Representations of Operator Algebras

Abstract

The following thesis is divided into two main chapters. In Chapter 2 we study isometric representations of product systems of correspondences over the semigroup 𝐍ᵏ which are minimal dilations of finite dimensional, fully coisometric representations. We show the existence of a unique minimal cyclic coinvariant subspace for all such representations. The compression of the representation to this subspace is shown to be a complete unitary invariant. For a certain class of graph algebras the nonself-adjoint WOT-closed algebra generated by these representations is shown to contain the projection onto the minimal cyclic coinvariant subspace. This class includes free semigroup algebras. This result extends to a class of higher-rank graph algebras which includes higher-rank graphs with a single vertex.

In chapter 3 we move onto semicrossed product algebras. Let 𝒮 be the semigroup 𝒮=Σ𝒮ᵢ, where 𝒮ᵢ is a countable subsemigroup of the additive semigroup 𝐑₊ containing 0. We consider representations of 𝒮 as contractions {Tᵣ }ᵣ on a Hilbert space with the Nica-covariance property: Tᵣ*Tᵤ=TᵤTᵣ* whenever t^s=0. We show that all such representations have a unique minimal isometric Nica-covariant dilation.

This result is used to help analyse the nonself-adjoint semicrossed product algebras formed from Nica-covariant representations of the action of 𝒮 on an operator algebra 𝒜 by completely contractive endomorphisms. We conclude by calculating the C*-envelope of the isometric nonself-adjoint semicrossed product algebra (in the sense of Kakariadis and Katsoulis).