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| Format: | Preprint |
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2023
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| Online Access: | https://arxiv.org/abs/2310.04175 |
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| _version_ | 1866929609776300032 |
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| author | Dessi, Joseph A. Kakariadis, Evgenios T. A. |
| author_facet | Dessi, Joseph A. Kakariadis, Evgenios T. A. |
| contents | We parametrise the gauge-invariant ideals of the Toeplitz-Nica-Pimsner algebra of a strong compactly aligned product system over $\mathbb{Z}_+^d$ by using $2^d$-tuples of ideals of the coefficient algebra that are invariant, partially ordered, and maximal. We give an algebraic characterisation of maximality that allows the iteration of a $2^d$-tuple to the maximal one inducing the same gauge-invariant ideal. The parametrisation respects inclusions and intersections, while we characterise the join operation on the $2^d$-tuples that renders the parametrisation a lattice isomorphism.
The problem of the parametrisation of the gauge-invariant ideals is equivalent to the study of relative Cuntz-Nica-Pimsner algebras, for which we provide a generalised Gauge-Invariant Uniqueness Theorem. We focus further on equivariant quotients of the Cuntz-Nica-Pimsner algebra and provide applications to regular product systems, C*-dynamical systems, strong finitely aligned higher-rank graphs, and product systems on finite frames. In particular, we provide a description of the parametrisation for (possibly non-automorphic) C*-dynamical systems and row-finite higher-rank graphs, which squares with known results when restricting to crossed products and to locally convex row-finite higher-rank graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_04175 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Equivariant Nica-Pimsner quotients associated with strong compactly aligned product systems Dessi, Joseph A. Kakariadis, Evgenios T. A. Operator Algebras Functional Analysis 46L08, 47L55, 46L05 We parametrise the gauge-invariant ideals of the Toeplitz-Nica-Pimsner algebra of a strong compactly aligned product system over $\mathbb{Z}_+^d$ by using $2^d$-tuples of ideals of the coefficient algebra that are invariant, partially ordered, and maximal. We give an algebraic characterisation of maximality that allows the iteration of a $2^d$-tuple to the maximal one inducing the same gauge-invariant ideal. The parametrisation respects inclusions and intersections, while we characterise the join operation on the $2^d$-tuples that renders the parametrisation a lattice isomorphism. The problem of the parametrisation of the gauge-invariant ideals is equivalent to the study of relative Cuntz-Nica-Pimsner algebras, for which we provide a generalised Gauge-Invariant Uniqueness Theorem. We focus further on equivariant quotients of the Cuntz-Nica-Pimsner algebra and provide applications to regular product systems, C*-dynamical systems, strong finitely aligned higher-rank graphs, and product systems on finite frames. In particular, we provide a description of the parametrisation for (possibly non-automorphic) C*-dynamical systems and row-finite higher-rank graphs, which squares with known results when restricting to crossed products and to locally convex row-finite higher-rank graphs. |
| title | Equivariant Nica-Pimsner quotients associated with strong compactly aligned product systems |
| topic | Operator Algebras Functional Analysis 46L08, 47L55, 46L05 |
| url | https://arxiv.org/abs/2310.04175 |