Operator theory on noncommutative varieties

By Gelu Popescu. 2005

106 citations recorded by the source corpus.

We develop a dilation theory on noncommutative varieties determined by row contractions T: = [T 1 ,..., T n ] subject to constraints such as p(T 1 ,...,T n ) = 0, p ∈ P, where T is a set of noncommutative polynomials. The model n-tuple is the universal row contraction [B 1 ,...,B n ] satisfying the same constraints as T, which turns out to be, in a certain sense, the maximal constrained piece of the n-tuple [S 1 ,..., S n ] of left creation operators on the full Fock space on n generators. The theory is based on a class of constrained Poisson kernels associated with T and representations of the C*-algebra generated by B 1 ,...,B n and the identity. Under natural conditions on the constraints we have uniqueness for the minimal dilation. A characteristic function Θ T is associated with any (constrained) row contraction T and it is proved that I - Θ T Θ* T = K T K* T , where K T is the (constrained) Poisson kernel of T. Consequently, for pure constrained row contractions, we show that the characteristic function is a complete unitary invariant and provide a model. We show that the curvature invariant and Euler characteristic asssociated with a Hilbert module generated by an arbitrary (resp. commuting) row contraction T can be expressed only in terms of the (resp. constrained) characteristic function of T. We provide a commutant lifting theorem for pure constrained row contractions and obtain a Nevanlinna-Pick interpolation result in our setting.

Subfields: Algebra and Number Theory, Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/281a311458aba20829b8d34de4d5f49c3b4d05e5

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In Algebra and Number Theory

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Citations in the corpus

The isomorphism problem for some universal operator algebras (2010, background): “years that AI can be naturally identified with the compression of Ad to the coinvariant subspace FI := F(E) [ĨF(E)] (see, in increasing order of generality, [4,8,36], and [17,30]).”
Free holomorphic automorphisms of the unit ball of B(ℋ) n (2008, methodology): “We showed in [28] that the curvature and the Euler characteristic of an arbitrary row contraction can be expressed only in terms of the standard characteristic function.”
Operator algebras for analytic varieties (2012, methodology): “Therefore, by (a slight modification of) Popescu’s results [25], AV is the universal operator algebra generated by a commuting row contraction subject to the relations in IV = JV ∩ Ad.”
Operator Theory on Noncommutative Domains (2007, unknown): “, Tn) ∈ Dp(H) is a commuting n-tuple of operators, we obtain the characteristic function introduced in [84] and [24], and further studied in [85], [25] and [20].”
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].”
Free holomorphic functions on the unit ball of B(H)n (2006, unknown): “We should mention that this operator plays an important role in noncommutative multivariable operator theory (see [39], [40]).”
Subproduct systems (2009, unknown): “Using this tool we derive some basic results (obtained previously by Popescu in [38]) which allow us to identify the operator algebra AX generated by the X-shift as the universal unital operator algebra generated by a row contraction subject to homogeneous polynomial identities.”
Free holomorphic functions and interpolation (2007, background): “In the last two decades, significant progress has been made in noncommutative multivariable operator theory regarding noncommutative dilation theory and its applications to interpolation in several variables ([24], [28], [40], [38], [30], [4], [36]), and unitary invariants for n-tuples of operators ”
Operator Theory and Function Theory in Drury–Arveson Space and Its Quotients (2013, background): “The results of [100] discussed in the previous paragraph can be compressed to the commutative case, yielding the following model theory for d-contraction satisfying polynomial relations.”