On density of transitive operator algebras
Let \(G\) be a discrete group and let \({\mathcal H}\) be the Hilbert space \(l^{2}(G)\) with the usual inner product. For each \(g\in G\), let \(\lambda_{g}\) denote the left translation of functions in \({\mathcal H}\) by \(g^{-1}\). Let \(L(G)\) be the von Neumann algebra generated by \(\{ \lambda_{g}:g\in G\} \). Similarly, let \(R(G)\) be the corresponding von Neumann algebra generated by right translations. Then \(R(G)=L(G)'\). Let \(\mathbf{F}_{\infty}\) be the free group with countable infinite set of generators \(\{ a_{i}:i=1,2,\dots\}\). Consider two sums \(\lambda_{a_{1}}+\lambda_{a_{2}}\) and \(\lambda_{a_{3}}+\lambda_{a_{4}}\) of unitary generators in the free von Neumann algebra \(L(\mathbf{F}_{\infty})\). Then the main result of the paper says that the set \( \{ \lambda_{a_{1}}+\lambda_{a_{2}}, \lambda_{a_{3}}+\lambda_{a_{4}} \} \) together with the commutant \(R(\mathbf{F}_{\infty})\) generate \({\mathcal B}({\mathcal H})\) as a von Neumann algebra. The author also discusses relations between the transitive algebra problem and the invariant subspace problem relative to some von Neumann algebras.
scientific article; zbMATH DE number 5673280
scientific article