Generalisation of the equivalence between $C^*(H)$ and $C_0(G/H) \rtimes G$; induction of group actions on C*-algebras

Question

There is a well known Morita equivalence between the group C*-algebra $C^*(H)$ and $C_0(G/H) \rtimes G$, where $H$ is a subgroup of $G$. The corresponding equivalence of representations is an incarnation of Mackey's imprimitivity theorem, see Rieffel's `Morita Equivalence for Operator Algebras', Example 1.

In topology, one can essentially treat a $H$-space $X$ as equivalent to the $G$ space $X \times_H G$. In particular, the $H$-equivariant K-theory of $X$ is isomorphic to the $G$-equivariant K-theory of $X \times_H G$. When $X$ is a point, this corresponds to the Morita equivalence of the previous paragraph.

My question is whether this Morita equivalence is a special case of an equivalence between $C_0(X) \rtimes H$ and $C_0(X \times_H G) \rtimes G$. When $X$ is a point, one uses $C^*(G)$ as the equivalence bimodule, but I don't know what to replace this with in the more general setting.

In case this is true: is there a general construction in $C^*$-algebras that is analogous to the construction of $X \times_H G$ from $X$. Explicitly, if $A$ is a $C^*$-algebra with $H$-action, is there a canonical $C^*$-algebra $\tilde A$ with $G$-action such that $A \rtimes H$ is Morita equivalent to $\tilde A \rtimes G$?

Answer 1

To answer the question in the final paragraph: yes, there is such a construction. If $H$ is a closed subgroup of $G$, and if $H$ acts on a $C^*$-algebra $A$, then one defines the induced $C^*$-algebra $\operatorname{Ind}_H^G A$ to be the collection of all continuous, bounded functions $f:G\to A$ satisfying:

1. $f(gh)=h^{-1}f(g)$ for all $h\in H$ and $g\in G$; and
2. the function $gH\mapsto \lVert f(g)\rVert$ vanishes at infinity on $G/H$.

$\operatorname{Ind}_H^G A$ is a $C^*$-algebra under pointwise operations and the supremum norm, and it carries an action of $G$ by $*$-automorphisms (coming from the action of $G$ on itself by left translation). This construction really is a generalisation of the situation considered in the earlier part of the question: if $X$ is a locally compact $H$-space then we have $\operatorname{Ind}_H^G C_0(X) \cong C_0(G\times_H X)$, $G$-equivariantly.

Green (The local structure of twisted covariance algebras, Zbl 0407.46053) proved, essentially, that there is a canonical Morita equivalence between the crossed products $(\operatorname{Ind}_H^G A)\rtimes G$ and $A\rtimes H$. An equivalence bimodule can be constructed from a suitable completion of the space of compactly supported continuous functions from $G$ to $A$, similarly to what is done for $A=\mathbb{C}$.

(Incidentally, a small comment on the second-last paragraph of the question: when $X$ is a point I believe that the imprimitivity bimodule is the one that implements unitary induction of representations from $H$ to $G$, as constructed by Rieffel. When $H=G$ this is indeed $C^*(G)$ but I'm not sure this holds in general. For instance, when $H$ is the trivial subgroup the Morita equivalence is between $\mathbb{C}$ and $C_0(G)\rtimes G$, with the equivalence bimodule being $L^2(G)$ (on which $C_0(G)\rtimes G$ acts faithfully as the full $C^*$-algebra of compact operators, per Mackey's generalisation of the Stone–von Neumann theorem).

A good place to learn about all of this, including the history and many related results, is Echterhoff's survey now published as Chapter 2 in Cuntz, Echterhoff, Li, and Yu - $K$-theory for group $C^*$-algebras and semigroup $C^*$-algebras, Zbl 1390.46001 (also available on the arXiv). See Theorem 2.6.4 in the published version, and Theorem 6.4 in the arXiv version.

Answer 2

Here is a proof that $C(X)\rtimes H$ is Morita equivalent to $C(X\times_H G)\rtimes G$. I'm assuming here that $G,H$ are countable discrete groups and $X$ is second countable. First of all, since I prefer to work with left actions, I will assume that $H$ acts on the left of $X$ and I will write instead $G\times_H X$ for $(G\times X)/H$ where the action of $H$ is on the right via $(g,x)h = (gh,h^{-1}x)$. I hope that is fine.

I will show that the corresponding transformation etale groupoids are Morita equivalent groupoids. See Sims - Hausdorff étale groupoids and their $C^*$ algebras section 3.4. It is a well-known result of Renault that Morita equivalent Hausdorff groupoids have strongly Morita equivalent $C^*$-algebras and the proof of Theorem 3.4.4 in the above reference tells you how to build the imprimitivity bimodule from the groupoid equivalence.

I will write $g\otimes x$ to denote the class of $(g,x)$ in $G\times_H X$ since this is like a tensor product in that $gh\otimes x=g\otimes hx$ for $h\in H$. Then $H\ltimes X$ is the groupoid with unit space $X$ and arrow space $H\times X$ where the arrow $(h,x)\colon x\to hx$. The product is $(h,h'x)(h',x)= (hh',x)$ and the topology is the product topology on $H\times X$ and the usual topology on $X$. The inverse is $(h,x)^{-1}=(h^{-1},hx)$. The groupoid $G\ltimes (G\times_H X)$ is defined similarly but has object space $G\times_H X$ and arrow space $G\times (G\times_H X)$.

To find a Morita equivalence I need a principal bibundle for these groupoids with the appropriate properties. The precise definition is on page 23 of the linked file.

Let $Z=G\times X$ with the product topology. We have open maps $p\colon Z\to G\times_H X$ and $q\colon Z\to X$ given by the quotient map in the first case and the projection in the second case. These can be use as anchors (or moment maps) for a left action of $G\ltimes (G\times_H X)$ and a right action of $H\ltimes X$ which commute. The left action of $G\ltimes (G\times H_X)$ is given by $(g_1,g_0\otimes x)(g_0,x) = (g_1g_0, x)$ and the right action of $H\ltimes X$ is given by $(g_0,x)(h,h^{-1}x) = (g_0h,h^{-1}x)$. It is easy to check that these are free and proper commuting actions. Also the quotient of $Z$ by the action of $G\ltimes (G\times_H X)$ is homeomorphic to $X$ via $q$ and the quotient of $Z$ by the action of $H\ltimes X$ is $G\times_H X$ by construction. Thus this bibundle gives a Morita equivalence of groupoids and hence a Morita equivalence of $C^*$-algebras.