Journal of Operator Theory

Volume 42, Issue 1, Summer 1999 pp. 103-119.

Correspondence of groupoid

$C^*$ -algebras

Authors: Marta Macho Stadler

$1$ , and Moto O'uchi

$2$
Author institution:

$1$ Departamento de Matematicas, Facultad de Ciencias, Universidad del Pais Vasco, Euskal Herriko Unibertsitatea, Apartado 644--48080 Bilbao, Spain

$2$ Department of Applied Mathematics, Osaka Women's University, Sakai City, Osaka 590-0035, Japan

Summary: Let

$G_1$ and

$G_2$ be topological groupoids. We introduce a notion of correspondence from

$G_1$ to

$G_2$ . We show that there exists a correspondence from

$C_{\rm r}^*(G_2)$ to

$C_{\rm r}^*(G_1)$ if there exists a correspondence from

$G_1$ to

$G_2$ . Let

$f$ be a homomorphism of

$G_1$ onto

$G_2$ . We show that there is a correspondence from

$G_1$ to

$G_2$ if

$f$ satisfies certain conditions. Moreover we show that it gives an element of

$\K(C_{\rm r}^*(G_2),C_{\rm r}^*(G_1))$ if

$f$ satisfies an additional condition . We study three examples where groupoids are topological spaces, topological groups and transformation groups respectively.

Keywords: Groupoid,

$C^*$ -algebra, correspondence, Hilbert module, Kasparov module,

$\rm KK$ -group

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