Constructing new C*-algebras – Universal C*-algebras (1)

Universal C*-algebras are C*-algebras defined implicitly by relations and generators, much like group presentations. Contrary to the case with groups where the presentation can be constructed as a quotient of the free group, there is no analogous construction for C*-algebras, and we are not always guarantied the existence of a universal C*-algebras for any presentation.

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.

Universal C*-algebra

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:

Definition 1 (Universal C*-algebra)
A C*-algebra $\mathcal{A}$ is universal with respect to a family $\{ u_i \}_I$ of indeterminates and relations $\mathcal{R}$ if for every other C*-algebra $\mathcal{B}$ generated by a family $\{v_i\}_I \subset \mathcal{B}$ of elements satisfying relations $\mathcal{R}$ there exists a unique surjective morphism $\phi: \mathcal{A} \to \mathcal{B}$ such that
$$\phi(u_i) = v_i$$
(henceforth referred to as “mapping generators to generators” )

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.

Proposition 1
The C*-algebra $C^*(\{ v_i\}, \mathcal{R})$ has the desired universal property of Definition 1.
Proof
Let $\mathcal{A}$ be a C*-algebra generated by the elements $\{ a_i \} \subset \mathcal{A}$ satisfying the relations $\mathcal{R}$. We know there exists a surjective *-homomorphism from the (dense) free *-algebra $F(\{ u_i \}, \mathcal{R})$ of $C^*(\{ v_i\}, \mathcal{R})$ to the (dense) *-subalgebra of $\mathcal{A}$ consisting of non-commuting complex polynomials in the variables $\{ a_i \}$ and $\{a_i^* \}$. Restricting the respective norms on these dense sub *-algebras, this *-homomorphism is norm decreasing (by construction), hence it lifts to a C*-algebra morphism of the completion with the desired properties.

Generated by unitaries

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,

Proposition 2
The C*-algebra given by unitaries $\{u_i\}$ and relations $\mathcal{R}$ is the (full) group C*-algebra $$C^*(\langle \{ v_i\} ~|~ \mathcal{R} \rangle),$$ where $\langle \{ v_i\} ~|~ \mathcal{R} \rangle$ is the group presentation given by the indeterminates $v_i$ and relations $\mathcal{R}$.
Proof
The proof hinges on the universal property of the group presentation $\langle \{ v_i\} ~|~ \mathcal{R} \rangle$, which states that for ever other group $G$ generated elements $\{ g_i\}$ satisfying relations $\mathcal{R}$ there exists a surjective gruop homomorphism $\phi: \langle \{ v_i\} ~|~ \mathcal{R} \rangle \to G$ mapping generators to generators. This group homomorphism is a unitary representation of $\langle \{ v_i\} ~|~ \mathcal{R} \rangle$ hence we can lift it to a C*-homomorphism of the (full) group C*-algebra $C^*(\langle \{ v_i\} ~|~ \mathcal{R} \rangle)$, which is easily seen to be the required surjective C*-homomorphism.

Examples

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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