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# 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