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In 2015, E. Katsoulis and C. Ramsey introduced the construction of a non-self-adjoint crossed product that encodes the action of a group of automorphisms on an operator algebra. They did so by realizing a non-self-adjoint crossed product as the subalgebra of a C*-crossed product when dynamics of a group acting on an operator algebra by completely isometric automorphisms can be extended to self-adjoint dynamics of the group acting on a C*-algebra by ∗-automorphisms. We show that this extension of dynamics is highly dependent on the representation of the given algebra and we define a lattice structure for an operator algebra's completely isometric representation theory. We characterize when a self-adjoint extension of dynamics exists in terms of the boundary ideal structure for the given operator algebra in its maximal representation. We use this characterization to produce the first example of dynamics on a finite dimensional non-self-adjoint operator algebra that are not extendable in a given representation and the first examples of always extendable dynamics for a family of operator algebras in a non-extremal representation. We give a partial crossed product construction to extend dynamics on a family of operator algebras, even when the operator algebra is represented degenerately. We connect W. Arveson's crossed product with that of E. Katsoulis and C. Ramsey by giving a partial answer to a recent open problem.
Adviser: Allan Donsig