2017/06/26 by Crespo, Jonathan
#FOS: Mathematics #K-Theory and Homology (math.KT) #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.1706.08516
In this article, we generalize to the case of measured quantum groupoids on a finite basis some important results concerning equivariant Kasparov theory for actions of locally compact quantum groups [S. Baaj and G. Skandalis, 1989, 1993]. To every pair (A,B) of C*-algebras continuously acted upon by a regular measured quantum groupoid on a finite basis \cal G, we associate a \cal G-equivariant Kasparov theory group \sf KK\cal G(A,B). The Kasparov product generalizes to this setting. By applying recent results concerning actions of regular measured quantum groupoids on a finite basis [S. Baaj and J. C., 2015; J. C., 2017], we obtain two canonical homomorphisms J\cal G:\sf KK\cal G(A,B)→\sf KK_\widehat\cal G(A\rtimes\cal G,B\rtimes\cal G) and J_\widehat\cal G:\sf KK_\widehat\cal G(A,B)→\sf KK\cal G(A\rtimes\widehat\cal G,B\rtimes\widehat\cal G) inverse of each other through the Morita equivalence coming from a version of the Takesaki-Takai duality theorem [S. Baaj and J. C., 2015; J. C., 2017]. We investigate in detail the case of colinking measured quantum groupoids. In particular, if \mathbbG1 and \mathbbG2 are two monoidally equivalent regular locally compact quantum groups, we obtain a new proof of the canonical equivalence of the associated equivariant Kasparov categories [S. Baaj and J. C., 2015].