## Abstract

We revisit tensor algebras of subproduct systems with Hilbert space fibers, resolving some open questions in the case of infinite dimensional fibers.We characterizewhen 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 showthat-in contrast to the finite dimensional case-in the case of infinite dimensional fibers thisNullstellensatz 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.

Original language | English |
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Journal | Canadian Journal of Mathematics |

DOIs | |

State | Accepted/In press - 1 Jan 2023 |

Externally published | Yes |

## Keywords

- noncommutative function theory
- nonselfadjoint operator algebras
- Subproduct systems

## ASJC Scopus subject areas

- General Mathematics