Journal of Operator Theory

Volume 80, Issue 2, Fall 2018 pp. 295-348.

Graded $C^*$ -algebras, graded $K$ -theory, and twisted $P$ -graph $C^*$ -algebras

Authors: Alex Kumjian (1), David Pask (2), and Aidan Sims (3)
Author institution: (1) Department of Mathematics (084), Univ. of Nevada, Reno NV 89557-0084, U.S.A.
(2) School of Mathematics and Applied Statistics, University of Wollongong, NSW 2522, AUSTRALIA
(3) School of Mathematics and Applied Statistics, University of Wollongong, NSW 2522, AUSTRALIA

Summary: We develop methods for computing graded $K$ -theory of $C^*$ -algeb\-ras as defined in terms of Kasparov theory. We establish graded versions of Pimsner's six-term exact sequences for graded Hilbert bimodules whose left action is injective and by compacts, and a graded Pimsner--Voiculescu sequence. We introduce the notion of a twisted $P$ -graph $C^*$ -algebra and establish connections with graded $C^*$ -algebras. Specifically, we show how a functor from a $P$ -graph into the group of order two determines a grading of the associated $C^*$ -algebra. We apply our graded version of Pimsner's exact sequence to compute the graded $K$ -theory of a graph $C^*$ -algebra carrying such a grading.

DOI: http://dx.doi.org/10.7900/jot.2017sep28.2192
Keywords: $KK$ -theory, graded $K$ -theory, $C^*$ -algebra, $P$ -graph, twisted $C^*$ -algebra, graded $C^*$ -algebra

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