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

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 $$\tilde A \rtimes G$$?

• I think the answer to your first question is yes but I'm not a C*-algebraist.The action of $H$ on $X$ extends to a partial action of $G$ on $X$ in the sense of Exel by leaving the action of the other elements of $G$ undefined.I believe that $C(X)\rtimes H$ is the same as the partial action crossed product of $G$ with $C(X)$ coming from this partial action. The enveloping action or globalization of this partial action is $X\times_H G$.It's a result of Abadie that if the enveloping action is Hausdorff then there is Morita equivalence of the partial crossed product with the full crossed product. Jul 28, 2022 at 10:47
• So this should give the Morita equivalence you seek. Jul 28, 2022 at 10:48
• Actually I think it is not too bad to give a morita equivalence of the associated transformation groupoids which via Renault will give your Morita equivalence Jul 28, 2022 at 11:45
• I think you can use $X\times G$ as a bibundle Jul 28, 2022 at 11:53

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.

• This is a wonderful answer, thank you! Aug 4, 2022 at 6:57

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.