# 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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