The ring structure for equivariant twisted K-theory

We prove, under some mild conditions, that the equivariant twisted K-theory group of a crossed module admits a ring structure if the twisting 2-cocycle is 2-multiplicative. We also give an explicit construction of the transgression map for any crossed module N → and prove that any element in the image is ∞-multiplicative. As a consequence, we prove, under some mild conditions, for a crossed module N → and any , that the equivariant twisted K-theory group admits a ring structure. As an application, we prove that for a compact, connected and simply connected Lie group G, the equivariant twisted K-theory group , defined as the K-theory group of a certain groupoid C*-algebra, is endowed with a canonical ring structure , where . The relation with Freed-Hopkins-Teleman theorem Loop groups and twisted K-theory, III, math.AT/0312155 still needs to be explored.