Tensor algebras of subproduct systems and noncommutative function theory

TL;DR

This paper investigates tensor algebras of subproduct systems with infinite dimensional fibers, characterizing their structure, conditions for isomorphism, and the applicability of Nullstellensatz in noncommutative function theory.

Contribution

It extends the understanding of tensor algebras of subproduct systems to infinite dimensions, providing new characterizations and resolving the isomorphism problem.

Findings

Characterization of when tensor algebras are noncommutative function algebras

Identification of conditions for residual finite dimensionality

Demonstration of Nullstellensatz failure in infinite dimensional case

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.