Geodesic
Tensor algebras of subproduct systems and noncommutative function theory
Canadian journal of mathematics, Tome 76 (2024) no. 5, pp. 1587-1608

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DOI

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.
DOI : 10.4153/S0008414X2300055X
Mots-clés : Subproduct systems, non-self-adjoint operator algebras, noncommutative function theory 
Hartz, Michael; Shalit, Orr Moshe. Tensor algebras of subproduct systems and noncommutative function theory. Canadian journal of mathematics, Tome 76 (2024) no. 5, pp. 1587-1608. doi: 10.4153/S0008414X2300055X
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     title = {Tensor algebras of subproduct systems and noncommutative function theory},
     journal = {Canadian journal of mathematics},
     pages = {1587--1608},
     year = {2024},
     volume = {76},
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