# 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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