## 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) 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.

Original language | English |
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Pages (from-to) | 801-868 |

Number of pages | 68 |

Journal | Documenta Mathematica |

Volume | 14 |

DOIs | |

State | Published - 1 Jan 2009 |

Externally published | Yes |

## Keywords

- Product system
- dilation
- e-dilation
- homogeneous polynomial identities
- q-commuting
- row contraction
- semigroups of completely positive maps
- subproduct system
- subshift C algebra
- universal operator algebra
- ∗automorphic dilation

## ASJC Scopus subject areas

- General Mathematics