Dilation methods in semigroup dynamics and noncommutative convexity

Abstract

Since seminal work of Stinespring, Arveson, and others, dilation theory has been an indispensable tool for understanding operator algebras. Dilations are fundamental to the representation theory of operator systems and (non-selfadjoint) operator algebras. This thesis is a compilation of three research papers in operator algebras and noncommutative convexity linked by their use of dilations and operator systems.

A semicrossed product is a non-selfadjoint operator algebra encoding the action of a semigroup on an operator or C*-algebra. In Chapter 2, we describe the C*-envelopes of a large class of semicrossed products. We prove that, when the positive cone of a discrete lattice ordered abelian group acts on a C*-algebra, the C*-envelope of the associated semicrossed product is a full corner of a crossed product by the whole group. After dilating the semigroup action to an automorphic action of the whole group using a direct product construction, we explicitly compute the Shilov ideal and therefore compute the C*-envelope. This generalizes a result of Davidson, Fuller, and Kakariadis from Z_+^n to the class of all discrete lattice ordered abelian groups.

Chapters 3 and 4 present results in noncommutative (or ``matrix") convexity. By the noncommutative Kadison duality of Webster-Winkler and Davidson-Kennedy in the unital setting, and Kennedy-Kim-Manor in the nonunital setting, the category of compact noncommutative (nc) convex sets is dual to the category of operator systems. Thus nc convexity allows a new avenue to study operator systems geometrically. In Chapter 3, we prove a noncommutative generalization of the classical Jensen's Inequality for multivariable nc functions which are convex in each variable separately. The proof involves a sequence of dilations resembling a noncommutative analogue of Fubini's Theorem. This extends a single-variable nc version of Jensen's Inequality of Davidson and Kennedy. We demonstrate an application of the multivariable separate nc Jensen's Inequality to free semicircular systems in free probability.

In Chapter 4, we discuss duals of operator systems. Recently, C.K. Ng obtained a nice duality theory for operator systems. Call a (possibly nonunital) operator system S dualizable if its dual S* embeds into B(H) via a complete order embedding and complete norm equivalence. Through the nonunital noncommutative Kadison duality of Kennedy, Kim, and Manor, we characterize dualizability of S using geometric conditions on its associated nc convex quasistate space K in two ways. Firstly, in terms of an nc affine embedding of K into the nc unit ball of a Hilbert space satisfying a certain extension property. Secondly, we show that Ng's characterization is dual to a normality condition between K and the nc cone R_+ K. As applications, we obtain some permanence properties for dualizability, and give a new nc convex-geometric proof of Choi's Theorem.