# 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