On Toeplitz algebras of product systems
Date
2025
Authors
Katsoulis, Elias G.
Laca, Marcelo
Sehnem, Camila F.
Journal Title
Journal ISSN
Volume Title
Publisher
Journal of the Australian Mathematical Society
Abstract
In the setting of product systems over group-embeddable monoids, we consider nuclearity of the associated Toeplitz C*-algebra in relation to nuclearity of the coefficient algebra. Our work goes beyond the known cases of single correspondences and compactly aligned product systems over right least common multiple (LCM) monoids. Specifically, given a product system over a submonoid of a group, we show, under technical assumptions, that the fixed-point algebra of the gauge action is nuclear if and only if the coefficient algebra is nuclear; when the group is amenable, we conclude that this happens if and only if the Toeplitz algebra itself is nuclear. Our main results imply that nuclearity of the Toeplitz algebra is equivalent to nuclearity of the coefficient algebra for every full product system of Hilbert bimodules over abelian monoids, over ax + b-monoids of integral domains and over Baumslag–Solitar monoids BS+(m, n) that admit an amenable embedding, which we provide for m and n relatively prime.
Description
Keywords
Toeplitz algebra, product systems, nuclearity
Citation
Katsoulis, E. G., Laca, M., & Sehnem, C. F. (2025). On Toeplitz algebras of product systems. Journal of the Australian Mathematical Society, 119(3), 436–463. https://doi.org/doi:10.1017/S1446788725101146