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Proceedings of the American Mathematical Society

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Some abelian Banach algebras of operators on the matricial $ {\rm II}\sb 1$ factor


Author: A. Guyan Robertson
Journal: Proc. Amer. Math. Soc. 109 (1990), 1063-1068
MSC: Primary 46L10; Secondary 22D25, 43A30
MathSciNet review: 1009999
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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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DOI: https://doi.org/10.1090/S0002-9939-1990-1009999-X
Article copyright: © Copyright 1990 American Mathematical Society