Subproduct systems

By Orr Shalit, B. Solel. Documenta Mathematica, 2009

88 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/3e5f7f77fbcedf69fbd52dc87e9e20b641ee72be

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Citations in the corpus

The isomorphism problem for some universal operator algebras (2010, methodology): “Since in [36] a proof was not provided, we include one for the commutative case.”
Operator Theory and Function Theory in Drury–Arveson Space and Its Quotients (2013, unknown): “ractions satisfying the relations in Jby compressing the noncommutative d-shift Lto the co-invariant subspace FJ = [J]⊥. See [101] for details; similar results for special classes of ideals appear in [36, 110]. 6.7. Constrained dilations in the commutative case. The results of [101] discussed in the”
Dilation theory in finite dimensions: the possible, the impossible and the unknown (2010, background): “-Nagy’s Dilation Theorems is deep (see Section 5 in [26]).”
Algebras of bounded noncommutative analytic functions on subvarieties of the noncommutative unit ball (2017, methodology): “We obtain a functional calculus for multiplier algebras on nc varieties, versions of which were observed in [68,78].”