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Tensor algebras of subproduct systems and noncommutative function theory

Michael Hartz, Orr Shalit

TL;DR

The paper addresses the problem of classifying tensor algebras of subproduct systems, including those with infinite-dimensional fibers, and identifies when these algebras realize as algebras of uniformly continuous noncommutative functions on nc homogeneous varieties. It develops a noncommutative function theory framework, relating the tensor algebra $\mathcal{A}_X$ to the quotient $\mathfrak{A}_d/J_X$ and to nc varieties $V(J)$, and analyzes finite-dimensional representations to connect to residual finite dimensionality and a Nullstellensatz. A key result is that bounded (or cb) isomorphisms of tensor algebras correspond to similarities of the underlying subproduct systems, while isometric (or completely isometric) isomorphisms correspond to actual isomorphisms of the subproduct systems; the so-called disc trick is used to promote to vacuum-preserving maps. The paper also shows that in the infinite-variable setting the Nullstellensatz can fail, demonstrating a richer class of operator algebras than previously thought, and provides a complete classification framework for the isomorphism problem via subproduct systems, including the role of finite-dimensional representations and nc function theory. Overall, the work extends finite-dimensional results to infinite-dimensional fibers, linking operator algebraic structure with nc analytic geometry and functional calculus.

Abstract

We revisit tensor algebras of subproduct systems with Hilbert space fibers, resolving some open questions in the case of infinite dimensional fibers. We characterize when a tensor algebra can be identified as the algebra of uniformly continuous noncommutative functions on a noncommutative homogeneous variety or, equivalently, when it is residually finite dimensional: this happens precisely when the closed homogeneous ideal associated to the subproduct system satisfies a Nullstellensatz with respect to the algebra of uniformly continuous noncommutative functions on the noncommutative closed unit ball. We show that - in contrast to the finite dimensional case - in the case of infinite dimensional fibers this Nullstellensatz may fail. Finally, we also resolve the isomorphism problem for tensor algebras of subproduct systems: two such tensor algebras are (isometrically) isomorphic if and only if their subproduct systems are isomorphic in an appropriate sense.

Tensor algebras of subproduct systems and noncommutative function theory

TL;DR

The paper addresses the problem of classifying tensor algebras of subproduct systems, including those with infinite-dimensional fibers, and identifies when these algebras realize as algebras of uniformly continuous noncommutative functions on nc homogeneous varieties. It develops a noncommutative function theory framework, relating the tensor algebra to the quotient and to nc varieties , and analyzes finite-dimensional representations to connect to residual finite dimensionality and a Nullstellensatz. A key result is that bounded (or cb) isomorphisms of tensor algebras correspond to similarities of the underlying subproduct systems, while isometric (or completely isometric) isomorphisms correspond to actual isomorphisms of the subproduct systems; the so-called disc trick is used to promote to vacuum-preserving maps. The paper also shows that in the infinite-variable setting the Nullstellensatz can fail, demonstrating a richer class of operator algebras than previously thought, and provides a complete classification framework for the isomorphism problem via subproduct systems, including the role of finite-dimensional representations and nc function theory. Overall, the work extends finite-dimensional results to infinite-dimensional fibers, linking operator algebraic structure with nc analytic geometry and functional calculus.

Abstract

We revisit tensor algebras of subproduct systems with Hilbert space fibers, resolving some open questions in the case of infinite dimensional fibers. We characterize when a tensor algebra can be identified as the algebra of uniformly continuous noncommutative functions on a noncommutative homogeneous variety or, equivalently, when it is residually finite dimensional: this happens precisely when the closed homogeneous ideal associated to the subproduct system satisfies a Nullstellensatz with respect to the algebra of uniformly continuous noncommutative functions on the noncommutative closed unit ball. We show that - in contrast to the finite dimensional case - in the case of infinite dimensional fibers this Nullstellensatz may fail. Finally, we also resolve the isomorphism problem for tensor algebras of subproduct systems: two such tensor algebras are (isometrically) isomorphic if and only if their subproduct systems are isomorphic in an appropriate sense.
Paper Structure (17 sections, 14 theorems, 53 equations)

This paper contains 17 sections, 14 theorems, 53 equations.

Table of Contents

  1. Introduction and background
  2. Subproduct systems and their tensor algebras
  3. The goals of this paper, briefly
  4. The isomorphism problem for tensor algebras
  5. The main results
  6. Noncommutative functions and subproduct systems
  7. Noncommutative functions and the noncommutative unit ball
  8. Homogeneous ideals and varieties
  9. Subproduct systems and closed homgeneous ideals
  10. Representations of tensor algebras
  11. Universality of tensor algebras
  12. Finite dimensional representability of tensor algebras
  13. Failure of the Nullstellensatz
  14. Classification by subproduct systems
  15. Reduction of the isomorphism problem to existence of graded maps
  16. ...and 2 more sections

Key Result

Proposition 1.3

For every $n \in \mathbb{N}$, there is a completely contractive surjective idempotent $\Phi_n : \mathcal{A}_X \to \mathcal{A}_X^{(n)}$. Every $T \in \mathcal{A}_X$ can be written uniquely as a Cesàro norm-convergent sum $T = \sum_{n=0}^\infty T_n$ where $T_n = \Phi_n(T) \in \mathcal{A}_X^{(n)}$ is $

Theorems & Definitions (44)

  • Definition 1.1
  • Definition 1.2
  • Proposition 1.3: Proposition 9.3, ShaSol09; Proposition 6.2 DorMar14
  • Example 1.4
  • Example 1.5
  • Definition 1.6
  • Theorem 1.8
  • Remark 2.1
  • Definition 2.2
  • Definition 2.3
  • ...and 34 more