Subproduct systems over ℕ×ℕ

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We develop the theory of subproduct systems over the monoid ℕ×ℕ, and the non-self-adjoint operator algebras associated with them. These are double sequences of Hilbert spaces {X(m,n)} m,n=0 equipped with a multiplication given by coisometries from X(i, j) ⊗. X(k, l) to X(i+. k, j+. l). We find that the character space of the norm-closed algebra generated by left multiplication operators (the tensor algebra) is homeomorphic to a complex homogeneous affine algebraic variety intersected with a unit ball. Certain conditions are isolated under which subproduct systems whose tensor algebras are isomorphic must be isomorphic themselves. In the absence of these conditions, we show that two numerical invariants must agree on such subproduct systems. Additionally, we classify the subproduct systems over ℕ×ℕ by means of ideals in algebras of non-commutative polynomials.

Original languageEnglish
Pages (from-to)4270-4301
Number of pages32
JournalJournal of Functional Analysis
Issue number10
StatePublished - 15 May 2012


  • Character space
  • Homogeneous non-commutative polynomials
  • Non-self-adjoint operator algebras
  • Subproduct systems

All Science Journal Classification (ASJC) codes

  • Analysis


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