Subproduct Systems

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) Nullstellensatz, 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.

2000 Mathematics Subject Classification: 46L55, 46L57, 46L08, 47L30.

Keywords and Phrases: Product system, subproduct system, semigroups of completely positive maps, dilation, $e_0$-dilation, $*$-automorphic dilation, row contraction, homogeneous polynomial identities, universal operator algebra, $q$-commuting, subshift C$^*$-algebra.

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