Subproduct systems

TL;DR

This paper introduces subproduct systems, establishing a correspondence with cp-semigroups, and develops a dilation theory that addresses open problems in operator algebras and noncommutative geometry.

Contribution

It defines subproduct systems and demonstrates their role in characterizing cp-semigroups and dilations, solving longstanding open problems in the field.

Findings

Established a one-to-one correspondence between cp-semigroups and subproduct system pairs.

Developed a dilation theory including automorphic dilations and conditions for endomorphic dilations.

Provided examples and classifications of operator algebras related to subproduct systems.

Abstract

The notion of a subproduct system, a generalization of that of a product system, is introduced. We show that there is an essentially 1 to 1 correspondence between cp-semigroups and pairs (X,T) where X is a subproduct system and T is an injective subproduct system representation. A similar statement holds for subproduct systems and units of subproduct systems. This correspondence is used as a framework for developing a dilation theory for cp-semigroups. Results we obtain: (i) a *-automorphic dilation to semigroups of *-endomorphisms over quite general semigroups; (ii) necessary and sufficient conditions for a semigroup of CP maps to have a *-endomorphic dilation; (iii) an analogue of Parrot's example of three contractions with no isometric dilation, that is, an example of three commuting, contractive normal CP maps on B(H) that admit no *-endomorphic dilation (thereby solving an open problem raised by Bhat in 1998). Special attention is given to subproduct systems over the semigroup N, which are used as a framework for studying tuples of operators satisfying homogeneous polynomial relations, and the operator algebras they generate. As applications we obtain a noncommutative (projective) Nullstellansatz, a model for tuples of operators subject to homogeneous polynomial relations, a complete description of all representations of Matsumoto's subshift C*-algebra when the subshift is of finite type, and a classification of certain operator algebras -- including an interesting non-selfadjoint generalization of the noncommutative tori.