Admissibility of C*-Covers for Operator Algebra Dynamical Systems

Abstract

We characterize when a C*-cover admits a C*-dynamical extension of dynamics on an operator algebra in terms of the boundary ideal structure for the operator algebra in its maximal representation and show that the C*-covers that admit such an extension form a complete lattice. We study dynamical systems arising from groups acting via inner automorphisms in a C*-cover and produce an example of a C*-cover that admits no extension of dynamics on a finite-dimensional non-self-adjoint operator algebra. We construct a partial action on a class of C*-covers that recovers the crossed product of an operator algebra as a subalgebra of the partial crossed product, even when the C*-cover admits no dynamical extension.