Right submodules of finite rank for von Neumann dynamical systems

Details Right submodules of finite rank for von Neumann dynamical systems Right submodules of finite rank for von Neumann dynamical systems

2010-09-03

arxiv.org/abs/1009.0684v3

Mathematics

Operator Algebras

Statements have been extended to arbitrary actions; 7 pages ; typos fixed

Scientific paper

Let $(M,\tau,\sigma,\Gamma)$ be a (finite) von Neumann dynamical system and let $N$ be a $\Gamma$-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 $\Gamma$-invariant, then we prove that, for every $\epsilon>0$, one can find a $\Gamma$-invariant submodule $W\subset V$ which has finite rank and such that $Tr(p_V-p_W)<\epsilon$. Furthermore, we also construct a $\sigma$-cocycle that gives the action of $\Gamma$ on a basis of $W$. In particular, this answers a question of T. Austin, T. Eisner and T. Tao.

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