Some Results on Semicrossed Products and Related Operator Algebras

Abstract

We investigate various properties of two classes of operator algebras: directed graph operator algebras and semicrossed products. First we consider analytic structure in the form of derivations and point derivations on these algebras. Our two main results describe the structure of derivations on graph operator algebras and point derivations on semicrossed product operator algebras. We then investigate multivariate semicrossed products and the maps on the associated, underlying compact Hausdorff space. We consider potential generalizations of classical 1-dimensional variants and look for which of our multivariate analogs have nice structure with a proposed invariant for multivariate dynamical systems. We close by developing a component-wise look at the maximal C∗-algebra of the n×n matrices, the simplest of the direct graph operator algebras. This is the first concrete example of a maximal C∗-algebra since the one example that accompanied the definition in the original paper about maximal C∗-envelopes.