# An operator algebraic proof of Agler's factorization theorem

01 Nov 2009-Vol. 137, Iss: 11, pp 3741-3748

TL;DR: In this paper, a short direct proof of Agler's factorization theorem using the Blecher-Ruan-Sinclair characterization of operator algebras is given. But this proof is restricted to polynomials.

Abstract: We give a short direct proof of Agler's factorization theorem that uses the Blecher-Ruan-Sinclair characterization of operator algebras. The key ingredient of this proof is an operator algebra factorization theorem. Our proof provides some additional information about these factorizations in the case of polynomials.

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Abstract: The Schur–Agler class consists of functions over a domain satisfying an appropriate von Neumann inequality. Originally defined over the polydisk, the idea has been extended to general domains in multivariable complex Euclidean space with matrix polynomial defining function as well as to certain multivariable noncommutative-operator domains with a noncommutative linear-pencil defining function. Still more recently there has emerged a free noncommutative function theory (functions of noncommuting matrix variables respecting direct sums and similarity transformations). The purpose of the present paper is to extend the Schur–Agler-class theory to the free noncommutative function setting. This includes the positive-kerneldecomposition characterization of the class, transfer-function realization and Pick interpolation theory. A special class of defining functions is identified for which the associated Schur–Agler class coincides with the contractivemultiplier class on an associated noncommutative reproducing kernel Hilbert space; in this case, solution of the Pick interpolation problem is in terms of the complete positivity of an associated Pick matrix which is explicitly determined from the interpolation data.

50 citations

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TL;DR: In this article, the authors present general theorems about operator algebras, i.e., functions on sets that can be represented as scalar multipliers of a reproducing kernel Hilbert space.

27 citations

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TL;DR: In this paper, the authors studied representations of positive definite kernels in a general setting, but with view to applications to harmonic analysis, to metric geometry, and to realizations of certain stochastic processes.

Abstract: We study representations of positive definite kernels K in a general setting, but with view to applications to harmonic analysis, to metric geometry, and to realizations of certain stochastic processes. Our initial results are stated for the most general given positive definite kernel, but are then subsequently specialized to the above mentioned applications. Given a positive definite kernel K on $$S\times S$$
where S is a fixed set, we first study families of factorizations of K. By a factorization (or representation) we mean a probability space $$\left( B,\mu \right) $$
and an associated stochastic process indexed by S which has K as its covariance kernel. For each realization we identify a co-isometric transform from $$L^{2}\left( \mu \right) $$
onto $$\mathscr {H}\left( K\right) $$
, where $$\mathscr {H}\left( K\right) $$
denotes the reproducing kernel Hilbert space of K. In some cases, this entails a certain renormalization of K. Our emphasis is on such realizations which are minimal in a sense we make precise. By minimal we mean roughly that B may be realized as a certain K-boundary of the given set S. We prove existence of minimal realizations in a general setting.

12 citations

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TL;DR: In this paper, the authors studied a family of positive definite kernels with respect to their role as covariance kernels of a variety of stochastic processes, and gave necessary and sufficient conditions for K to have realizations and factorizations in a fixed sigma-finite measure space.

Abstract: Given a fixed sigma-finite measure space $$\left( X,\mathscr {B},
u \right) $$
, we shall study an associated family of positive definite kernels K. Their factorizations will be studied with view to their role as covariance kernels of a variety of stochastic processes. In the interesting cases, the given measure $$
u $$
is infinite, but sigma-finite. We introduce such positive definite kernels $$K\left( \cdot ,\cdot \right) $$
with the two variables from the subclass of the sigma-algebra $$\mathscr {B}$$
whose elements are sets with finite $$
u $$
measure. Our setting and results are motivated by applications. The latter are covered in the second half of the paper. We first make precise the notions of realizations and factorizations for K, and we give necessary and sufficient conditions for K to have realizations and factorizations in $$L^{2}\left(
u \right) $$
. Tools in the proofs rely on probability theory and on spectral theory for unbounded operators in Hilbert space. Applications discussed here include the study of reversible Markov processes, and realizations of Gaussian fields, and their Ito-integrals.

10 citations

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TL;DR: In this paper, a refined Agler decomposition for bounded analytic functions on the bidisk is presented, which can be used to reprove an interesting result of Guo et al. related to extending holomorphic functions without increasing their norm.

Abstract: We prove a refined Agler decomposition for bounded analytic functions on the bidisk and show how it can be used to reprove an interesting result of Guo et al. related to extending holomorphic functions without increasing their norm. In addition, we give a new treatment of Heath and Suffridge's characterization of holomorphic retracts on the polydisk.

7 citations

##### References

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21 Apr 2003TL;DR: In this article, the reader is provided with a tour of the principal results and ideas in the theories of completely positive maps, completely bounded maps, dilation theory, operator spaces and operator algebras, together with some of their main applications.

Abstract: In this book, first published in 2003, the reader is provided with a tour of the principal results and ideas in the theories of completely positive maps, completely bounded maps, dilation theory, operator spaces and operator algebras, together with some of their main applications. The author assumes only that the reader has a basic background in functional analysis, and the presentation is self-contained and paced appropriately for graduate students new to the subject. Experts will also want this book for their library since the author illustrates the power of methods he has developed with new and simpler proofs of some of the major results in the area, many of which have not appeared earlier in the literature. An indispensable introduction to the theory of operator spaces for all who want to know more.

1,530 citations

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01 Mar 2002TL;DR: In this article, the universal Pick kernel with the complete Pick redux was defined and the spectral theorem for normal $m$-tuples was shown to be equivalent to the result of the Pick problem.

Abstract: Prerequisites and notation Introduction Kernels and function spaces Hardy spaces $P^2(\mu)$ Pick redux Qualitative properties of the solution of the Pick problem in $H^\infty(\mathbb{D})$ Characterizing kernels with the complete Pick property The universal Pick kernel Interpolating sequences Model theory I: Isometries The bidisk The extremal three point problem on $\mathbb{D}^2$ Collections of kernels Model theory II: Function spaces Localization Schur products Parrott's lemma Riesz interpolation The spectral theorem for normal $m$-tuples Bibliography Index.

597 citations

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TL;DR: In this paper, the authors give a characterization of unital operator algebras in terms of their matricial norm structure, and show that the quotient of an operator algebra by a closed two-sided ideal is again an unital algebra up to complete isometric isomorphism.

199 citations