Operator theory on noncommutative varieties

Details Operator theory on noncommutative varieties Operator theory on noncommutative varieties

2005-07-07

arxiv.org/abs/math/0507158v2

Indiana Univ. Math. J. 55 (2) (2006), 389--442

Mathematics

Operator Algebras

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Scientific paper

We develop a dilation theory for row contractions subject to constraints determined by sets of noncommutative polynomials. Under natural conditions on the constraints, we have uniqueness for the minimal dilation. A characteristic function is associated with any (constrained) row contraction. For pure constrained row contraction, we show that the characteristic function is a complete unitary invariant and provide a model. We also show that the curvature invariant and Euler characteristic asssociated with a Hilbert module generated by an arbitrary (resp. commuting) row contraction can be expressed only in terms of the corresponding (resp. constrained) characteristic function. We provide a commutant lifting theorem for pure constrained row contractions and obtain a Nevanlinna-Pick interpolation result in this setting. Our theory includes, in particular, the commutative and q-commutative cases, while the standard noncommutative dilation theory for row contractions serves as a "universal model".

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