6
$\begingroup$

Let $A$ be a C*-algebra, let $G$ be a locally compact group, and let $\alpha\colon G\to\mathrm{Aut}(A)$ be a (strongly) continuous action. It is well known that there is a natural map $\iota_G\colon C^*(G)\to M(A\rtimes_\alpha G)$.


Question: Is the natural map $\iota_G\colon C^*(G)\to M(A\rtimes_\alpha G)$ injective?


This is true for reduced group $C^*$-algebras and reduced crossed products, and I wonder whether it holds in the universal case too.

One would need to know something about what group representations $v\colon G\to \mathcal{U}(H_v)$ can be part of a covariant representation $(\pi,v)$ of the dynamical system $(G,A,\alpha)$ on $H_v$. My question can be restated as follows:


Reformulation: Let $w$ be the direct sum of all unitary representations $v\colon G\to \mathcal{U}(H_v)$ such that there exists a representation $\pi\colon A\to \mathcal{B}(H_v)$ so that $(\pi,v)$ is a covariant pair. Is $u$ the universal representation of $G$?


$\endgroup$
6
$\begingroup$

The answer is no. For a counter example, take an amenable action $\alpha$ of a non-amenable discrete group G on a unital C*-algebra $A$. Then $A\rtimes_\alpha G$ coincides with the reduced crossed product and hence the standard conditional expectation is faithful. If your map were injective, the standard conditional expectation on $C^*(G)$ would also be faithful, but it isn't.

| cite | improve this answer | |
$\endgroup$
  • 1
    $\begingroup$ Perhaps I should add that, by a result of Ozawa (arXiv:math/0002185), every exact group, such as $F_2$, admits an amenable action on a compact space. $\endgroup$ – Ruy Sep 14 '16 at 21:16
  • $\begingroup$ If I am not mistaken your argument shows that $\iota_G$ is never injective when $A$ is exact and $\alpha$ is an amenable action of a nonamenable group. It will also be interesting to know whether $\iota_G$ can be injective for nonamenable actions. $\endgroup$ – Adrián González-Pérez Nov 23 '16 at 14:58
  • $\begingroup$ @Adrian, if $A$ is unital, the argument works regardless of exactness. In the nonunital case I guess one would have to extend the conditional expectation to the multiplier algebra and, if this works, exactness of $A$ will not be needed. Can you explain your argument? $\endgroup$ – Ruy Nov 23 '16 at 18:58
  • $\begingroup$ My apologies, @Ruy. I missread your argument. I just read the beginning and thought that you were using that if $A$ is exact and $\alpha$ amenable, $A \rtimes_\alpha G$ is exact and that $C^\ast G$ is not exact for nonamenable $G$. You cannot embbed a non exact algebra in an exact one. Your argument works fine since we are not considering all possible embeddings, just $\iota_G$. My bad. $\endgroup$ – Adrián González-Pérez Nov 23 '16 at 20:50
0
$\begingroup$

No. Let $G$ be a locally compact group acting on a compact Hausdorff topological space $X$. Then the canonical $*$-homomorphism $C^*(G)\rightarrow C(X)\rtimes G$ is injective iff there is an invariant measure on $X$. The later can be equivalent to amenability of $G$ for some $X=\beta^{lu}(G)$. When $G$ is discrete, then $\beta^{lu}(G)$ is the Stone-Cech compactification of $G$.

| cite | improve this answer | |
$\endgroup$
  • $\begingroup$ Isn't this just a special case of @Ruy's answer? $\endgroup$ – Yemon Choi Sep 22 '16 at 19:41
  • $\begingroup$ @YemonChoi: I just want to indicate the converse implication. You can replace C(X) by a unital $C^*$-algebra $A$ and replace an invariant measure on $X$ by a $G$-invariant state on $ A$ to obtain a generalization. $\endgroup$ – m07kl Sep 24 '16 at 10:50

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.