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 $\tilde A \rtimes G$?



Generalisation of the equivalence between $C^*(H)$ and $C_0(G/H) \rtimes G$; induction of group actions on C*-algebras
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](https://zbmath.org/?q=an%3A0407.46053)) 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](https://zbmath.org/?q=an%3A1390.46001) (also available on the [arXiv](https://arxiv.org/pdf/1006.4975.pdf)). See Theorem 2.6.4 in the published version, and Theorem 6.4 in the arXiv version.


