$C^*$-Crossed-Products by an Order-Two Automorphism

http://dx.doi.org/10.4153/CMB-2010-008-4
Canad. Math. Bull. 53(2010), 37-50

  Published:2009-12-04
 Printed: Mar 2010

We describe the representation theory of $C^*$-crossed-products of a unital $C^*$-algebra A by the cyclic group of order~2. We prove that there are two main types of irreducible representations for the crossed-product: those whose restriction to A is irreducible and those who are the sum of two unitarily unequivalent representations of~A. We characterize each class in term of the restriction of the representations to the fixed point $C^*$-subalgebra of~A. We apply our results to compute the K-theory of several crossed-products of the free group on two generators.