Quasi-invariant states
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866914640595779584 |
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| author | Accardi, Luigi Dhahri, Ameur |
| author_facet | Accardi, Luigi Dhahri, Ameur |
| contents | We develop the theory of quasi--invariant (resp. strongly quasi--invariant) states under the action of a group $G$ of normal $*$--automorphisms of a $*$--algebra (or von Neumann alegbra) $\mathcal{A}$. We prove that these states are naturally associated to left--$G$--$1$--cocycles. If $G$ is compact, the structure of strongly $G$--quasi--invariant states is determined. For any $G$--strongly quasi--invariant state $φ$, we construct a unitary representation associated to the triple $(\mathcal{A},G,φ)$. We prove, under some conditions, that any quantum Markov chain with commuting, invertible and hermitean conditional density amplitudes on a countable tensor product of type I factors is strongly quasi--invariant with respect to the natural action of the group $\mathcal{S}_{\infty}$ of local permutations and we give the explicit form of the associated cocycle. This provides a family of non--trivial examples of strongly quasi--invariant states for locally compact groups obtained as inductive limit of an increasing sequence of compact groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_12717 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Quasi-invariant states Accardi, Luigi Dhahri, Ameur Mathematical Physics Operator Algebras We develop the theory of quasi--invariant (resp. strongly quasi--invariant) states under the action of a group $G$ of normal $*$--automorphisms of a $*$--algebra (or von Neumann alegbra) $\mathcal{A}$. We prove that these states are naturally associated to left--$G$--$1$--cocycles. If $G$ is compact, the structure of strongly $G$--quasi--invariant states is determined. For any $G$--strongly quasi--invariant state $φ$, we construct a unitary representation associated to the triple $(\mathcal{A},G,φ)$. We prove, under some conditions, that any quantum Markov chain with commuting, invertible and hermitean conditional density amplitudes on a countable tensor product of type I factors is strongly quasi--invariant with respect to the natural action of the group $\mathcal{S}_{\infty}$ of local permutations and we give the explicit form of the associated cocycle. This provides a family of non--trivial examples of strongly quasi--invariant states for locally compact groups obtained as inductive limit of an increasing sequence of compact groups. |
| title | Quasi-invariant states |
| topic | Mathematical Physics Operator Algebras |
| url | https://arxiv.org/abs/2209.12717 |