Tensor algebras of subproduct systems and noncommutative function theory

Abstract

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 with 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.