Homomorphisms of transformation group C*-algebras – A worked example

In this blog post we work through some examples of computations with crossed product C*-algebras, particularly we look at characterizing *-homomorphisms from a transformation group C*-algebra to a commutative C*-algebra. Some familiarity with C*-crossed products will be assume. Trying to determine all homomorphisms between two given C*-algebra, their algebraic structure, their homotopy classes and so forth, can be quite challenging in general. Most of the standard references on crossed products show how to construct a *-homomorphism of crossed products from an equivariant *-homomorphism of the underlying C*-algebra, but rarely show examples of morphisms that don’t arise in this way.

The example we will look at in this blog post is rather simple though so everything can be computed explicitly. I will be focusing on maps

$C(X)\rtimes \Gamma \to C(Y)$

for two compact Hausdorff spaces ${X,Y}$ and discrete groups ${\Gamma}$ . In particular, this covers all characters of ${C(X)\rtimes \Gamma}$ , and when ${\Gamma}$ is abelian, also maps of the form

$C(X)\rtimes \Gamma \to \C\rtimes \Gamma = C(\hat{\Gamma})$

between the C*-algebra ${C(X)\rtimes \Gamma}$ and ${C(\hat{\Gamma})}$ where ${\Gamma}$ is a discrete abelian group and ${\hat{\Gamma}}$ its Pontryagin dual. The reason one may want to study these maps is that for each fixed point of the ${\Gamma}$ action in ${X}$ we get a *-homomorphism ${\Gamma}$ -equivariant map ${ev_x\otimes id: C(X)\rtimes \Gamma \to C^*(\Gamma) = C(\hat{\Gamma})}$ , hence they show up quite naturally for actions with fixed points.

For many pathological examples of transformation group C*-algebras see .

1. Notation

Throughout this post ${X}$ will denote a compact Hausdorff space with a continuous action of ${\Gamma}$ . The crossed product C*-algebra of the dynamical system ${C(X)\rtimes \Gamma = C^\star(X\rtimes \Gamma)}$ will be referred to as the transformation group C*-algebra. We will denote by ${Fix_{\Gamma}(X)}$ the fixed points of ${\Gamma}$ in ${X}$ , meaning the points ${\{x\in X~|~ \gamma x = x ~ \text{for all } \gamma\in \Gamma\}}$ . For now we make no assumptions on this set, but in most cases it will be assumed to be discrete.

An element ${f}$ in the dense subalgebra ${C_c(\Gamma, C(X))\subset C(X)\rtimes \Gamma}$ will be denoted by ${f = \sum_{z\in \Gamma} f_z u_z}$ where ${f(z) = f_z}$ .

We denote by ${\alpha}$ or ${\beta}$ the action of the group ${\Gamma}$ on a topological space, is continuous function algebra or arbitrary C*-algebras.

For a *-homomorphism ${\phi: C(X)\to C(Y)}$ between two ${\Gamma}$ -dynamical systems, commuting with the ${\Gamma}$ -action, we denote by ${\phi\otimes id: C(X)\rtimes \Gamma \to C(Y)\rtimes\Gamma}$ the induced *-homomorphism on the transformation group C*-algebra, defined for a finitely supported function ${\sum f_z u_z}$ by

$(\phi\otimes id)(\sum f_z u_z ) = \sum \phi(f_z) u_z.$

If we endow ${\C}$ with the trivial action of of ${\Gamma}$ we get, by functionality of the crossed product, a surjective map ${ev_x\otimes id: C(X)\rtimes \Gamma \to \C\rtimes \Gamma = C^\star(\Gamma) = C(\hat{\Gamma})}$ for each ${\Gamma}$ -fixed point ${x\in X}$ , where ${ev_x: C(X)\to \C}$ is the evaluation at ${x}$ map.

As an example to have in mind, if ${\Gamma = \Z^n}$ we have ${\hat{\Gamma} = \mathbb{T}^n}$ and here the homotopy classes of endomorphisms of ${C(\mathbb{T}^n)}$ is given by

$End(C(\mathbb{T}^n)) = M_n(\Z)$

as any continuous map on ${\mathbb{T}^n}$ can be approximated by a linear map on the universal cover. This endows the homotopy classes of maps $[C(X)\rtimes \Z^n, C(\matbb{T}^n)]$ with a right $M_n(\Z)$ action.

Let us first prove the following basic but important result –

Lemma 1 Let ${\Gamma}$ be any discrete group, ${Y}$ a compact Hausdorff space and ${\phi : C(X)\rtimes \Gamma \to C(Y)}$ a non-zero *-homomorphism. Then the restriction ${\phi_e = \phi|_{C(X)} : C(X)\to C(Y)}$ is a ${\Gamma}$ -invariant *-homomorphism. In particular if any continuous function ${Y\to Fix_{\Gamma}(X)}$ is constant (say ${Fix_{\Gamma}(X)}$ is discrete and ${Y}$ connected) we have

$\phi_e = 1\cdot ev_x$

for some ${x\in Fix_{\Gamma}(X)}$ .

Proof

Since

$u_z f u_z^* = \alpha_z(f)$

we can write

$\begin{align*} \phi(fu_z) & = \phi_e(f)\phi(u_z) & = \phi(u_z)\phi_e(f) & = \phi(u_z f) & = \phi(\alpha_z(f) u_z) & = \phi_e(\alpha_z(f)) \phi(u_z) \end{align*}$

This has to hold for any $z$ and ${f}$ which forces ${\phi_e}$ to be ${\Gamma}$ -invariant.

Since we can write ${\phi_e(f) = f\circ \hat{\phi_e}}$ for some continuous map ${\hat{\phi_e}: Y \to X}$ , equivariance of ${\phi}$ is equivalent to

$\hat{\phi_e}(Y)\subset Fix_\Gamma(X).$

It follows that ${\phi_e}$ is just the evaluation map if ${\hat{\phi_e}}$ is constant.

Now if ${\phi: \aa \to \bb}$ is a ${\alpha-\beta}$ -equivariant *-homomorphism between two C*-algebras with respect to two actions of a discrete group ${\Gamma}$ and ${\rho : \Gamma \to \Gamma}$ is a group homomorphism, we can define a linear map

$\phi \otimes \rho: C(X)\rtimes_\alpha \Gamma \to C(X)\rtimes_\beta \Gamma$

for any equivariant map ${\phi: C(X)\to C(Y)}$ by

$\phi\otimes \rho(\sum f_\gamma u_\gamma) = \sum \phi(f)u_{\rho(\gamma)}.$

Lemma 2 The map ${\phi\otimes \rho}$ determines a *-homomorphism ${ \aa\rtimes_\alpha \Gamma \to \bb \rtimes_\beta \Gamma}$ if and only if $\Gamma$ is unimodular and the following equivalent conditions hold

${\beta_{\rho(\gamma)\gamma^{-1}} = id}$ for all ${\gamma \in \Gamma}$ , where ${\beta}$ is the action of ${\Gamma}$ on ${\bb}$ .
${\rho}$ descends to a the identity map ${id = [\rho]: G/H \to G/H}$ , where ${H = \bigcap_{x\in X}G_x}$ is the intersection of all stabilizer subgroups.

In particular if the action ${\beta}$ is effective this only holds if ${\rho = id}$ .

Proof

It is a good exercise to check that it preserves products of compactly supported functions, which holds even without the conditions on ${\beta}$ . Let us show what goes wrong for the involution, assuming for simplicity that ${\Gamma}$ is unimodular and fill out the details for the general case yourself

$\begin{align*} \phi\otimes \rho ((\sum f_\gamma u_\gamma)^* ) & = \phi\otimes \rho (\sum \alpha_{\gamma^{-1}}(f_\gamma) u_\gamma) \\ & = \sum \phi( \alpha_{\gamma^{-1}}(f_\gamma)) u_{\rho(\gamma)} \\ & = \sum \beta_{\gamma^{-1}}(\phi( f_\gamma)) u_{\rho(\gamma)} \end{align*}$

However, we also have

$\begin{align*} (\phi\otimes \rho (\sum f_\gamma u_\gamma) )^* & = \sum \beta_{\rho(\gamma^{-1})}(\phi(f_\gamma)) u_{\rho_\gamma} \end{align*}$

Hence we have a ${*}$ -preserving map if and only if ${\beta_{\rho(\gamma)} = \beta_{\gamma}}$ , which proves the lemma (modulo details left to the reader).

Exercise) Using the above Lemma and our friend the group C*-algebra, show that not all *-homomorphisms of crossed product C*-algebras are induced from equivariant *-homomorphisms of the C*-algebra.

We have so far tacitly ignored the question of boundedness of the above induced maps, since we have been working with the full crossed product. It should be mentioned that for non-effective actions, the question of norm boundedness of the induced map in the above Lemma becomes relevant. For the record, the induced map is bounded in the reduced crossed product if and only if the representation given by composing ${\phi_\Gamma}$ with a faithful representation of ${C(Y)}$ is “weakly contained” in the regular representation of $C_c(\Gamma, C(X)))$ on $l^2(\Gamma, C(X))$ .

Let us prove the following Lemma –

Lemma 3 Let ${\Gamma}$ be any discrete group. Let ${\phi_e:C(X)\to C(Y)}$ be any ${\Gamma}$ -invariant *-homomorphism and ${\phi_\Gamma: \Gamma \to U(C(U))= C(Y, \mathbb{T})}$ an arbitrary group homomorphism. Then the map ${\phi}$ on ${C_c(\Gamma, C(X))}$ given by

$\phi(\sum f_\gamma u_\gamma) := \sum \phi_e(f_\gamma) \phi_\Gamma(u_\gamma)$

determines a *-homomorphism of ${C(X)\rtimes \Gamma \to C(Y)}$ . Conversely, all *-homomorphisms ${\phi: C(X)\rtimes \Gamma \to C(Y)}$ are of this form for a pair of morphisms ${\phi_e}$ and ${\phi_\Gamma}$

Proof

Let ${\phi:C(X)\rtimes \Gamma \to C(Y)}$ be a *-homomorphism and ${\phi_e}$ , ${\phi_\Gamma}$ its restriction to ${C(X)}$ and ${\Gamma \simeq \{u_\gamma\}_{\gamma \in \Gamma}}$ respectively. Since ${C(X)}$ and ${\Gamma}$ generate ${C(X)\rtimes \Gamma}$ clearly any *-homomorphism is determined by its values on these subsets and

$\phi(\sum f_\gamma u_\gamma) = \sum\phi(f_\gamma) \phi(u_\gamma) = \sum \phi_e(f_\gamma) \phi_\Gamma (u_\gamma).$

Now let ${\phi_e: C(X)\to C(Y)}$ be an arbitrary ${\Gamma}$ -invariant *-homomorphism and ${\phi_\Gamma \to U(C(Y))}$ an arbitrary group homomorphism. Then let us show the ${\phi}$ defined in the lemma is a *-homomorphism. It is linear by construction. Let us check multiplicativity

$\begin{align*} \phi(\sum f_\gamma u_\gamma \star \sum f'_\gamma u_\gamma) &= \phi(\sum_\gamma \sum_\eta f_\eta \alpha_\eta(f'_{\eta^{-1}\gamma} )u_\gamma) \\ & = \sum_\gamma \sum_\eta \phi_e( f_\eta) \phi_e( \alpha_\eta(f'_{\eta^{-1}\gamma})) \phi_\Gamma (u_\gamma) \\ &= \sum_\gamma \sum_\eta \phi_e( f_\eta) \phi_e(f'_{\eta^{-1}\gamma}) \phi_\Gamma (u_\gamma)\\ & = (\sum_\gamma \phi_e( f_\gamma) \phi_\Gamma (u_\gamma)) (\sum_\gamma \phi_e( f'_\gamma) \phi_\Gamma (u_\gamma) )\\ &= \phi(\sum_\gamma f_\gamma u_\gamma) \phi(\sum_\gamma f'_\gamma u_\gamma ). \end{align*}$

And *-preserving:

$\begin{align*} \phi((f_\gamma u_\gamma)^*) \\ &= \phi( \alpha_{\gamma^{-1}}(f_\gamma^*) u_\gamma^*) \\ & = \phi_e(\alpha_{\gamma^{-1}}(f_\gamma^*)) \phi_\Gamma(u_\gamma^*) \\ & = \phi_e(f_\gamma^*) \phi_\Gamma(u_\gamma^*) \\ & = [\phi_e(f_\gamma)\phi_\Gamma(u_\gamma)]^* = \phi(f_\gamma u_\gamma)^\star \end{align*}$

The next Lemma tells us a little more about the homotopy classes of *-homomorphisms:

Lemma 4 Let ${\Gamma}$ be a discrete group with discrete fixed point set and assume ${Y}$ is connected. Then two *-homomorphisms ${\phi, \phi': C(X)\rtimes \Gamma \to C(Y)}$ with corresponding group homomorphisms

$\phi_\Gamma, \phi'_\Gamma : \Gamma \to U(C(Y))$

and algebra *-homomorphisms

$\phi_e, \phi'_e :C(X)\to C(Y)$

are homotopic if and only if ${\phi_e = \phi'_e}$ and ${\phi_\Gamma}$ is homotopic to ${\phi_\Gamma'}$ (i.e. there is a family of group homomorphisms ${\phi_\Gamma^t: \Gamma \to U(C(Y)) = C(Y, \mathbb{T})}$ , for ${t\in [0,1]}$ such that the map ${t \mapsto \phi^t_\Gamma(u_\gamma)}$ is continuous for all ${u_\gamma \in \Gamma}$ and ${\phi^0_\Gamma = \phi_\Gamma}$ , ${\phi_\Gamma^1 = \phi'_\Gamma}$ ).

Proof

Let ${\phi^t}$ be a homotopy between ${\phi^0 := \phi}$ and ${\phi^1 = \phi'}$ . Then ${\phi^t_e := \phi^t|_{C(X)}}$ is a homotopy between ${\phi_e}$ and ${\phi'_e}$ . Since these are evaluation maps and ${Fix_\Gamma(X)}$ is discrete, we must have that ${\phi_e = \phi^t_e = \phi'_e}$ for all ${t \in[0,1]}$ . By a similar logic, we have that ${\phi^t_\Gamma = \phi^t|_\Gamma}$ is a homotopy between ${\phi_\Gamma}$ and ${\phi'_\Gamma}$ .

Conversely, let ${\phi_\Gamma, \phi'_\Gamma: \Gamma \to U(C(Y))}$ is a group homomorphism for which there is a homotopy ${\phi^t_\Gamma}$ between them. Then define ${\phi^t(\sum f_\gamma u_\gamma):= \sum \phi_e^t(f_\gamma) \phi^t_\Gamma(u_\gamma)}$ for some ${\Gamma}$ -invariant ${\phi_e: C(X)\to C(Y)}$ . By the previous Lemma, ${\phi^t}$ extends to a *-homomorphism ${\phi^t: C(X)\rtimes \Gamma \to C(Y)}$ for each ${t}$ .

Let ${t_i \to t}$ and fix ${x\in C(X)\rtimes \Gamma}$ . If ${x \in C_c(\Gamma, C(X))}$ , then it is easy to check that ${\phi^{t_i}(x)\to \phi^{t}(x)}$ , hence ${t\mapsto \phi^t(x)}$ is continuous if ${x\in C_c(\Gamma, C(X))}$ . Assume now that ${x = \lim_s \sum f_\gamma^s u_\gamma}$ for some sequence ${\sum f_\gamma^s u_\gamma \in C_c(\Gamma, C(X))}$ . For an arbitrary ${\epsilon > 0}$ we pick ${s}$ large enough so that ${||x_s - x|| < \epsilon}$ . Then since *-homomorphisms are contractive, we have

$||\pi^{t_i}(x_s) - \pi^{t_i}(x)|| \leq ||x_s - x|| < \epsilon$

for all ${i}$ , hence the convergence is ${s}$ is uniform in ${i}$ . We can thus us Moore-Osgood’s theorem to interchange the limits

$\begin{align*} \phi^t(x) = \lim_s \phi^{t}(x_s) = \lim_s \lim_i \phi^{t_i}(x_s) = \lim_i \lim_s \phi^{t_i }(x_s) = \lim_i \phi^{t_i}(x) \end{align*}$

which shows that ${t\mapsto \phi^t(x)}$ is continuous for any ${x\in C(X)\rtimes \Gamma}$ .

Armed with the above lemmas, we can now prove the following Proposition

Proposition 5 Assume ${\Gamma}$ is finitely generated group with discrete fixed point set and ${Y}$ a connected compact Hausdorff space. Let ${[C(X)\rtimes \Gamma, C(Y)]}$ denote the homotopy classes of *-homomorphisms ${C(X)\rtimes \Gamma \to C(Y)}$ . Then there is a 1-1 correspondence

$[C(X)\rtimes \Gamma, C(Y)] \leftrightarrow \{(\rho_x)_{x\in Fix_\Gamma(X)} ~|~ \rho_x: \Gamma \to \pi^1(Y)\}$

where

${\pi^1(Y):=U(C(Y))/U(C(Y))_0}$

is the cohomotopy group of ${Y}$ .

Proof

Let ${\gamma_i}$ be the set of generators of ${\Gamma}$ . If ${\phi, \phi' : \Gamma \to U(C(Y))}$ are two group homomorphisms with ${\phi(\gamma_i)}$ and ${\phi'(\gamma_i)}$ in the same connected component of ${U(C(Y))}$ we can continuously deform ${\phi}$ to a group homomorphism ${\hat{\phi}}$ for which ${\hat{\phi}(\gamma_i) = \phi'(\gamma_i)}$ . Induction on the set of generators shows $\phi$ and $\phi'$ are homotopic. The claim now follows easily.

Examples Let us go through some examples. Assume the conditions in the above Corollary are met, then:

If ${Y = \{pt\}}$ and ${\Gamma}$ is abelian, then

${[C(X)\rtimes \Gamma, \C] \leftrightarrow Fix_\Gamma(X) \times S}$
where ${S}$ is the number of connected components of ${\hat{\Gamma}}$ . This is because all morphisms are of the form ${ev_x\otimes \rho}$ for some character ${\rho: \Gamma \to \mathbb{T} \subset \C}$ . Consequently, if ${\Gamma}$ is also torsion-free then

$[C(X)\rtimes \Gamma, \C] \leftrightarrow Fix_\Gamma(X).$
Similarly, if either ${\Gamma}$ is perfect or $K^1(Y) = 0$ then

$[C(X)\rtimes \Gamma, C(Y)] \leftrightarrow Fix_\Gamma(X)$
in the first case because there are no non-trivial group homomorphisms from ${\Gamma}$ to any abelian group, in the latter because ${\pi^1(Y)\subset K^1(Y)} = 0$ , (see this book Section 8.3).
Assume $\pi^1(Y)$ (or $K^1(Y)$ ) is finitely generated, $\Gamma$ is torsion-free with $H^1(\Gamma, \Z) =0$ , then

$[C(X)\rtimes \Gamma, C(Y)] \leftrightarrow Fix_\Gamma(X)$
To see this, note the conditions imply there are no non-trivial group homomorphisms $\phi: \Gamma \to \Z$ , hence there can be no non-trivial group homomorphism $\phi: \Gamma \to \pi^1(Y) = \bigoplus_{i} \Z_{n_i} = \pi^1(Y)$ where $n_i\in \mathbb{N}\cup \{\infty\}$ .

Here is an more interesting application of the above, for those familiar with symmetric spaces of non-compact type:

Corollary 6 Assume ${G}$ is a linear algebraic group and ${K\subset G}$ a maximal compact such that ${G/K}$ is a symmetric space of non-compact type. Let ${P\subset G}$ be a minimal parabolic subgroup and let ${A}$ be a maximal flat totally geodesic submanifold of ${G/K}$ with ${\Gamma_A \subset A}$ is a lattice in ${A}$ , hence ${\Gamma_A \simeq \Z^n}$ where ${n}$ is the ${\R}$ -rank of ${G}$ . Let ${W}$ denote the Weyl group of the (restricted) roots of ${A}$ . Then

$|W| = \#[C(G/P)\rtimes \Gamma_A, \C]$

hence the cardinality of the Weyl group is an isomorphism invariant of $C(G/P)\rtimes \Gamma_A$ .

Proof

It is well known that the Weyl group is in 1-1 correspondence with the fixed points of ${A}$ in ${G/P}$ . Since ${P}$ is Zariski closed and ${\Gamma_A\subset A}$ is Zariski dense, and the action of ${A}$ is Zariski continuous it follows that

$Fix_A(G/P) = Fix_{\Gamma_A}(G/P).$

The claim now follows from Proposition 5, see also Example 2 above.

More examples of this kind will come shortly, but for this post I think this is a good place to stop.

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1. Notation

Bibliography

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