Some abelian Banach algebras of operators on the matricial 𝐼𝐼₁ factor

Robertson, A. Guyan

doi:10.1090/S0002-9939-1990-1009999-X

Some abelian Banach algebras of operators on the matricial $\textrm {II}_ 1$ factor
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by A. Guyan Robertson PDF

Proc. Amer. Math. Soc. 109 (1990), 1063-1068 Request permission

Abstract:

We show that, if $G$ is an amenable discrete group, then the set of completely bounded [completely positive] multiplication operators on $VN(G)$ is maximal abelian and norm-$1$ complemented in various sets of bounded [positive] operators on $VN(G)$. Since there are many different amenable discrete groups $G$ which generate the matricial $II_{1}$ factor $R$, this shows the set of completely bounded normal operators on $R$ contains uncountably many non-isomorphic maximal abelian subalgebras each of which is complemented by a positivity-preserving projection of norm one. Our results are closely related to [16], which considers the case of group ${C^ * }$-algebras of general locally compact groups.

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