Non-unital semigroup crossed products (Q2732524)

An Ore semigroup is a semigroup \(S\) which can be embedded into a group \(G\) such that \(G=S^{-1}S\). The author studies crossed products for proper \(C^*\)-dynamical systems \((A,S,\alpha)\). Among the results there are the following. If \(I\) is an ideal in \(A\) (suitably compatible with \(\alpha\)), then there is a short exact sequence for the crossed products for \((I,S,\alpha\lceil I)\) and \((A,S,\alpha)\). The crossed product of the maximal tensor product of two dynamical systems \((A,S,\alpha)\) and \((B,T,\beta)\) is (under a mild technical condition) the maximal tensor product of the crossed products. The author proves an existence result for a demposition series and, finally, applies it to the commutative case.