Operator Product States on Tensor Powers of $C^\ast$-Algebras
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arXiv
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| Format: | Preprint |
| Publié: |
2021
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| _version_ | 1866913578249879552 |
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| author | Prodan, Emil |
| author_facet | Prodan, Emil |
| contents | The program of matrix product states on tensor powers $\mathcal A^{\otimes \mathbb Z}$ of $C^\ast$-algebras, initiated in Comm. Math. Phys. {\bf 144}, 443-490 (1992), is re-assessed in a context where $\mathcal A$ is a generic nuclear $C^\ast$-algebra. For any shift invariant state $ω$, we demonstrate the existence of an order kernel ideal $\mathcal K_ω$, whose quotient action reduces and factorizes the initial data $(\mathcal A^{\otimes \mathbb Z}, ω)$ to the tuple $(\mathcal A,\mathcal B_ω= \mathcal A^{\otimes \mathbb N^\times}/\mathcal K_ω, \mathbb E_ω: \mathcal A \otimes \mathcal B_ω\to \mathcal B_ω, \bar ω: \mathcal B_ω\to \mathbb C)$, where $\mathcal B_ω$ is an operator system and $\mathbb E_ω$ and $\bar ω$ are unital and completely positive maps. Reciprocally, given a (input) tuple $(\mathcal A,\mathcal S,\mathbb E,ϕ)$ that shares similar attributes, we supply an algorithm that produces a shift-invariant state on $\mathcal A^{\otimes \mathbb Z}$. We give sufficient conditions in which the so constructed states are ergodic and they reduce back to their input data. As examples, we formulate the input data that produces AKLT-type states, this time in the context of infinite site algebras, such as the group algebra of discrete amenable groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2108_13932 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Operator Product States on Tensor Powers of $C^\ast$-Algebras Prodan, Emil Operator Algebras Strongly Correlated Electrons Mathematical Physics The program of matrix product states on tensor powers $\mathcal A^{\otimes \mathbb Z}$ of $C^\ast$-algebras, initiated in Comm. Math. Phys. {\bf 144}, 443-490 (1992), is re-assessed in a context where $\mathcal A$ is a generic nuclear $C^\ast$-algebra. For any shift invariant state $ω$, we demonstrate the existence of an order kernel ideal $\mathcal K_ω$, whose quotient action reduces and factorizes the initial data $(\mathcal A^{\otimes \mathbb Z}, ω)$ to the tuple $(\mathcal A,\mathcal B_ω= \mathcal A^{\otimes \mathbb N^\times}/\mathcal K_ω, \mathbb E_ω: \mathcal A \otimes \mathcal B_ω\to \mathcal B_ω, \bar ω: \mathcal B_ω\to \mathbb C)$, where $\mathcal B_ω$ is an operator system and $\mathbb E_ω$ and $\bar ω$ are unital and completely positive maps. Reciprocally, given a (input) tuple $(\mathcal A,\mathcal S,\mathbb E,ϕ)$ that shares similar attributes, we supply an algorithm that produces a shift-invariant state on $\mathcal A^{\otimes \mathbb Z}$. We give sufficient conditions in which the so constructed states are ergodic and they reduce back to their input data. As examples, we formulate the input data that produces AKLT-type states, this time in the context of infinite site algebras, such as the group algebra of discrete amenable groups. |
| title | Operator Product States on Tensor Powers of $C^\ast$-Algebras |
| topic | Operator Algebras Strongly Correlated Electrons Mathematical Physics |
| url | https://arxiv.org/abs/2108.13932 |