum_{h \in G }||\xi(h)||^2 = ||\xi||^2$. Additionally
\begin{align*}
((\lambda^H_g\tilde{\pi}(a) (\lambda^H_g)^*) \xi)(s)
&= (\tilde{\pi}(a) (\lambda^H_g)^*) \xi)(g^{-1}s) = (\tilde{\pi}(a) (\lambda^H_{g^{-1}})) \xi)(g^{-1}s) \\
& = [ \pi(\alpha_{(g^{-1}s)^{-1}}(a))] (\lambda^H_{g^{-1}})(\xi)(g^{-1}s)
= \pi(\alpha_{s^{-1}}( \alpha_g(a)) )\xi(s)\\
& = (\tilde{\pi}(\alpha_g(a))\xi)(s)
\end{align*}
for all $\xi\in l^2(G, H)$ and $s\in G$, where we used that $(\lambda^H_g)^* = \lambda_{g^{-1}}^H$. So the covariance relation (\ref{eq:covariance-relation}) also holds.
Before stating the definition of the crossed product, note that if $(\pi, U)$ is a covariant representation of $(\mathcal{A}, G, \alpha)$, by the covariance relation, we have the equalities
$$\left( \pi(a) U_g \right)^\star = \pi(\alpha_{g^{-1}}(a^\star))U_{g^{-1}}~ \text{ and }~ (\pi(a)U_g)(\pi(b)U_h) = \pi(a\alpha_g(b))U_{gh}.$$
Consequently to any covariant representation $(\pi, U)$ of a dynamical system $(\mathcal{A}, G, \alpha)$ we can associate the C*-algebra
$$C^\star(\pi, U) = \overline{ \text{span}\{ \pi(a)U_g ~: ~ a\in \mathcal{A}, ~ g\in G\} }.$$
A natural question would then be if the above defined $C^*$-algebra could be a working definition for the crossed product of $\mathcal{A}$ and $G$. One issue is that we have not address the arbitrariness in the choice of covariant representation.
To remedy this, let’s instead define the crossed product $G \rtimes_\alpha \mathcal{A}$ by the following universal property which was first introduced by Raeburn in a slightly more general form. It relies on the concept of a covariant homomorphism of a dynamical system $(\mathcal{A}, G, \alpha)$ into a $C^*$-algebra $\mathcal{B}$ which is defined as a pair $(j_\mathcal{A}, j_G)$, where $j_\mathcal{A}: \mathcal{A} \to \mathcal{B}$ is a $\star$-homomorphism and $j_G: G\to U\mathcal{B}$ (the unitaries) is a group homomorphisms satisfying the relation $j_\mathcal{A}(\alpha_g(a)) = j_G(g)j_\mathcal{A}(a)j_G(g)^*$.
and is unique in the following sense; If $(\mathcal{B}, j_\mathcal{A}’, j_G’)$ also satisfy the above conditions for some unital C*-algebra $\mathcal{B}$, then there exists an isomorphism $\phi: \mathcal{A}\rtimes_\alpha G\to B $ such that $j_\mathcal{A} = \phi\circ j_\mathcal{A}’$ and $j_G = \phi \circ j_G$.
Below we show the existence of such a crossed product C*-algebra. As one would expect, historically that definition preceded the the one given in Definition 3, but working with the universal definition above has its benefits.
To highlight the similarities with the semidirect product of groups, I will try to give the algebraic intuition behind the the construction of the concrete realization of the crossed product $\mathcal{A}\rtimes_\alpha G$.
Let’s start with the vector space $W = \{ \sum_{(a,g)\in I}(a, g) ~| ~ I\subset \mathcal{A}\times G, I \text{ is finite} \}$ over $\mathbb{C}$ consisting of all finite formal sums of elements in $\mathcal{A}\times G$ with scalar multiplication distribuiting over sums and defined elementwise by $\lambda (a, g) = (\lambda a, g)$.
Addition is treated formally, that is $\sum_{(a,g)\in I}(a, g) + \sum_{(a,g)\in J}(a, g) = \sum_{(a,g)\in I \sqcup J }(a, g)$ where $\sqcup$ is the disjoint union of the index sets.
Just as with the semidirect product, we may define a product operation on $W$, by
\begin{align*}
(a, g_1)\cdot (b, g_2) = (a \alpha_{g_1}(b), g_1g_2).
\end{align*}
making $W$ an algebra over $\mathbb{C}$.
The subspace $V \subset W$ defined as the span
$$V = \text{span}\{ (a_1, g) + (a_2, g) – (a_1 – a_2, g ) ~ | ~ (a_1, g), (a_2, g) \in \mathcal{A} \times G\}$$ is a two sided ideal of $W$ since
\begin{array}{cc}
(b, h) \cdot [(a_1, g) + (a_2, g) – (a_1 – a_2, g)] & = (b\alpha_{h}(a_1),hg) + (b\alpha_{h}(a_2), hg) \\ & – (b\alpha_{h}(a_1) +b\alpha_{h}(a_2) , hg) \in V\\
[(a_1, g) + (a_2, g) – (a_1 – a_2, g)] \cdot (b, h) & = (a_1\alpha_{g}(b), gh) + (a_2\alpha_{g}(b), gh) \\ & – (a_1\alpha_{g}(b) + a_2\alpha_{g}(b), gh) \in V
\end{array}
The quotient algebra $W/V$, consisting of finite sums on the form $\sum_{g\in J\subset G}(a_g, g)$, contains a copy of $\mathcal{A}$, (by the inclusion $a \mapsto (a, e)$) and a copy of $G$ (by the inclusion $g\mapsto (1, g)$), and is called the skew algebra.
One can now identify this algebra with $C_c(G, \mathcal{A})$, the algebra of finitely supported functions from $G$ to $\mathcal{A}$ (well… compactly supported, but recall that $G$ is discrete, so all compact sets are finite), in a natural way by the map
$$ \sum_{g\in G} (a_g, g) \mapsto f \qquad \text{where}\qquad f(g) = a_g.$$
Under this identification, the above defined product induce a product on $C_c(G, \mathcal{A})$ given by:
\begin{array}{ll}
&(f_1\star f_2)(h) = \sum_{g\in G} f_1(g) \alpha_g(f_2(g^{-1}h)).
\end{array}
We define the involution $ f^\star(g) = \alpha_g(f(g^{-1})^*)$, making $C_c(G, \mathcal{A})$ a unital *-algebra. In addition, with $\delta_{a, g} \in C_c(G, \mathcal{A})$ given by
\begin{equation}
\delta_{a, g}(h) = \begin{cases} a & g = h \\ 0 & g\neq h\end{cases}
\end{equation}
the previously defined inclusion maps of $\mathcal{A}$, and $G$, lift to the maps
\begin{equation}
\iota_\mathcal{A}(a) = \delta_{a, e} \qquad \iota_G(g) = \delta_{1, g}
\end{equation}
Some authors write $a$ and $u_g$ for the inclusions $\delta_{a, e}$ and $\delta_{1, g}$ respectively, but I have chosen to stick with this $\delta$ notation.
Now let $(\pi, U)$ be a covariant representation of $(\mathcal{A}, G, \alpha)$ on a Hilbert space $H$. Define $\pi \times U: C_c(G, \mathcal{A}) \to B(H)$ by
$$\pi\times U (f) = \sum_{g\in G}\pi(f(g))U_g \qquad \text{for all }f\in C_c(G, \mathcal{A})$$
then $\pi\times U$ is a representation of $C_c(G, \mathcal{A})$ on $H$, which is non-degenerate, since $\pi\times U(\delta_{1, e}) = \pi(1)U(e) = I_H$ for the identity element $\delta_{1, e}$ of $C_c(G, \mathcal{A})$. Here we needed the group G to be discrete for the sum to be finite. For general locally compact groups one needs to use an integral with respect to the Haar measure, and take into account the “modular” term which arises when the left an right Haar measures don’t coincide. For a thorough exposition to the general theory of crossed products consult the book of Dana Williams ( Products of C*-Algebras).
Next we wish to complete this *-algebra with respect to a C*-norm, and here there are two common choices. For the first, we need the following lemma
The definition now reads
It is straightforward to check that $||\cdot||_r$ is indeed a C*-seminorm, and employing Lemma 1 we see that $||f||_r = 0 \Rightarrow f = 0$, hence it is a C*-norm. The above definition can also be shown to be independent of choice of faithful representation $\pi$ (see prop. 4.1.5 of Brown and Osawa’s “C∗-Algebras and Finite-Dimensional Approximations”)
Now we introduce a second norm on $C_c(G, \mathcal{A})$, called the universal norm, defined by
\begin{equation}
||f||_u = \sup ||\pi\times U (f)||
\end{equation}
where the supremum is taken over all covariant representations $(\pi, U)$ of $(\mathcal{A}, G, \alpha )$.
We now prove that the above constructed C*-algebra together with the inclusion maps $\iota_\mathcal{A}$ and $\iota_G$ defined by equation (TODO), is the universal object of Definition 3.
For a C*-algebra $\mathcal{B}$ we may define the category $\text{rep}(\mathcal{B})$ of all non-degenerate representations of $\mathcal{B}$\footnote{ This is clearly not a well defined set, as we have not fixed a Hilbert space for our representations. Such categories are called large, and the ways to overcome the set theoretic paradoxes is outside the scope of this thesis, and the competence of the author.}. If $\pi$ and $\pi’$ are two non-degenerate representations of $\mathcal{B}$ on Hilbert spaces $H$ and $H’$ respectively, then we define the morphisms from $\pi$ to $\pi’$ to be the bounded equivariant linear maps $\phi: H\to H’$, where “equivariant” means
$$\pi'(a)(\phi v) = \phi(\pi(a) v) \qquad \text{for all $a\in \mathcal{B}$ and $v\in H$}.$$
We denote the set of such morphisms $\text{hom}_\mathcal{B}(\pi, \pi’)$. As usual we define composition of morphisms as the composition of maps, and the identity morphisms are the identity maps.
Similarly for a discrete group $G$, we let $\text{rep}(G)$ denote the category of all unitary representations of a group $G$ whose morphisms are again the bounded equivariant maps. We denote the set of all morphism from $U$ to $U’$ by $\text{hom}_{G}(U, U’)$.
For a dynamical system $(\mathcal{A}, G, \alpha)$ we let $\text{rep}(A, G, \alpha)$ be the category whose objects are the covariant representations of $(\mathcal{A}, G, \alpha)$. As for the morphisms, if $(\pi, U)$ and $(\pi’, U’)$ are covariant representation on Hilbert spaces $H$ and $H’$ respectively, then we simply define
$$\text{hom}_{(A, G, \alpha)}((\pi, U), (\pi’, U’)) = \text{hom}_G(U, U’) \cap \text{hom}_{\mathcal{A}}(\pi, \pi’)$$
That is, the continuous linear maps from $H$ to $H’$ that commute with both actions.
Next, we define a map
$$L :\text{rep}(A, G, \alpha) \to \text{rep}(\mathcal{A}\rtimes_\alpha G) $$
by sending the objects $(\pi, U) \mapsto \pi\times U$ and acting as the identity on the morphisms.
If $\pi\times U = \pi’\times U’$ then we have that
\begin{align*}
\pi(a) &= (\pi\times U)(\iota_\mathcal{A}(a)) = (\pi’\times U’)(\iota_\mathcal{A}(a)) = \pi'(a) \\
U(g) &= (\pi\times U)(\iota_G(g)) = (\pi’\times U’)(\iota_G(g)) = U'(g). \\
\end{align*}
Thus $L$ is also injective.
Lastly, we need to check that
$$\text{hom}_{(A, G, \alpha)}((\pi, U), (\pi’, U’)) = \text{hom}_{{\mathcal{A}\rtimes_\alpha G}}(\pi\times U, \pi’\times U’).$$
For the first inclusion, let $\phi \in \text{hom}_{(A, G, \alpha)}((\pi, U), (\pi’, U’))$, then
\begin{align*}
\phi((\pi\times U)(f)(v)) &= \phi(\sum_{g\in G} \pi(f(g)) U_g(v))\\
& = \sum_{g\in G} \phi((\pi(f(g)) U_g)(v))) \\
& = \sum_{g\in G} (\pi'(f(g)) U’_g)(\phi(v))\\
& = (\pi’\times U’)(f)(\phi(v)).
\end{align*}
We can conclude that $\phi \in \text{hom}_{{\mathcal{A}\rtimes_\alpha G}}(\pi\times U, \pi’\times U’)$ and
$$\text{hom}_{(A, G, \alpha)}((\pi, U), (\pi’, U’)) \subset \text{hom}_{{\mathcal{A}\rtimes_\alpha G}}(\pi\times U, \pi’\times U’).$$
Conversely, if $\phi\in \text{hom}_{{\mathcal{A}\rtimes_\alpha G}}(\pi\times U, \pi’\times U’)$, we have that
\begin{align*}
\phi(\pi(a)(v)) &= \phi( (\pi\times U)(\iota_\mathcal{A}(a))(v)) \\
& = \pi’\times U'(\iota_\mathcal{A}(a))(\phi(v)) \tag{by equivariance}\\
& = \pi'(a)(\phi(v))
\end{align*}
so $\phi \in \text{hom}_\mathcal{A}(\pi, \pi’)$. A similar argument shows that $\phi \in \text{hom}_G(U, U’)$, which means $\phi\in \text{hom}_{(\mathcal{A}, G, \alpha)}((\pi, U), (\pi’, U’))$ and
$$ \text{hom}_{{\mathcal{A}\rtimes_\alpha G}}(\pi\times U, \pi’\times U’) \subset \text{hom}_{(\mathcal{A}, G, \alpha)}((\pi, U), (\pi’, U’)). $$
Clearly for the identity maps (morphisms) we have $L(id) = id$ and $L(f\circ g) = L(f)\circ L(g)$ for any two morphisms (when compositions make sense). Hence $L$ is a functor with inverse $L^{-1}$ sending $\pi\times U \mapsto (\pi, U)$, and acting as the identity on the morphisms.
This is just a restatement of Proposition 2.40 of Williams book (Crossed products of C*-algebras)[p.59]. If a representation $\pi$ of $\mathcal{A}$ is not unitary equivalent to $\pi’$, then $\text{hom}_\mathcal{A}(\pi, \pi’)$ contains no unitary operator, and so neither will $\text{hom}_G(U, U’) \cap \text{hom}_\mathcal{A}(\pi, \pi’) = \text{hom}_{\mathcal{A}\rtimes_\alpha G}(\pi\times U, \pi’\times U’)$, so we get the immediate corollary,
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]]>Continue reading "Constructing new C*-algebras – Injective and Projective limits"
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]]>First we will need to get our hands dirty with some categorical definitions, whilst keeping things slightly informal. For nets or sequences, one starts with a map that assigns to each element in the index set, an element in some topological space. Completely analogously, if we want to define a notion of limits in a category $C$, one first defines an “index set”, or rather index category, and determines a function, or rather a functor, from this index category to into $C$. This is called a diagram in $C$, and is what one uses to define a limit. By an abuse of terminology, and to make the definition as “functor-free” as possible, I will often refer to the objects in the diagram as the diagram.
Next we need the notion of a cone. A cone over a diagram $D$ in a category $C$ is an object $c$ in $C$ with a collection of “projection” morphisms $p_x$ from $c$ to each element $x$ in the diagram, such that if $f: x\to y$ is a morphism in the diagram $D$, then $p_y = f\circ p_x$. The name stems from the usual depiction of a cone as in the diagram below, and the condition on the projection maps is equivalent to the requirement that each triangle not entirely contained in $D$ commutes.
An analogous (or dual) notion of a cone is a co-cone, which is just like a cone in which all the arrows not entirely contained in the diagram are reversed. That is, a co-cone is an object $c$ together with a family of “inclusion” morphisms $\iota_x: x\to c$, one for each object $x\in D$, satisfying, for each morphism $f:x\to y$ between objects in $D$, the equality
$$\iota_x = \iota_y \circ f. $$
Among all cones over a diagram $D$ one can choose (if it exists) a special cone which is universal in the sense that any other cone factors uniquely through it. More precisely, if $(c, \{ p_x\})$ is this universal cone over $D$, and $(c’, \{\phi_x\})$ is any other cone over the diagram $D$, then there exists a unique morphism $\alpha: c’ \to c$ such that $\phi_x = p_x\circ \alpha $ (see the picture below). This universal cone is what is referred to as the limit of the diagram $D$. Convince yourself this limit is unique (up to unique isomorphism).
There is also a universal co-cone, which is what is called a colimit in $\mathcal{C}$. It is uniquely determined (if it exists) as the object $c$ in $C$ and the family of morphisms $\iota_x$ from $x \in D$ to $c$, by the requirement that for every other co-cone $(c’, \{\phi_x\})$, there exists a unique morphism $\alpha: c \to c’ $ such that $\phi_x = \alpha \circ \iota_x$.
The category of C*-algebras, like the category $Ban_1$ of Banach spaces with contractive maps as morphisms, is complete and co-complete, meaning that all diagrams which are not too large, have limits and colimits. Not too large here means that the index category can be treated as a set and not a class. I will adhere to the following quite standard notation throughout this post:
\begin{array}{cc}
\varinjlim A_i & \text{for the limit}\\
\varprojlim B_i & \text{for the colimit.}
\end{array}
Lastly, it should be mentioned that the above definition is often stated by means of natural transformations of functors. Now enough of this abstract extravagances, let’s get back to the task at hand.
In the next two sections a concrete realization of a colimit and a limit will be defined in cases where the index category is an upward directed set, for some basic categories like sets, topological spaces and most algebraic objects. The proofs of universality will not be supplied in these sections. The third section will deal with limits and colimits of C*-algebras.
Among all limits and colimits, two pop up all the time, the injective (or Direct or Inductive) limit, which is a colimit, and the projective (or inverse) limit, which is a limit. They are characterized by the structure of their index category. Unfortunately the terminology does vary, so be on the alert. Let’s define the projective limit first, as this is somewhat simpler to define.
Let $(\mathcal{A}_i)_{i\in I}$ be a net of objects in a category $\mathcal{C},$ and assume there is a family of morphisms of morphisms $f_{i, j}: A_j \to A_i$ whenever $i\prec j$, such that
then the the net $\{ A_i \}$ together with the family of morphisms $f_{i, j}$ is called an inverse system (this is a special type of diagram in $\mathcal{C}$). The second condition above captures the idea of being “path independent”. When the index set is linearly ordered one usually define the morphisms for each consecutive index, and drops the above condition, as there is only one path between two indices. The limit over this system is called the projective (or inverse) limit of the system, and is denoted
$$\varprojlim (\mathcal{A}_i, f_{i,j}).$$
The name “inverse limit” is justified by the fact that the morphisms $f_{i,j}$ point in the direction of “decreasing” indices. The name “projective limit” stems from the fact that this is a limit in the categorical sense, hence is equipped with a family of “projections” $$p_r: \varprojlim (\mathcal{A}_i, f_{i,j}) \to \mathcal{A}_r.$$ The intuition behind the construction is that the projective limit defines an object by piecing together the collection $\mathcal{A_i}$ and using the family of morphisms to define (and identify) the “overlap”.
For many algebraic objects (rings, groups, (left/right) R-modules, etc ), topological space or sets, one has a concrete realization of the projective limit, given as as follows. In the category of sets the limit is defined as the set,
$$ \varprojlim (\mathcal{A}_i, f_{i,j}) = \{ (a_i) \subset \prod_{i\in I} \mathcal{A_i}~ | ~ a_j = f_{i,j}(a_i) \text{ for all } i\prec j \} $$
with projections $p_r: \varprojlim (\mathcal{A}_i, f_{i,j}) \to \mathcal{A}_r$ given as the usual coordinate projections $(a_i) \mapsto a_r$. For algebraic objects, this set can be made to inherit the algebraic structure form the $\mathcal{A}_i$’s by pointwise operations, making the projections $p_r$ morphisms in the corresponding category. For topological spaces, the set theoretic limit is endowed with the weak topology induced by the projection maps $p_r$. All these objects can be shown to be universal (as defined above) in their respective categories.
Completely analogously to the projective limit, with $(\mathcal{A}_i)_{i\in I}$ a net of objects in a category $\mathcal{C},$ assume there is a family of morphisms $f_{i, j}: \mathcal{A}_i\to \mathcal{A}_j$ for each $i\prec j$ (note that we now map in the direction of increasing indices!), satisfying the following conditions
Then the net $(\mathcal{A}_i)$ together with the family of morphisms $(f_{i,j})$ is called a directed system of objects in $\mathcal{C}.$ The colimit of this system is called the injective limit (or direct limit, or inductive limit) and is denoted $$\varinjlim (\mathcal{A}_i, f_{i,j}).$$ Again, there is no fixed terminology, and one needs to determine from the context what the author means when he/she uses these terms. Similarly to the projective limit, one has a concrete realization of the injective limit for most algebraic objects (groups, rings, (left/right) R-modules, etc.), sets and topological spaces given as follows.
Let $\mathcal{A}’ = \bigsqcup A_i$ be the disjoint union of all $\mathcal{A_i}$’s, and define an equivalence relation on $\mathcal{A}’$ by
$$a \sim b \qquad \text{ if there exists a } k\in I \text{ such that } f_{i,k}(a) = f_{j, k}(b)$$
where we have tacitly assumed $a\in \mathcal{A}_i$, $b\in \mathcal{A}_j$ and we have identified $\mathcal{A}_i$ and $\mathcal{A}_j$ with their image in $\mathcal{A}’$. The quotient $\mathcal{A} = \mathcal{A}’/\sim$, together with the inclusion maps $\iota_i : \mathcal{A}_i \to \mathcal{A} $ sending $a_i$ to the equivalence class in $\mathcal{A}$ containing $a_i$, forms a co-cone over the directed system, which can be shown to be the colimit in the category of sets. This is the (set theoretic) injective limit of the system.
For algebraic objects we would like the set theoretic injective limit $\mathcal{A} $ to inherits the algebraic structure from the $\mathcal{A}_i$’s. This is done in the following way: Let $a_i, a_j \in \mathcal{A}$ and $k$ be such that $i,j \prec k$ (here we finally need the index set to be directed). We define an operation $$\cdot: \mathcal{A} \times \mathcal{A} \to \mathcal{A}$$ by $a_i \cdot a_j := f_{i,k}(a_i) \cdot f_{j, k}(a_j)$, where the later $\cdot$ is the operation in the object $\mathcal{A}_k$. It is not that hard to verify that this operation is well defined on $\mathcal{A}$ (i.e. independent of choice of $k$). The resulting object is the injective limit of the system.
For topological spaces, one endows $\mathcal{A}$ with strong (or final) topology induced by the inclusion maps $\iota_i$. That is the finest or strongest topology for which all $\iota_i$’s are continuous.
We conclude this section by a introducing a notion of isomorphisms om directed systems. A directed system $(\mathcal{A}_i, f_{i, j})$ and $(\mathcal{B}_i, g_{i,j})$ are said to be isomorphic if there exists a family of isomorphisms $\phi_{i}: \mathcal{A}_i \to \mathcal{B}_i$ such that for each $i\prec j$ the following diagram commutes
$$
\newcommand{\ra}[1]{\!\!\!\!\!\!\!\!\!\!\!\!\xrightarrow{\quad#1\quad}\!\!\!\!\!\!\!\!}
\newcommand{\da}[1]{\left\downarrow{\scriptstyle#1}\vphantom{\displaystyle\int_0^1}\right.}
%
\begin{array}{llll}
\mathcal{A}_i & \ra{f_{i, j}} &\mathcal{A}_{j} \\
\da{\phi_i} & & \da{\phi_j} \\
\mathcal{B}_i & \ra{g_{i, j}} &\mathcal{B}_{j} \\
\end{array}
$$
A completely analogous definition exists for inverse systems. As one would expect, isomorphic directed (or inverse) systems have isomorphic colimits (or limits), but there may be non-isomorphic directed systems (inverse systems) which have isomorphic injective limits (projective limits).
Now that we have a gist of how projective and injective limits are formed for algebraic objects, most of the heavy lifting has already been done, and we are ready to turn our attention to C*-algebras.
Let $(\mathcal{A}_i, f_{i,j})$ be a directed system of C*-algebras and $\mathcal{A}’$ the injective limit of the underlying algebras (see the previous section). If $a \in \mathcal{A}’$ is the equivalence class of the element $a \in \mathcal{A}_i$, one defines a C*-seminorm
$$ ||a|| = \lim_{j>i } || f_{i, j}(a) ||. $$
It is well defined since the maps $f_{i, j}$ are all norm decreasing. Lifting this to the quotient space $\mathcal{A}’/ \{ a ~: ~ ||a|| = 0\}$ we have a C*-norm. The injective limit of the directed system is defined as the C*-algebra $\mathcal{A}$ given by the completion of $\mathcal{A}’/ \{ a ~: ~ ||a|| = 0\}$ with respect to this norm.
Now here is the definition of the projective limit of an inverse system of C*-albegras $(\mathcal{A}_i, f_{i,j})$. Let $\mathcal{A}’$ be the projective limit of the underlying algebras (defined above). We define $\mathcal{A} \subset \mathcal{A}’$ to be the subalgebra given by
$$\mathcal{A} = \{ (a_i) \subset \mathcal{A}’ ~ : ~ \sup_{i\in I} ||a_i|| < \infty \}$$
A closely related notion of the above, is that of a pro-C*-algebra (or local C*-algebras) which is just the projective limit of an inverse system of C*-algebras taken in the category of topological *-algebras. Another related concept is that of a $\sigma$-C*-algebra, where the inverse system of C*-algebras is required to be countable and the limit is again taken in the category of topological *-algebras. These are not necessarily C*-algebras per se, and I will not delve into the theory here.
Lastly, I would also like to point out that there exists a notion of a generalized injective limit of C*-algebras, where the morphisms between the objects are not necessarily *-homomorphisms of C*-algebras, but their “asymptotic behavior” mirrors that of an ordinary *-homomorphism. For the precise definition consult the last section of Blackader’s book (Operator Algebras).
There is also another way to exhibit an isomorphism between limit C*-algebras $\mathcal{A}$ and $\mathcal{B}$, which is to find an interwining of the algebras. For injective limits when the index set is a sequence these are maps $\phi_i: \mathcal{B}_i \to \mathcal{A}_i$ and $\psi_i: \mathacal{A}_i \to \mathcal{B}_{i+1}$ (note the incremented index!) such that every “triangle” commutes when one sketches out the diagram. As one would expect the increment of 1 is arbitrary, one only needs a subsequence of such interwinings for the limit algebras to be isomorphic. One can make this ever weaker by introducing approximate interwining. Similar constructions could be carried out for projective limits directed index sets in the obvious way.
Some common injective limits are,
And some common projective limits are
There often is a correspondence between colimit objects and limits of dual objects, that is, if $G$ is the injective limit of locally compact abelian groups $(G_i, f_i)$, the dual group $G^*$ of $G$ is the projective limit of the inverse system $(G_i^*, f_i^*)$ where the $G_i^*$ is the dual groups of $G_i$ and $f_i^*$ is the lift of $f_i$.
The abstract justification for this is that the Pontryagin dual functor, which sends a group to its dual group, is (by its very definition) a contravariant representable functor hence sends colimits to limits. As such we could have written the p-adic solenoid above as a certain projective limit of so called Prüfer p- groups (the dual groups of the p-adic integers), but no, let’s not. The same holds for the functor sending a vector space to its dual.
For another example of duality in action, more relevant to the scope of the blog, the assignment of a compact Hausdorff space to a commutative unital C*-algebra (via the Gelfand transform) is known to be a (contravariant) equivalence of categories and hence preserves both limits and colimits. As a consequence if $(X_i, f_{i, j})$ s a directed system compact Hausdorff with $$X = \varinjlim (X_i, f_{i, j}),$$ then for the corresponding inverse system $(C(X_i), f^*_{i, j})$ of C*-algebras we have the projective limit $$C(X) = \varprojlim (C(X_i), f_{i, j}^*)$$ where $f_{i, j}^*$ is the usual lift of $f_{i, j}$.
In the future I hope to be able to cover some more concrete properties of these limits, which might be more useful in applications, and show how they are employed to define so called AF algebras (to my knowledge first studied by Ola Bratteli here at the university of Oslo) and UHF algebras. If you find any mistakes, or have a nice example of use of projective/injective limits that gets used in analysis, let me know by pm or in the comments below and I will add them to the list.
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]]>The post A tour of functional analysis 1 – Locally convex vector spaces and the Hahn-Banach theorem(s) appeared first on That Can't Be Right.
]]>
Locally convex topological vector spaces (LCTVS) can be characterized either as a topological vector space $X$ where the origin has a neighborhood basis consisting of convex open sets $U$ each of which satisfy
\begin{array}{lll}
& \lambda x \in U & \text{for all }\lambda \leq 1 \text{ and all }x\in U & \textbf{(Balanced)}\\
& \bigcup^{\infty}_{t=1}tU = X & & \textbf{(Absorbent)}. \\
\end{array}
(recall that on topological vector spaces, translation is a homeomorphism, by definition, so we only need to specify a local neighborhood basis at one point to determine the topology). Note also that any neighborhood of 0 of a topological vector space must be absorbent, this can be shown since “scaling” is continuous.
Alternatively, one can define it as a topological vector spaces whose topology is “induced” by a family of seminorms, (this will be made precise shortly). Here I will follow the latter route.
Let $X$ be a topological vector space, we say that a map $\sigma : X\to \mathbb{R}$ is a seminorm if for all $x, y \in X$ and $\lambda \in \mathbb{C}$, it satisfies
\begin{array}{ll}
& \sigma(x) \geq 0 \\
& \sigma(x + y) \leq \sigma(x) + \sigma(y) \\
& \sigma(\lambda x) = |\lambda|\sigma(x)
\end{array}
That is, a seminorm is a norm that might not be “positive definite”, e.i. $\sigma(x) = 0 \not\Rightarrow x= 0$.
The topology induced by an arbitrary family of seminorms $\{ \sigma_\alpha \}$ is the weak topology induced by the seminorm maps $\sigma_\alpha: X\to \mathbb{R}$ which is the weakest (or coarsest) topology for which all the seminorms are continuous. Explicitly, it is the topology induced by the neighborhood subbasis elements of the origin given by
$$V_{\epsilon, \alpha} = \{ x\in X ~ |~ \sigma_\alpha(x) < \epsilon \}.$$
Norms are of course also seminorms, and Banach spaces are LCTVS’s since their topologies are weak topologies induced by a single “seminorm” (an actual norm this time).
Let $C$ be a convex set containing $0$. The point 0 is said to be an internal point of $C$ if the intersection of every line through 0 with $C$ is a non-trivial interval. Note that this is satisfied if $0$ is an interior point of $C$, or if $C$ is balanced and absorbent. This is the minimum structure we need to define the following (non-linear) functional, as it ensures the infimum is taken over a non-empty set:
The Minkowski functionals (sometimes called the Gauge functionals or the support functions for C) bridges the gap between the two definitions of LCTVS’s by assigning to each convex absorbent and balanced neighborhood basis element $U$ of $0$ a functional $\rho_U$, and employing the following fact, which I state here without proof,
The following Lemma in addition to its intrinsic value will be of use later on in the proof of the Hahn-Banach theorem.
It is an easy exercise to check that same conclusion holds if $Re(l)$ or $|l|$ are bounded in the above lemma (using the formula $l(v) = Re(l(v)) + i Re(l(iv))$), and the lemma holds for real vector spaces as well.
Since every metrizable space must be (at the very least) first countable and Hausdorff, Proposition 1 has the consequence that if a LCTVS is metrizable it must be possible to induce it’s topology by a countable family of seminorms that separate points, namely the Minkowski functionals associated with the (countable) neighborhood basis of the origin. The first condition comes from the requirement that the space be first countable (i.e. has a countable neighborhood basis at 0) the second from the fact that it must be Hausdorff. Recall that a family $\mathcal{F}$ of seminorms on $V$ is said to separate points if for each distinct $x,y \in V$ there exists a seminorm $\sigma \in \mathcal{F}$ such that $\sigma_n(x -y) > 0$. The important fact worth remembering is that this is actually a sufficient requirement:
So for LCTVS metrizability is equivalent to being Hausdorff and second countable, which is equivalent to the existence of a countable family of separating seminorms which induce the topology. Be warned, some authors define topological vector spaces to be Hausdorff out of the box, so they might not even mention the “separates points” requirement, as that is baked into the definition of topological vector spaces.
Now that we got some of the basic theory of LCTVS’s out of the way, let’s proceed to the main topic of this post. Here is the original Hahn-Banach theorem for real vector spaces. A function $p: V\to \mathbb{R}$ on a real vector space $V$ is said to be sublinear if it satisfies
\begin{array}{cl}
&p(u + v) \leq p(u) + p(v), & \text{for all } u, v\in V\\
&p(r v) = rp(v)& \text{for all } r\geq 0.
\end{array}
If the vector space is equipped with a countable (Hamel) basis, it may be tempting to rewrite the above proof as an induction argument on the basis, and hope to get away without relying on any of the choice axiom (in particular the axiom of dependent choices). But unfortunately there is no canonical way to choose the value of $k$ at each step in the iteration, so we would still need the axiom. Be warned that the reliance on the axiom of dependent choices is deemed so innocuous that many authors understate or completely miss it.
The Hahn-Banach theorem was first discovered by Hahn and later independently by Banach. At this stage we have not had any use for the condition that the extension be dominated by a sublinear functional, and it may seem strange that both authors came up with this seemingly arbitrary condition, but there is a method to the madness. First of, note that norms, seminorms and the Minkowski functionals are examples of sublinear functionals, so one can think the sublinear functionals as distilling the essential features of these maps, which ensures that an extension of a continuous linear functional on a normed space is also continuous (see the next corollary). Secondly, Hahn never actually formulated the theorem by means of sublinear functionals at all, but something to the extent of the next corollary. The above theorem is due to Banach. Thirdly, both authors likely knew about a previous result of Helly which showed that continuous linear functionals on normed spaces could be extended to continuous linear functionals on the space obtained by adjoining a single basis vector to the domain. So the theorem should be seen as an attempt to generalize Helly’s result to arbitrary continuous extensions of continuous linear functionals on normed spaces.
The above theorem can be extended to complex vector spaces. First we define a complex sublinear functional on $V$ to be a functional satisfying
\[
\begin{array}{l}
& p(u + v) \leq p(u ) + p(v) & \text{for all $u, v \in V$} \\
& p(\alpha u) = |\alpha| p(u) & \text{for all $\alpha \in \mathbb{C}$ and all $u\in V$}
\end{array}
\]
that is, $p$ is just like a seminorm on $V$ that can attain negative values.
The next theorem is sometimes called the analytic version of the Hahn-Banach theorem, or simply just the Hahn-Banach theorem. All vector spaces will henceforth be assumed to be complex.
The first supremum is always attained.
In the above corollary, if $V$ happens to be a Banach space (that is, it is complete with respect to its norm), the second supremum is attained if and only if the space is reflexive, which just means it is isomorphic to its second dual under the canonical imbedding. This is called James Theorem, and is a deep result of functional analysis which will not be covered here, but it should definitely be mentioned.
Now that some of the corollaries to the Hahn-Banach theorem for normed spaces have been covered we turn to the case of LCTVS. Here we will need the following analogue to Theorem 1,
The preceding “separation” theorem, sometimes called the Hahn-Banach separation theorem, though to the best of my knowledge neither Hahn nor Banach actually stated it, has the following interesting consequence on locally convex spaces
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]]>Continue reading "Topological Complements"
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]]>The first steps outside the comforts of the category of Hilbert spaces, the safe space for of functional analysis, into the unruly world of topological vector spaces, can be a troubling experience for any student, myself included. To easy the passage, here are a few tips and results regarding the existence of complementary subspaces in the general setting of topological vector spaces. For Hilbert spaces it is known that every closed subspace has a preferred (topologically) complementary subspace, namely the orthogonal complement, but any two (algebraically) complementary closed subspaces are automatically (topologically) complementary (by Theorem 1).
First, lets get the definitions out of the way. Two subspaces $U, V \subset X$ of a vector spaces $X$ are said to be algebraic complements (denote $X = U \oplus_A V$) if any $x \in X$ can be written uniquely as a sum $x = u + v $, where $u \in U$ and $v \in V$. This is equivalent to the condition $U \cap V = \{0\}$ and $X = U + V$ (as a set).
Here two subspaces $U, V \subset X$ of a toplogical vector space $X$ are (topologically) complementary, written $X = U \oplus V$, if either of the following equivalent conditions hold:
By the codimension of a subspace $V \subset X$, we mean the dimension of the quotient space $X/V$. This is equivalent to the, slightly more intuitive, definition of the dimension of the algebraic complement of a subspace since these have the same dimension, which can be verified using the first isomorphism theorem for vector spaces. All subspaces have algebraic complements, just augment the basis of the subspace to a basis of the whole space. I don’t intend to make a habit of pointing out what result relies on the axiom of choice here, since one really cannot have much fun without it, but yes… it does rely on the axiom of choice.
To the best of my knowledge the next theorem requires the Hahn-Banach separation theorem, so we are restricted to locally convex spaces.
Let $\{ e_i\}_{i=1}^n$ be a basis for the finite dimentional subspace $U \subset X$, with the notation
$$\{ e_1, …\hat{e_k}, … e_n \} = \{e_i\}_{i=1}^n\backslash \{ e_k \}$$ we pick
$\phi_k \in span(\{ e_1, …\hat{e_k}, … e_n \})^\perp$ with $\phi_{k}(e_k) = 1$. It is now easy to check that the linear operator $P_U$ defined by by the equation
$$P_U(x) = \sum_{i=1}^n \phi_i(x)e_i $$ for all $x\in X$ is a bounded projection onto $U$. Setting $$V = Ker(P)$$ we have our complementing subspaces.
Note the generality of the following statement.
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]]>Continue reading "Three different entropies, variational principle and the degree formula."
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]]>
We give the definitions in chronological order.
Let $ (X, \mathcal{B})$ be a measurable space, and $ T:X\to X$ a measurable map. The collection of all probability measures will be denoted $M(X)$, and the T-invariant probability measures, that is, measures $\mu \in M(X)$ for which $\mu(A) = \mu(T^{-1}A)$ for all measurable $A$, will be denoted $ M(X, T)$. A measurable partition is defined as a collection $ \xi \subset \mathcal{B}$ of disjoint measurable sets that cover $ X$. We define the join of two measurable partitions as
\[
\xi \vee \eta = \{ C\cap D ~ | ~ C \in \xi, ~ D\in \eta \}
\]
The entropy of a measurable partition is defined as the (generalized) sum
$$H_\mu(\xi) = -\sum_{C\in \xi} \mu(C)log(\mu(C))$$
with the conventions that $H_\mu(\xi) = \infty$ if the union of all sets of measure zero in $ \xi$ have non-zero measure, and we let $\mu(C)log(\mu(C)) = 0$ if $\mu(C) = 0$.
Now the entropy of T with respect to a T-invariant probability measure $ \mu$ and a partition $ \xi$ with $H_\mu(\xi) < \infty$ is defined as
$$h_{\mu}(T, \xi) = \inf_{n\in \mathbb{N}} \left\{ \frac{1}{n} H(\bigvee^{n-1}_{k=0}T^{-k}\xi ) \right\} = \lim_{n\to\infty} \frac{1}{n} H_\mu\left(\bigvee^{n-1}_{k=0} T^{-k}\xi\right).$$
It should be noted that some authors define a measurable partition as a collection of measurable sets whose intersection have measure zero and which cover the space almost everywhere. Personally, I prefer the definition of a measurable partition to be independent of choice of measure. It is also customary to remove sets of measure zero in stead (as an alternative to the convention $0log(0) = 0$ we adopt here), but this again requires a choice of measure, which is unfortunate.
The map sending an invariant probability measure $ \mu \mapsto h_\mu(T)$ is called the entropy map. It is affine, but in general not continuous.
We also will need the following definition in the proof of the Variational principle below
Analogously, we may define a notion of entropy in the category of compact topological spaces, which we will call the topological entropy. It was first given in the article of Adler et. al. [2]. For this we need the following preliminary definitions (here $ X$ will denote a compact space).
For any open cover $ \mathcal{U}$ of $X$ the quantity $ N(\mathcal{U})$ is the minimal cardinality of any subcover of $\mathcal{U}$. We define the entropy of the cover $\mathcal{U}$ to be the number $$H_{top}(\mathcal{U}) = log(N(\mathcal{U}))$$
The join of two open covers $ \mathcal{U}, \mathcal{W}$ of $ X$ is defined as the open cover $$ \mathcal{U}\vee \mathcal{W} = \{ U\cap W ~ | ~ U\in \mathcal{U}, ~ W\in \mathcal{W} \} .$$
The topological entropy of a continuous map $ \phi: X\to X$, with respect to the open cover $ \mathcal{U}$ of $ X$ is defined as the number
$$h_{top}(\phi, \mathcal{U}) = \lim_{n\to \infty}\frac{1}{n} H_{top}\left(\bigvee^{n-1}_{i=0} \phi^{-1}\mathcal{U}\right). $$
Though we will not delve into this, it should be mentioned that there are more definitions of topological entropy in the literature for general topological spaces which all reduce to the above definition if the space is compact.
Lastly we have an entropy which (to the best of my knowledge) was first introduced in [3] by Bowen. It makes explicit use of a metric, hence the name. Some authors call it the topological entropy, since, as we will see, they are actually equivalent on compact metric spaces.
Let $(X, d)$ be a compact metric space, and $T: X\to X$ a continuous map. To define the metric entropy, we need the following preliminary definitions
We define the metric $ d_T^n $ on $X$ by
$$d_T^n(x, y )= \max_{0\leq i \leq n-1} d(T^ix, T^iy).$$
The metrics $ d$ and $ d_T^n$ are (strongly) equivalent Why?
A collection $ F \subset X$ of points is said to be $(n, \epsilon)$-separated if $$d^n_T(x,y) \geq \epsilon \qquad \text{for all }x,y \in F, \text{ with } x\neq y. $$ By compactness this set must be finite.
We let $N_d(T, \epsilon, n)$ denote the maximum carnality all $ (n, \epsilon)$-separated subsets of $ X$. This number is necessarily finite
Why?
Define the quantity $h_d(T, \epsilon)$ to be
$$h_d(T,\epsilon) = \limsup_{n\to \infty} \frac{1}{n} \log(N_d(T, \epsilon, n)).$$
$ h_d(T)$ is is well defined by the following lemma
Lemma 1 If $ \epsilon_1 < \epsilon_2$ then $ N_d(T, \epsilon_2, n) \leq N_d(T, \epsilon_1, n) < \infty$, hence $ h_d(T, \epsilon)$ is monotone non-increasing as $ \epsilon \to 0$ and bounded below by $ 0$ (by inspection).
We can also treat the topological and measure theoretic entropies as limits over appropriate directed index sets. Let’s see how this plays out in the case of topological entropy, bearing in mind that the same could just as well be done with the measure theoretic entropy.
With $X$ a compact Hausdorff space, the family $$O = \{ \mathcal{U} ~ | ~ \mathcal{U} \text{ is an open cover of } X \}$$ can be made into a directed set with respect to the order relation of refinement, that is $\mathcal{U}_1 \prec \mathcal{U}_2$ if $\mathcal{U}_2$ is a refinement of $\mathcal{U}_1$. It is not hard to check $\prec$ is a partial ordering on the collection of open covers, and for any open cover $\mathcal{U}_1$ and $ \mathcal{U}_2$ we have $$ \mathcal{U}_1, \mathcal{U}_2 \prec \mathcal{U}_1\vee \mathcal{U}_2.$$
The same can be done for the measure theoretic entropy with the order relation $\xi\prec \eta$ if for all $U\in \eta $ there is some $V\in \xi$ such that $U\subset V$. Again $\xi, \eta \prec \xi\vee \eta$ so in both these cases, $\prec$ determines a directed set.
Now we can define the topological entropy and measure theoretic entropy of a map $\phi$ by
The situation for $h_\mu$ is completely analogous, using Proposition 3 (3) in place of Proposition 6 (2).
Note that for compact metric spaces, we know that the above defined net of open covers has a cofinal subsequence, that is, a subnet which is also a sequence, and since all subnets of a convergent net converge to the same limit, we may as well take the limit over this sequence.
To construct the sequence we need the Lebesgue number lemma, which asserts that for any open cover $\mathcal{W}$ of a compact metric space there exists a number $\delta >0 $ such that any disc of radius $\delta$ is completely contained in some element in the cover. Let $\mathcal{U}_n = \{ B_{\frac{1}{n}}(x) ~ | ~ x \in X\}$ be the open cover of all balls of radius $\frac{1}{n}$. Now it follows that for any open cover $\mathcal{W}$ of $X$ with a Lebesgue number $\delta$, we have $\mathcal{W} \prec \mathcal{U}_n$ for all $n\in \mathbb{N}$ such that $\frac{1}{n} \leq \delta$, hence the sequence is cofinal.
Here we list some of the basic properties of the entropy. First and foremost let’s check that the definition given above makes sense.
With $H_m = H_\mu\left(\bigvee^{m-1}_{k=0}T^{-k}\xi \right)$ we will show that the sequence $\{ H_n/n\}$ is decreasing as $n$ increases. This suffices to prove the theorem since the sequence is bounded below by 0. Noting that $H_{m+n} \leq H_m + H_n$, we may, for some fixed $p$, partition the index set into steps of $p$, that is, we may write $n = i+ kp$ where $0\leq i< p $. Now
\begin{align*}
\frac{H_n}{n} & = \frac{H_{i + kp}}{i + kp} \leq \frac{H_{i}}{i+kp} + \frac{H_{kp}}{i+kp} \leq \frac{H_{i}}{kp} + \frac{H_{kp}}{kp}\\
&\leq \frac{H_{i}}{kp} + \frac{k H_{p}}{kp} = \frac{H_{i}}{kp} + \frac{H_{p}}{p}.
\end{align*}
The result now follows by noting that as $n\to \infty$, $k\to \infty$.
Proposition 2
Other noteworthy properties of the measure theoretic entropy are listed in chapter 4.5 in [1]. We state them here without proof. Let $\xi, \eta$ be finite entropy measurable partitions of $X$.
Proposition 3
The following theorem is used repeatedly, is easy to state, and has an elegant one-line proof, so there is no good reason not to include it here.
It is not too hard to check that by concavity of the function $x\mapsto xlog(x)$ we get the inequality
$$ 0 \leq H_{p\mu + (1 – p )m}(\xi) – pH_\mu(\xi) – (1-p)H_m(\xi).$$
By log-sum arithmetics, one can also produce the inequality
$$ H_{p\mu + (1 – p )m}(\xi) – pH_\mu(\xi) – (1-p)H_m(\xi) \leq log(2)$$
From this we get that
\begin{align*}
\frac{H_{p\mu + (1 – p )m}(\bigvee_{i=0}^{n-1} T^{-1} \xi)}{n} & – \frac{ pH_\mu(\bigvee_{i=0}^{n-1} T^{-1} \xi)}{n} & \\ & – \frac{(1-p)H_m(\bigvee_{i=0}^{n-1} T^{-1} \xi)}{n} \leq \frac{log(2)}{n}
\end{align*}
which, as $n\to 0$ yields
$$h_{p\mu + (1 – p )m}(T, \xi) = ph_\mu(T, \xi)+ (1-p)h_m(T, \xi)$$
Deviating a bit from Walter’s proof, we may take the limit over the net of all measurable partitions (of finite entropy) in the above equality, which produces the equality
$$h_{p\mu + (1 – p )m}(T) = ph_\mu(T)+ (1-p)h_m(T).$$
As the next proposition shows, the above proposition can be extended to arbitrary convex combinations of ergodic probability measures, which are defined as measures $\mu \in M(T,X)$ for which $T^{-1}A = A \Rightarrow \mu(A) \in \{ 0, 1 \}$. It relies on the ergodic decomposition theorem which I hope to cover in a subsequent post.
Consult Theorem 4.11 of [1] for the proof of the following important result,
In this section $X, Y$ are compact metrizable space, the maps in question will always be continuous and $\mathcal{U}$ will denote an open cover of $X$. First we collect some basic results, whose proofs are all more or less straightforward. Recall that the order relation of refinement, which we denoted $\prec$ is used to determine a partial order on the collection of open covers.
Proposition 6
The first thing we should check is that the definition of the topological entropy given above is well defined,
We have the following collection of basic properties relating to the topological entropy.
Proposition 8
In this section $(X,d)$ is a compact metric space.
Some properties of Metric entropy
Two metrics $d$ and $d’$ are said to be equivalent (denoted $d \sim d’$) if they induce the same topology. On a compact metric space $X$, this is equivalent to the existence of two fixed real numbers $\alpha, \beta > 0$ such that $\alpha d(x, y) \leq d'(x,y) \leq \beta d(x,y)$ for all $x, y \in X$. On a general metric space, $\alpha $ and $\beta$ are allowed to depend $x$ or $y$.
ProofIn the literature the topological and metric entropies are used interchangeably, the reason for which is the following theorem.
Step 1: First let’s produce the following inequality
$$N_d(T, n, \epsilon) \leq N(\mathcal{U}^n).$$
Letting $V\in \mathcal{U}^n$, that is $V = C_0 \cap T^{-1}C_1 \cap … \cap T^{-n+1}C_{n-1}$ for some $C_i \in \mathcal{U}$. Note that we have $ diam(T^{i}V) < \epsilon$ since
\begin{align*} $diam(T^iV) & = diam(T^{i}C_0 \cap… \cap C_i\cap ….\cap T^{-n + i+1}C_{n-1})\\ & \leq diam(C_i) < \epsilon.
\end{align*}
It follows that if $x,y \in V$, then
\begin{align*}
& d(x,y)< \epsilon \tag{since $x,y \in C_0$ }\\
& d(Tx, Ty) < \epsilon \tag{since $Tx, Ty \in C_1$}\\
& …
\end{align*}
hence $d_T^n(x,y) = \max_{i=0, …, n-1} d(T^ix, T^iy) < \epsilon$. So if $F$ is any $(n, \epsilon)$-separated it can contain at most one point in each element of a subcover $\mathcal{U}^n$. This proves the inequality and consequently we get that
\begin{align*}
h_{top}(T, \mathcal{U}) & = \lim_{n\to \infty} \frac{1}{n} log(N(\mathcal{U}^n)) \geq \limsup_{n\to \infty} \frac{1}{n} log(N_d(T, n, \epsilon)) \\ & = h_d(T,\epsilon)
\end{align*}
Step 2 : If $\delta$ is a Lebesgue number of $\mathcal{U}$, we will now prove that
$$ N(\mathcal{U}^n) \leq N_d(T, n, \frac{\delta}{2}).$$
First note that if $\delta$ is a Lebesgue number for $\mathcal{U}$ with respect to the metric $d$, then $\delta$ is the Lebesgue number for $\mathcal{U}^n$ with respect to the metric $d_T^n$. To see this, let $B_{\delta}^n(z)$ be a $\delta$-disc in the metric $d_T^n$ centred at some arbitrary $z\in X$. If $x, y \in B_{\delta}^n(z)$ then $d_T^n(x,y) < \delta$ from which we get
\begin{align*}
& d(x,y) < \delta \\
& d(Tx, Ty )< \delta \\
& d(T^2x, T^2y )< \delta \\
& ….
\end{align*}
So, for any $i$ we have that $\{ T^ix ~ |~ x \in B_{\delta}^n(z) \} $ is contained in some ball of radius $\delta$, hence also contained in some $C_i \in \mathcal{U}$ (since $\delta$ is the Lebesgue number of $\mathcal{U}$). Define $V\in \mathcal{U}^n$ by $V = C_0 \cap T^{-1}C_1 \cap… \cap T^{-n + 1} C_{n-1}$. It’s now easy to check that $B_{\delta}^n(x) \subset V$.
We can construct a cover $\mathcal{B}$ of $\delta$-neighbourhoods (in the metric $d_T^n$) inductively such that each ball is centered around points which are separated by a distance greater than $\frac{\delta}{2}$, and the have radius (in the metric $d_T^n$) less than $\delta$. This is a refinement of $\mathcal{U}^n$ and hence $N(\mathcal{U}^n) \leq N(\mathcal{B})$ and the center points form a collection of $(n, \frac{\delta}{2})$-separated points. The claim now follows, and we get the inequality
\begin{align*} h_{top}(T, \mathcal{U}) & = \lim_{n\to \infty} \frac{1}{n}log(N(\mathcal{U}^n)) \leq \limsup_{n} \frac{1}{n}log(N_d(T, n, \frac{\delta}{2})) \\ & = h_d(T, \frac{\delta}{2} )
\end{align*}
Letting $\mathcal{B}_{n} = \{\text{all balls of radius }\leq \frac{1}{n} \}$ be the (cofinal) sequence of open covers of $X$ defined earlier we have seen that $h_{top} = \lim_{n\to \infty} h_{top}(T, \mathcal{B}_n)$. But as $n\to \infty$ both $\epsilon $ and $\delta$ go to zero, hence the above inequalities become and equalities and we get
$$h_d(T) = h_{top}(T).$$
The variational principle relates the topological/metric entropy on a compact metric space with the measure theoretic entropy with respect to regular Borel measures. Explicitly we have
\begin{align*} H_\mu(\xi|\nu) & = -\sum_{j= 0}^k\sum_{i= 1}^k \mu(B_j) \phi(\frac{\mu(B_j\cap A_i)}{\mu(B_j)}) \\ & = -\mu(B_0) \sum_{i= 1}^k\phi(\frac{\mu(B_0\cap A_i)}{\mu(B_0)}) \\ & \underbrace{-\mu(B_1) \sum_{i= 1}^k\phi(\frac{\mu(B_1\cap A_i)}{\mu(B_1)}) – … -\mu(B_k) \sum_{i= 1}^k\phi(\frac{\mu(B_k\cap A_i)}{\mu(B_k)})}_{= 0} \\ & = -\mu(B_0) \sum_{i= 1}^k\phi(\frac{\mu(B_0\cap A_i)}{\mu(B_0)}) \end{align*} This follows, since, with $j > 0$, we get $\frac{\mu(B_j \cap A_i)}{\mu(B_j)} = \begin{cases} 0 & i\neq j \\ 1 & i=j \end{cases}$, either way $\phi(1) = 0$ and (by convention) $\phi(0) = 0$.
By Proposition 3 (7) we have that
$$H_\mu(\xi) \leq log(k).$$
Inserting this into the above equation, we get the inequality
\begin{align*}
H_\mu(\xi|\nu) & \leq \mu(B_0)log(k).
\end{align*}
Now $\mu(B_0) = \mu(X\backslash \bigcup_{j=1}^kB_j) = \mu(\bigcup_{j=1}^k (A_i \backslash B_i)) = k \epsilon$, hence
\begin{align*}
H_\mu(\xi|\nu) & \leq \mu(B_0)log(k) < k\epsilon \log(k) < 1. \end{align*} From Proposition 3 (6) we get $$ h_\mu(T, \xi) \leq h_\mu(T, \eta) + H_\mu(\xi | \eta) < h_\mu(T, \eta) + 1$$ Now define the partition $\beta = \{B_0\cup B_1, ..., B_0\cup B_k \}$. Note that $\beta$ is also an open cover of $X$! For each $a \in \bigvee_{i=1}^{n-1}T^{-i}\beta$ there are at most $2^n$ distinct elements from $\bigvee_{i=1}^{n-1}T^{-i} \nu$ contained in $a$. To see this, let $a = (B_0 \cup B_{j_0}) \cap T^{-1}(B_0 \cup B_{j_1}) ... \cap T^{-n+1}(B_0 \cup B_{j_{n-1}})$, and assume $b = B_{i_0}\cap T^{-1}(B_{i_1}) \cap ... \cap T^{-n +1}(B_{i_{n-1}}) \in \bigvee_{i=1}^{n-1}T^{-i} \nu $ is contained in $a$. It follows that \begin{align*} & B_{i_0} \subset B_0\cup B_{j_0} & \Rightarrow ~ B_{i_0} = \begin{cases} B_0 \\ B_{j_0} \end{cases}\\ & T^{-1}(B_{i_1}) \subset T^{-1}(B_0\cup B_{j_1}) & \Rightarrow ~ B_{i_1} = \begin{cases} B_0 \\ B_{j_1} \end{cases}\\ & ...& \\ & T^{-n +1 }(B_{i_{n-1}}) \subset T^{-n+1}(B_0\cup B_{j_{n-1}}) & \Rightarrow ~ B_{i_{n-1}} = \begin{cases} B_0 \\ B_{j_{n-1}} \end{cases}\\ \end{align*} which gives a total of (at most) $2^n$ possible combinations. Hence $$|\bigvee_{i=1}^{n-1}T^{-i} \nu| \leq 2^n|\bigvee_{i=1}^{n-1}T^{-i}\beta|$$ where $|\cdot|$ denotes the cardinality of the collections. Since $\beta$ is a minimal cover, that is, it has no proper subcovers, one can verify that $\bigvee^{n-1}_{i=0} \beta$ is also minimal, hence we have $N(\bigvee^{n-1}_{i=0} \beta) = |\bigvee^{n-1}_{i=0} \beta |$. Inserting this into the definitions, we get
\begin{align*}
h_\mu(T, \nu) & < h_\mu(T, \eta) + 1 = \lim_{n\to \infty} \frac{H_\mu(\bigvee^{n-1}_{i=0}T^{-i}\eta)}{n} + 1 \\
& \leq \lim_{n\to \infty} \frac{log(2^nN(\bigvee_{i=0}^{n-1} T^{-i}\beta))}{n} + 1 = h_{top}(T, \beta) + \log(2) +1 \\
& \leq h_{top}(T) + \log(2) +1.
\end{align*}
Substituting $T$ with $T^n$ we get that $n h_\mu(T) < n h_{top}(T) + \log(2) + 1,$ which, since $n$ is arbitrary, shows that $$h_\mu(T) \leq h_{top}(T)$$ for all continuous maps $T$ on $X$. This concludes the first part of the proof. Now we will show that we can find a $\mu \in M(X, T)$, with $h_\mu(T)$ arbitrarily close to $h_d(T)$ (the metric entropy) which has been shown to be equal to $h_{top}(T)$. To this end, fix $\epsilon > 0 $, let $E_n^\epsilon \subset X$ be a $(n, \epsilon)$-separated set of maximal cardinality (that is $|E_n^\epsilon | = N_d(T, n, \epsilon)$) and define
\begin{align*}
& \sigma_n = \left( \frac{1}{|E_n^\epsilon|}\right) \sum_{x\in E_n^\epsilon} \delta_x \qquad \qquad \text{where $\delta_x$ is the Dirac measures} \\
& \mu_n = \frac{\sum_{i=0}^{n-1} \sigma_n\circ T^{-i}}{n}.
\end{align*}
By compactness of $M(X, T)$, we may find a subsequences, indexed by $n_i$ such that $h_d(T, \epsilon) = \lim_{i\to \infty} \frac{1}{n_i} log(N_d(T, n_i, \epsilon))$, and $\mu_{n_i}$ converges in the vague (or weak*)-topology on $M(X, T)$. By definition of vague convergence we see that since all $\mu_{n_i}$ are $T$ invariant, it follows that
\begin{align*}
\int_{X} f\circ d(\mu \circ T^{-1}) &= \int_{X} f\circ T d\mu = \int = \lim_{n_i \to \infty} \int_{X} f \circ T d\mu_{n_i} \\
& = \lim_{n_i \to \infty} \int_{X} f d\mu_{n_i} = \int_{X} f d \mu
\end{align*}
so $\mu \in M(X, T)$.
We will show that
$$\limsup_{n\to \infty}log(N_d(T, n, \epsilon)) \leq h_\mu(T)$$
which, since $\epsilon$ was arbitrary, shows that we may approximate $h_{top}(T)$ from below by $h_\mu(T)$ where $\mu \in M(X, T)$ is a regular Borel measures.
For this we will need the following lemma
Lemma 1
Proof
Why do we care about sets with boundary measure zero? The reason is that if $\eta_n$ is a sequence of probability measures which converges to $\eta$ in the vague topology, and $B$ has $\eta_n(\partial B) = 0$ for all $n$, , then
$$ \eta(B) = \lim_{n\to \infty} \eta_n(B).$$
As a consequence for any partition $\xi$ consisting of sets with zero boundary measure, we have
\begin{equation}
\lim_{n\to \infty} H_{\mu_n}(\xi) = H_{\mu}(\xi)
\end{equation}
Employing this lemma, let $\xi = \{A_1, …, A_k \}$ be a measurable partition of $X$ such that $\mu(\partial A_i) = 0$ and $diam(A_i) < \epsilon$. We have
\begin{equation}
\label{eq:3}
H_{\sigma_n}(\bigvee_{i=0}^{n-1}T^{-i} \xi) = -\sum_{C\in \bigvee_{i=0}^{n-1}T^{-i} \xi} \sigma_n(C) log(\sigma_n(C)) = log(|E_n|)
\end{equation}
since each $C \in \bigvee_{i=0}^{n-1}T^{-i} \xi $ contains at most one $x \in supp(\sigma_n)$.
Here things get messy, but the idea is to split $\bigvee_{i=0}^{n-1}T^{-i} \xi$ into more manageable pieces. Fixing $p, n\in \mathbb{N}$, with $1 < q < n$ and let $\alpha : \{0, 1, …, q-1 \} \to \mathbb{N}$ be given by $\alpha(j) = \left[\frac{n-j}{q} \right]$ (i.e. the smallest integer greater than $\frac{n-j}{q}$). Now we decompose the set
$$\{0, 1, …, n \} = \{ j + rq -i ~| ~ 0 < r< \alpha(j), 0 < i \leq q \} \cup S$$
where $S = \{ 0, 1, …, j, j+a(j)q+1, …, n-1 \}$. By construction we have $|S| \leq 2q$. From this we deduce that
$$ \bigvee_{r=0}^{\alpha(j) -1}T^{-(rq + j)} \left( \bigvee_{i=0}^{q-1}T^{-i} \xi \right) \vee \left( \bigvee_{j\in S}T^{-j} \xi \right)$$
since, even though we have repeated some indices more than once on the right hand side, we exploit that for any measurable partition $\xi$ we have $\xi \vee \xi = \xi$, hence the repetition does not affect the join.
Now
\begin{align*}
log(|E_n|) & = H_{\sigma_n}(\bigvee_{i=0}^{n-1}T^{-i} \xi) \\
& = H_{\sigma_n}\left( \bigvee_{r=0}^{\alpha(j) -1}T^{-(rq + j)} \left( \bigvee_{i=0}^{q-1}T^{-i} \xi \right) \vee \left( \bigvee_{j\in S}T^{-j} \xi \right)\right) \\
& \leq \sum_{r= 0}^{\alpha(j)- 1} H_{\sigma_n}\left( \bigvee_{r=0}^{\alpha(j) -1}T^{-(rq + j)} \left( \bigvee_{i=0}^{q-1}T^{-i} \xi \right) \right) + \sum_{j\in S} H_{\sigma_n}(T^{-j}\xi) \\
& \leq \sum_{r=0}^{\alpha(j) – 1} H_{\sigma_n \circ T^{-(rq + j)}} \left( \bigvee^{q-1}_{i= 0}T^{-i}\xi \right) + 2q log(k) \\
\end{align*}
where the last two inequalities follow by Proposition 3 (2) and (7) respectively.
If we sum over all j’s in the interval $0\leq j\leq q-1$ we end up with
\begin{align*}
q log(|E_n|) & \leq \sum_{j= 0}^{n-1} \sum_{p=0}^{n-1} H_{\sigma_n \circ T^{-(rq + j)}}\left( \bigvee_{i=0}^{q-1} T^{-i} \xi \right) + 2q^2 log(k)\\
& \leq \sum_{p= 0}^{n-1} H_{\sigma_n \circ T^{-p}}\left( \bigvee_{i=0}^{q-1} T^{-i} \xi \right) + 2q^2 log(k)\\
\end{align*}
Divide both sides by $n_j$, and take the limit as $j\to \infty$ gives
\begin{align*}
q \limsup_{n\to \infty} \frac{log(N_d(T, n, \epsilon))}{n} & = q \lim_{j\to \infty}\frac{1}{n_j} \frac{log(N_d(T, n_j, \epsilon))}{n_j}\\
& \leq q\lim_{j\to \infty} \frac{1}{n_j}\sum_{p= 0}^{n_j-1} H_{\sigma_n \circ T^{-p}}\left( \bigvee_{i=0}^{q-1} T^{-i} \xi \right) \\ & + \frac{1}{n_j} 2q^2 log(k)\\
& = q\lim_{j\to \infty} H_{\mu_{n_j}}\left( \bigvee_{i=0}^{q-1} T^{-i} \xi \right) + \frac{1}{n_j} 2q^2 log(k)\\
& = H_{\mu}\left( \bigvee_{i=0}^{q-1} T^{-i} \xi \right)
\end{align*}
We used that
\begin{align*}
\frac{1}{n_j}\sum_{p=0}^{n_j-1}H_{\sigma_{n_j}\circ T^{-p}}\left( \bigvee_{i=0}^{q-1} T^{-i} \xi \right) & = H_{\sum_{p=0}^{n_j – 1} \frac{\sigma_{n_j}\circ T^{-p}}{n_j}}\left( \bigvee_{i=0}^{q-1} T^{-i} \xi \right) \\ & = H_{\mu_{n_j}} \left( \bigvee_{i=0}^{q-1} T^{-i} \xi \right)
\end{align*}
Now dividing by $q$ and taking limits as $q\to \infty$ on the right hand side we finally get the following inequality
$$ \limsup_{n\to \infty} \frac{log(N_d(T, n_j, \epsilon))}{n_j} \leq h_\mu(T, \xi) \leq h_\mu(T).$$
This section is devoted the proof a theorem due to Misiurewicz, Przytycki and Gromov, which relates the topological entropy of a holomorphic map on the complex $n$-sphere to the degree of the map. Here is the first result which is a shameless copy of the proof given Theorem 8.3.1 in [4].
Theorem 5 (Misiurewicz-Przytycki)
Let $M$ be a smooth compact orientable manifold and $f: M \to M$ a continuously differentiable map, then $h_d(f) \geq log(deg(f))$.
Let $L = \sup_{x\in M} |Jf(x)|$ (the (total) derivative of $f$), fix any $\alpha \in (0, 1)$ and define $\epsilon = L^{-\alpha/(\alpha -1)}$ (this will make sense shortly). Define the compact set
$$B= \{ x\in M ~ | ~ |Jf(x)| \geq \epsilon \}$$
and, using the inverse function theorem, cover $B$ by open sets where $f$ is injective. Let $\delta$ be a Lebesgue number of this covering. That is, in every $\delta$-disk of $B$ the function $f$ is injective. Define
$$A = \{ x\in M ~ |~ card(B\cap \{x, f(x), .., f^{n-1}(x) \}) \leq \alpha n \}$$
Note that by our choice of $\epsilon$ we have that $|Jf^n(x)| < 1$ for all $x\in A$, since
$$|Jf(x)| = \prod_{i=0}^{n-1} |Jf(f^i(x))| < \epsilon^{1-\alpha}n L^{\alpha n} = 1$$
hence $f^n$ is strictly contractive on $A$, so $Vol_\omega(M\backslash f^n(A)) < Vol_\omega(M)$, where $\omega$ is a normalized volume form. Since the critical values have measure zero (Sard’s theorem) there exists a regular value $x\in M\backslash f^n(A)$, that is values with preimages of cardinality $N$. For $i=1, .., n$ define the set
$$Q^x = \begin{cases} f^{-1}(\{x\}) & \text{if } f^{-1}(\{x\}) \subset B \\ \text{a single point of } f^{-1}(\{x \})\text{ outside of } B & \text{else.} \end{cases}$$
Now we construct the $(n, \delta)$-separated set by the following induction process
\begin{align*}
Q_1 &:= Q^x\\
Q_2 & = \bigcup_{y\in Q_1} Q^y\\
….\\
Q_n &= \bigcup_{y\in Q_{n-1}} Q^y
\end{align*}
We will show $Q_n$ is the set we are looking for.
$Q_n$ is $(\delta, n)$-separated: Let $y_1, y_2 \in Q_n$, and assume for contradiction that $d_f^n (y_1, y_2) = \max_{i = 0, 1, …, n-1} d(f^i(y_1), f^{i}(y_2)) < \delta$. Assume first that $f^{n-1}(y_1)$ and $ f^{n-1}(y_2)$ are distinct points in $Q_1$. By construction then we must have that $Q_1 \subset B$, but since $d( f^{n-1}(y_1), f^{n-1}(y_2)) < \delta$, and $f$ is injective on all $\delta$-discs in $B$, we must have $f(f^{n-1}(y_1)) \neq f(f^{n-1}(y_2))$, which is impossible since $f^{n-1}(y_1)$ and $f^{n-1}(y_2)$ are in the same fiber (namely $f^{-1}(\{x\})$). Hence $f^{n-1}(y_1) = f^{n-1}(y_2)$. Continuing the process inductively we eventually get $y_1 = y_2$.
To show $Q_n$ is large enough, first note that $Q_n \cap A = \emptyset$ since $Q_n \subset f^{-n}(M\backslash f^n(A)) \subset M\backslash A$, so we know that for any $y\in Q_n$ we must have
$$card(B \cap \{y, f(y), …, f^{n-1}(y) \}) \geq \alpha n.$$
There are hence at least $[\alpha n]$ numbers $0 \leq i \leq n-1$ such that $f^i(y) \subset B\cap Q_{n-i}$. For each such number there are, by construction, $N$ distinct elements in $Q_i$ in the same fiber as $f^{i}(y)$. Each of these numbers have at least one distinct preimage in $Q_n$. This shows that $card(Q_n) \geq N^{\alpha n}$ and concludes the proof.
Next we specialise to the case where the manifold is the complex n-sphere.
\begin{align*}
d_{+}^m(\tilde{x}, \tilde{y}) &= \sum_{i=0}^{m-1} d(x_i, y_i)\\
d^m(\tilde{x}, \tilde{y}) & = \max_{i=0,.., m-1} \{ d(x_i, y_i)\}.
\end{align*}
These metrics induce the same topology, since $d^m \leq d_+^m \leq m d^m$. Note that the metric $d_f^m$ defined in section “metric entropy” coincides with the metric $d^m$, by this we mean that
$$d_f^n(x) = \max_{i=0, .., m-1}\{ d(f^i(x), f^i(y)) \} = d^m(\tilde{x}, \tilde{y})$$
where $\tilde{x} = (x, f(x), …, f^{m-1}(x))$ and $\tilde{y}=(y, f(y), …, f^{m-1}(y))$ are in $\Gamma_m$.
Next we will need a result from geometry, which says that for any $\tilde{x} = (x_0, x_1, …, x_{m-1}) \in \Gamma_m$ we have a lower bound on the volume of the disc $B^{m, +}_\epsilon(x) = \{ \tilde{y}=(y_0, .., y_{m-1}) \in \Gamma_m, ~|~ d^m_{+}(\tilde{y}, \tilde{x}) < \epsilon \}$ which is independent of both $x$ and $m$. That is $$Vol_\omega(B_\epsilon(x)) \geq C_\epsilon, $$ where $C_\epsilon$ depends on $\epsilon$ but not on $x$ or $m$, and $\omega$ is the (normalized) volume form associated with the Riemann metric $d_{+}^m$. Be warned that this result requires some sort of compatibility between the metric and the complex structure, which $d^m$ does not satisfy.
Note that, since $d_{+}^m \geq d^m$, we have $B^{m, +}_\epsilon(\tilde{x}) \subset B^{m}_\epsilon(\tilde{x}) := \{ \tilde{y} \in \Gamma_m ~|~ d^m(\tilde{x}, \tilde{y}) <\epsilon \}.$ Let $E_m^\epsilon$ be a maximal $(m, \epsilon)$-separated set in $\mathbb{P}^n$, then we know the collection $\{ B^{m}_{\epsilon/2}(\tilde{x})~ |~ x\in E_m^\epsilon, ~ \tilde{x} = (x, f(x), .., f^{m-1}(x)) \}$ do not intersect, and hence neither will $\{ B^{m,+}_{\epsilon/2}(\tilde{x})~ |~ x\in E_m^\epsilon, ~ \tilde{x} = (x, f(x), .., f^{m-1}(x)) \}$. So we have
$$Vol_\omega(\Gamma_m) \geq \sum_{x\in E_\epsilon^m} Vol_{\omega}(B^m_\epsilon(\tilde{x})) \geq \sum_{x\in E_\epsilon^m} Vol_{\omega}(B^{m, +}_\epsilon(\tilde{x})) \geq C_{\epsilon} N_d(f, m, \epsilon)$$
Now $Vol(\Gamma_m) = \sum^m \int_{\mathbb{P}^n}|Jf^i| \omega = \int_{\mathbb{P}^n}(f^i)^*\omega $ where $\omega$ is a volume form on $\mathbb{P}^n$. By inspection we have
\begin{equation}
\label{eq:2}
Vol(\Gamma_m) = \sum^{m-1}_{i=0} \int_{\mathbb{P}^n}|Jf^i| \omega = \sum^{m-1}_{i=0} N^i
\end{equation}
where the last inequality follows from algebraic black magic.
Inserting this into the entropy formula, we get
\begin{align*}
h_d(f) & = \lim_{\epsilon \to 0} \limsup_{m\to \infty} \frac{log(N_d(f, m, \epsilon))}{m} \\
& \leq \lim_{\epsilon \to 0} \limsup_{m\to \infty} \frac{log( \frac{\sum_{i=0}^{m-1} N^i}{C_\epsilon}) }{m} \\
& = \lim_{\epsilon \to 0} \limsup_{m\to \infty} \frac{log(\sum_{i=0}^{m-1} N^i )}{m} \\
& = \lim_{\epsilon \to 0} \limsup_{m\to \infty} \frac{log(\frac{1 – N^m}{1-N})}{m} \qquad \text{Sum formula Geometric series}\\
& = log(N)
\end{align*}
To wrap things up, here are some useful examples to keep in mind. Any isometry or contractive map $f$ of a metric space has zero entropy, both topological, metric and measure theoretic (the measures on topological spaces are always assumed to be regular Borel measures). This is easiest verified for the metric entropy, since the metric $d_f^n = d$, we have that $N_d(f, n, \epsilon)$ is constant as $n$ increases. The result extends to the measure theoretic entropy and topological entropy by applying Theorem 1 and Theorem 2 respectively.
Previously it was shown that if $f: X\to X$ is a continuous map on a compact metric space $X$, and the family $\{ f^n \}_{n=0}^\infty$ is (uniformly) equicontinuous, then the $f$ has zero entropy.
In both the above examples the Fatou set the of the family $\{ f^n \}_{n=0}^\infty$ is whole domain. By this observation, at least we can deduce that if the entropy of a holomorphic map on a compact metrizable space has non-zero entropy, then it’s Julia set must be non-empty. This doesn’t really give us that much in the case of $\mathbb{P}^1$ since for any holomorphic map $f$ with $deg(f) \geq 2$ we know that $J(f) \neq \emptyset$, and for $deg(f) = 1$, we have $h(f)= log(1) = 0$.
In general not that much can be said about the entropy given the size and shape of its Julia set. Take for instance the maps $g, f : \mathbb{P}^1 \to \mathbb{P}^1$ defined by $f(z) \mapsto \frac{(z^2 + 1)^2}{4z(z^2 -1 )}$ (see [5] problem 7-g) and $g(z) = z^4$. It is know that the Julia set of $f$ is the entire sphere, and the Julia set of $g$ is the unit circle centered at zero. But the entropy is $h_{top}(f) = h_{top}(g) = log(4)$.
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