Constructing new C*-algebras – Crossed product.

In this post I want to cover the concept of a crossed product C*-algebras. These are algebras which arise naturally as C*-algebras associated with dynamical systems and (as the name suggests) has many similarities with the semidirect of groups. It is a wide area of active research. To simplify things, unless stated otherwise, $\mathcal{A}$ will denote a unital C*-algebra, $G$ a discrete group and $H$ will always denote a complex Hilbert space. I will give references for the more general definitions when the C*-algebra is non-unital or the group is only assumed to be locally compact, as this would require the use of multiplier algebras and Haar measures/integrals making the exposition less approachable and the notation a lot less readable.

Preliminaries

Definition (C*-dynamical system)
A C*-dynamical system is a triple $(\mathcal{A}, G, \alpha)$, where $\alpha:G \to Aut(\mathcal{A})$ is a group homomorphism into the group of $\star$-automorphisms of $\mathcal{A}$. Generally the homomorphism $\alpha$ is required to be “strongly continnous”, that is, the map $g\mapsto \alpha_g(a)$ is continuous for all $a\in \mathcal{A}$, but we dropped this requirement, since $G$ is assumed to be discrete.
Definition (Covariant Representation)
A covariant representation of a dynamical system $(\mathcal{A}, G, \alpha)$ is defined as a pair $(\pi, U)$ where $\pi: \mathcal{A} \to B(H)$ is a non-degenerate representation of $\mathcal{A}$ on some Hilbert space $H$, $U: G \to B(H)$ is a unitary representation of $G$ on $H$, satisfying the “covariance” relation:
\begin{equation}
\pi(\alpha(g)(a)) = U(g)\pi(a) U(g)^* \qquad \forall ~ g\in G, a \in \mathcal{A}
\end{equation}
Henceforth we will often denote $U(g)$ and $\alpha(g)$ simply by $U_g$ and $\alpha_g$ respectively.

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.

Proposition 1 (Existence of covariant representations)
For any dynamical system $(\mathcal{A}, G, \alpha)$ there exists at least one covariant representation associated with any non-degenerate representation $\pi:\mathcal{A}\to B(H)$ of $\mathcal{A}$, namely the \textbf{regular covariant representation}, which we will denote by $(\tilde{\pi}, \lambda^H)$.
Proof
Letting $\pi:\mathcal{A} \to B(H)$ be any non-degenerate representation of $\mathcal{A}$, we define $(\tilde{\pi}, \lambda^H)$ to be the covariant representation on the Hilbert space direct sum
$$ \bigoplus_{g\in G}H = \{ f:G \to H ~| ~ \sum_{g\in G} ||f(g)||^2 < \infty \} =: l^2(G, H), $$
given by
\begin{array}{llc}
(\tilde{\pi}(a)\xi)(s) & = \pi(\alpha_{s^{-1}}(a))\xi(s) & \text{for all $s\in G$, $a\in \mathcal{A}$ and $\xi\in l^2(G, H)$, }\\
(\lambda^H_g\xi)(s) & = \xi(g^{-1}s) & \text{for all $s, g \in G$, and $\xi \in l^2(G, H)$}.\\
\end{array}

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.

Universal property of the crossed-product C*-algebra

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 $j_\mathcal{A}(\alpha_g(a)) = j_G(g)j_\mathcal{A}(a)j_G(g)^*$.

Definition 3 (Raeburn)
The C*-crossed product associated with $(\mathcal{A}, G, \alpha)$, denoted $\mathcal{A}\rtimes_\alpha G$, is the C*-algebra satisfying the following universal properties

  • There exists a covariant homomorphism $(j_\mathcal{A}, j_G)$ of $(\mathcal{A}, G, \alpha)$ into $\mathcal{A}\rtimes_\alpha G$
  • For any covariant representation $(\pi, U)$ of $(\mathcal{A}, G, \alpha)$, there is a non-degenerate representation $L = L_{(\pi, U)}$ of $\mathcal{A}\rtimes_\alpha G$ such that
    \begin{array}{cc}
    L \circ j_\mathcal{A} &= \pi\\
    L\circ j_G &= U
    \end{array}
  • $\mathcal{A}\rtimes_\alpha G = \overline{\text{span}\{ j_\mathcal{A}(a)j_G(g) ~|~ a \in \mathcal{A}, ~ g\in G\}}$

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.

Skew Algebra

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.

The concrete crossed-product C*-algebra

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

Lemma 1
If $\pi:\mathcal{A} \to B(H)$ is a faithful non-degenerate representation, then $\tilde{\pi}\times \lambda^H $ is a non-degenerate faithful representation of $C_c(G, \mathcal{A})$, where $(\tilde{\pi}, \lambda^H)$ is the regular covariant representation of Proposition 1.
Proof
Let $\pi$ be a faithful non-degenerate representation of $\mathcal{A}$, let $f\in C_c(G, \mathcal{A})$ be non-zero and let $g_0 \in supp(f)$. Since $\pi$ is faithful and $\alpha_{g_0^{-1}}$ is an automorphism, we may pick an $x \in H$ such that $\pi(\alpha_{g_0^{-1}}(f(g_0)))x \neq 0$.
Define, $\Delta_{x, g}\in C_c(G, H) \subset l^2(G, H)$ by
$$\Delta_{x, g}(s) = \begin{cases} x & \text{if } s = g\\ 0 & \text{else} \end{cases}$$
Note that $\lambda_g^H\Delta_{x, e} = \Delta_{x,g}$ for all $g\in G$. Hence we get
\begin{array}{cc}
\left[\tilde{\pi}\times \lambda^H(f)(\Delta_{x, e})\right](g_0) & = \sum_{g\in G}\left[ (\tilde{\pi}(f(g))\lambda^H_g)(\Delta_{x, e})\right](g_0) \\
& = \sum_{g\in G} \left[\pi(\alpha_{g_0^{-1}}(f(g))) (\lambda^H_g(\Delta_{x, e}))\right](g_0)\\
& = \sum_{g\in G} \left[ \pi(\alpha_{g_0^{-1}}(f(g))) (\Delta_{x, g})\right](g_0) \\
& = \pi(\alpha_{g_0^{-1}}(f(g_0)))x \neq 0
\end{array}
So $ \tilde{\pi}\times \lambda^H(f) \neq 0$ and $\tilde{\pi}\times \lambda^H$ is injective.

The definition now reads

Reduced Crossed Product C*-algebra
The \textbf{reduced crossed product C*-algebra} associated with the dynamical system $(A, G, \alpha)$, denoted $\mathcal{A}\rtimes_{r, \alpha}G$, or $\mathcal{A}\rtimes_r G$, is defined as the C*-completion of $C_c(G, \mathcal{A})$ with respect to the following C*-norm
$$||f||_r = ||\tilde{\pi}\times \lambda^H(f)|| $$
Where $\tilde{\pi}\times \lambda^H$ is the regular representation associated with any faithful non-degenerate representation $\pi$ of $\mathcal{A}$.

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 $(\mathcal{A}, G, \alpha )$.

Proposition 2
$||\cdot||_u$ is a well defined C*-norm on $C_c(G, \mathcal{A})$, satisfying the C*-equality.
Proof
Submultiplicativity, and the C*-equality of $||\cdot||_u$ are straightforward to check. The supremum is finite, since we have a uniform bound given by
\begin{array}{cc}
||\pi\times U (f)|| & = || \sum_{g\in G} \pi(f(g))U_g || \leq \sum_{g\in G}||f (g)|| := ||f||_1 < \infty
\end{array}
since $f$ has finite support. Lastly since $||f||_u \geq ||f||_r$ we have that $||f||_u = 0 \Rightarrow ||f||_r = 0 \Rightarrow f = 0$, so it is indeed a C*-norm.
Definition 4 (Crossed Product C*-algebra)
The (full) crossed product C*-algebra associated with a dynamical system $(A, G, \alpha)$, denoted $\mathcal{A}\rtimes_\alpha G$, is defined as the C*-completion of $C_c(G, \mathcal{A})$ with respect to the universal norm defined above.

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.

Proof
By inspection we can see that
$$\delta_{\alpha_g(a), e} = \delta_{1, g} \star \delta_{a, e} \star \delta_{1, g}^* $$
making the previously defined inclusion maps $\iota_\mathcal{A}, \iota_G$ a covariant homomorphism of the system $(\mathcal{A}, G, \alpha)$.
If $(\pi, U )$ is a covariant representation of $(\mathcal{A}, G, \alpha)$, and $a\in \mathcal{A}$, $g\in G$ are arbitrary, we have
\begin{array}
& \pi\rtimes U \circ \iota_\mathcal{A}(a) = \pi(a) \\
& \pi\rtimes U \circ \iota_G(g) = U(g)
\end{array}
so with $L_{(\pi, U)} = \pi\rtimes U$ the two first conditions of Definition 3 are satisfied.
Lastly we have that $\delta_{a, e} \star \delta_{1, g} = \delta_{a, g}$, hence
$$\text{span}\{ \iota_\mathcal{A}(a)\iota_G(g) ~| ~ a\in \mathcal{A}, ~ g\in G \} = C_c(G, \mathcal{A})$$
which is dense in $\mathcal{A}\rtimes_\alpha G$.

Representations of Crossed products

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.

Proposition 3
The map $L$ is a functor and determines an isomorphism of categories $$\text{rep}(\mathcal{A}\rtimes_\alpha G) \simeq \text{rep}(A, G, \alpha).$$
Proof
First let’s show $(\pi, U) \mapsto \pi\times U$ is indeed a bijection.\\
If $\rho$ is a non-degenerate representation of $\mathcal{A}\rtimes_\alpha G$ on a $H$, let
\begin{align*}
\pi_\rho(a) &= \rho(\iota_\mathcal{A} (a))\\
U_\rho(g) &= \rho(\iota_G(g))
\end{align*}
where $\iota_\mathcal{A}$ and $\iota_G$ are the inclusion maps defined earlier. Note that $\pi(1_{\mathcal{A}}) = I_H$ so $\pi$ is also non-degenerate. One readily checks that $(\pi_\rho, U_\rho)$ is a covariant representation of $(\mathcal{A}, G, \alpha)$ with $\pi_\rho\times U_\rho = \rho$. So $L$ is surjective.

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,

If either $\pi\not\sim \pi’$ or $U\not\sim U’$ then $\pi\rtimes U \not\sim \pi’\rtimes U’$, where $\sim$ denotes unitary equivalence.

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