Inverse semigroups of separated graphs and associated algebras
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arXiv
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| Format: | Preprint |
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2024
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| author | Ara, Pere Buss, Alcides Costa, Ado Dalla |
| author_facet | Ara, Pere Buss, Alcides Costa, Ado Dalla |
| contents | In this paper we introduce an inverse semigroup $\mathcal{S}(E,C)$ associated to a separated graph $(E,C)$ and describe its internal structure. In particular we show that it is strongly $E^*$-unitary and can be realized as a partial semidirect product of the form $\mathcal{Y}\rtimes\mathbb{F}$ for a certain partial action of the free group $\mathbb{F}=\mathbb{F}(E^1)$ on the edges of $E$ on a semilattice $\mathcal{Y}$ realizing the idempotents of $\mathcal{S}(E,C)$. In addition we also describe the spectrum as well as the tight spectrum of $\mathcal{Y}$.
We then use the inverse semigroup $\mathcal{S}(E,C)$ to describe several "tame" algebras associated to $(E,C)$, including its Cohn algebra, its Leavitt-path algebra, and analogues in the realm of $C^*$-algebras, like the tame $C^*$-algebra $\mathcal{O}(E,C)$ and its Toeplitz extension $\mathcal{T}(E,C)$, proving that these algebras are canonically isomorphic to certain algebras attached to $\mathcal{S}(E,C)$. Our structural results on $\mathcal{S}(E,C)$ imply that these algebras can be realized as partial crossed products, revealing a great portion of their structure. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_05295 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Inverse semigroups of separated graphs and associated algebras Ara, Pere Buss, Alcides Costa, Ado Dalla Operator Algebras Rings and Algebras 46L55, 20M18 In this paper we introduce an inverse semigroup $\mathcal{S}(E,C)$ associated to a separated graph $(E,C)$ and describe its internal structure. In particular we show that it is strongly $E^*$-unitary and can be realized as a partial semidirect product of the form $\mathcal{Y}\rtimes\mathbb{F}$ for a certain partial action of the free group $\mathbb{F}=\mathbb{F}(E^1)$ on the edges of $E$ on a semilattice $\mathcal{Y}$ realizing the idempotents of $\mathcal{S}(E,C)$. In addition we also describe the spectrum as well as the tight spectrum of $\mathcal{Y}$. We then use the inverse semigroup $\mathcal{S}(E,C)$ to describe several "tame" algebras associated to $(E,C)$, including its Cohn algebra, its Leavitt-path algebra, and analogues in the realm of $C^*$-algebras, like the tame $C^*$-algebra $\mathcal{O}(E,C)$ and its Toeplitz extension $\mathcal{T}(E,C)$, proving that these algebras are canonically isomorphic to certain algebras attached to $\mathcal{S}(E,C)$. Our structural results on $\mathcal{S}(E,C)$ imply that these algebras can be realized as partial crossed products, revealing a great portion of their structure. |
| title | Inverse semigroups of separated graphs and associated algebras |
| topic | Operator Algebras Rings and Algebras 46L55, 20M18 |
| url | https://arxiv.org/abs/2403.05295 |