Equivariant KK-theory for semimultiplicative sets (Q1043784)

A semimultiplicative set is a set \(G\) equipped with a partially defined associative multiplication. In this paper, the author first establishes the notion of the left regular representation of a countable discrete semimultiplicative set and defines crossed product \(C^*\)-algebras. He then changes course and defines a equivariant Kasparov theory for semimultiplicative sets that are countable and discrete; for most of the paper, he does not even require that the multiplication be associative. Given a countable discrete semimultiplicative set \(G\) and \(G\)-\(C^*\)-algebras \(A\) and \(B\), the author takes the standard construction of \(KK^G\) when \(G\) is a group and mimics it to form \(KK^G(A, B)\) in this much more general context. Surprisingly, even with such weak conditions on \(G\), the author is able to show that \(KK^G(A, B)\) is an abelian group and that \(KK^G\) admits an associative Kasparov product. However, such generality comes at a price: for \(KK^G(A, B)\) to be functorially well-behaved, the author has to restrict to the cases where \(A\) and \(B\) are Hilbert \(C^*\)-algebras; a Hilbert \(C^*\)-algebra is a \(G\)-\(C^*\)-algebra that is also a \(G\)-Hilbert module over \(A\) with inner product \(\langle x, y\rangle = x^*y\) and action \(U_g(x) = g(x)\) for all \(x, y\) in \(A\) and \(g\) in \(G\).