Subproduct systems over \(\mathbb N \times \mathbb N\)
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 with a unit ball. He distinguishes subproduct systems having an inclusion-reversing bijection with ideals in algebras of non-commutative polynomials.
scientific article; zbMATH DE number 6044496
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