Subproduct systems over \(\mathbb N \times \mathbb N\)
type: Item
instance of: scholarly article
title: Subproduct systems over \(\mathbb N \times \mathbb N\)
zbMATH Open document ID: 1266.47103
published in: Journal of Functional Analysis
publication date: 2012-06-08T00:00:00Z
full work available at URL: https://arxiv.org/abs/1109.1393
reviewed by: E. Julien Atto
Mathematics Subject Classification ID: 47L30
Mathematics Subject Classification ID: 47L75
Mathematics Subject Classification ID: 47L22
zbMATH DE Number: 6044496
zbMATH Keywords: non-self-adjoint operator algebras
zbMATH Keywords: subproduct systems
zbMATH Keywords: Fourier coefficients
zbMATH Keywords: character space
zbMATH Keywords: homogeneous non-commutative polynomials
MaRDI profile type: Publication
OpenAlex ID: W1599516262
arXiv ID: 1109.1393
cites work: Noncommutative interpolation and Poisson transforms
cites work: INCLUSION SYSTEMS AND AMALGAMATED PRODUCTS OF PRODUCT SYSTEMS
cites work: Nevanlinna-Pick interpolation for non-commutative analytic Toeplitz algebras
cites work: Dilation Theory for Rank 2 Graph Algebras
cites work: The isomorphism problem for some universal operator algebras
cites work: Q3963718
cites work: The Analytic Algebras Of Higher Rank Graphs
cites work: Higher rank graph \(C^*\)-algebras
cites work: Operator algebras associated with unitary commutation relations
cites work: Subproduct systems
cites work: Covariant representations of subproduct systems
DOI: 10.1016/J.JFA.2012.02.017
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author: Maxim Gurevich
review text: A subproduct system (of finite dimensional Hilbert spaces) over an additive cancellative monoid \(\mathcal{S}\) is a familly \(\{ X(s)\}_{s \in \mathcal{S}}\) of finite dimensional Hilbert spaces equipped with a multiplication given by coisometries NEWLINE\[NEWLINE\{ U_{s,t}: X(s)\otimes X(t)\rightarrow X(s+t) \}_{s,t \in \mathcal{S}}NEWLINE\]NEWLINE such that \(\dim X(0)=1\), with additional properties. The norm-closed algebra \(\mathcal{A}_{X} \subset B(\mathcal{F}_{X})\) generated by \(\{ L^{(s)}_{\eta}\}_{\eta\in X(s), s \in \mathcal{S}}\) is called the tensor algebra of \(X\), where \(\mathcal{F}_{X}:= \bigoplus_{s\in \mathcal{S}}X(s)\) is called the \(X\)-Fock space and NEWLINE\[NEWLINE L^{(s)}_{\eta|X(t)}:=L^{s,t}_{\eta}:X(t)\rightarrow X(s+t), \quad \xi\mapsto U_{s,t}(\eta\otimes\xi).NEWLINE\]NEWLINENEWLINENEWLINEThe author develops the theory of subproduct systems when \(\mathcal{S}=\mathbb{N}\times\mathbb{N}\) as an extension of the works made by some authors such as [\textit{O. Shalit} and \textit{B. Solel}, ``Subproduct systems'', Doc. Math., J. DMV 14, 801--868 (2009; Zbl 1189.46056)] and [\textit{K. R. Davidson} et al., ``The isomorphism problem for some universal operator algebras'', Adv. Math. 228, No. 1, 167--218 (2011; Zbl 1228.47069)], in the particular case \(\mathcal{S}=\mathbb{N}\).NEWLINENEWLINEOne of the two formulas given by the author defining the Fourier coefficients of an operator in a tensor algebra \(\mathcal{A}_{X}\) of a subproduct system \(X\) over \(\mathbb{N}^{2}\) is NEWLINE\[NEWLINE \tilde{\Phi}(T):=\dfrac{1}{2\pi}\int^{2\pi}_{0}e^{-ikt}U_{t,t}TU^{*}_{t,t}\, dt. NEWLINE\]NEWLINENEWLINENEWLINEAmong the main results, he shows some conditions to have an isometric isomorphism between two subproduct systems when their tensor algebras are isomorphic. He proves the existence of a homeomorphism between the character space of \(\mathcal{A}_{X}\), denoted by \(\mathcal{M}(\mathcal{A}_{X})\), and a complex homogeneous variety intersected wi…
description: scientific article; zbMATH DE number 6044496