Right submodules of finite rank for von Neumann dynamical systems
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2010
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| _version_ | 1866909354571071488 |
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| author | Jolissaint, Paul |
| author_facet | Jolissaint, Paul |
| contents | Let $(M,τ,σ,Γ)$ be a (finite) von Neumann dynamical system and let $N$ be a $Γ$-invariant unital von Neumann subalgebra of $M$. If $V\subset L^2(M)$ is a right $N$-submodule whose projection $p_V$ has finite trace in $< M,e_N>$ and is $Γ$-invariant, then we prove that, for every $ε>0$, one can find a $Γ$-invariant submodule $W\subset V$ which has finite rank and such that $Tr(p_V-p_W)<ε$. Furthermore, we also construct a $σ$-cocycle that gives the action of $Γ$ on a basis of $W$. In particular, this answers a question of T. Austin, T. Eisner and T. Tao. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1009_0684 |
| institution | arXiv |
| publishDate | 2010 |
| record_format | arxiv |
| spellingShingle | Right submodules of finite rank for von Neumann dynamical systems Jolissaint, Paul Operator Algebras 46L10 Let $(M,τ,σ,Γ)$ be a (finite) von Neumann dynamical system and let $N$ be a $Γ$-invariant unital von Neumann subalgebra of $M$. If $V\subset L^2(M)$ is a right $N$-submodule whose projection $p_V$ has finite trace in $< M,e_N>$ and is $Γ$-invariant, then we prove that, for every $ε>0$, one can find a $Γ$-invariant submodule $W\subset V$ which has finite rank and such that $Tr(p_V-p_W)<ε$. Furthermore, we also construct a $σ$-cocycle that gives the action of $Γ$ on a basis of $W$. In particular, this answers a question of T. Austin, T. Eisner and T. Tao. |
| title | Right submodules of finite rank for von Neumann dynamical systems |
| topic | Operator Algebras 46L10 |
| url | https://arxiv.org/abs/1009.0684 |