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Matthew Daws
Matthew Daws
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Generalisation of the equivalence between $C*$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$C^*$-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*$C^*$-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

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

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 $\tilde A \rtimes G$?

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Motmot
Motmot
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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 $\tilde A \rtimes G$?