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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Almost periodic operators in $ {\rm VN}(G)$


Author: Ching Chou
Journal: Trans. Amer. Math. Soc. 317 (1990), 229-253
MSC: Primary 43A60; Secondary 22D10, 22D25
MathSciNet review: 943301
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Abstract: Let $ G$ be a locally compact group, $ A(G)$ the Fourier algebra of $ G$, $ B(G)$ the Fourier-Stieltjes algebra of $ G$ and $ {\text{VN}}(G)$ the von Neumann algebra generated by the left regular representation $ \lambda $ of $ G$. Then $ A(G)$ is the predual of $ {\text{VN}}(G)$; $ {\text{VN}}(G)$ is a $ B(G)$-module and $ A(G)$ is a closed ideal of $ B(G)$. Let $ {\text{AP}}(\hat G) = \{ T \in {\text{VN}}(G):u \mapsto u \cdot T$ is a compact operator from $ A(G)$ into $ {\text{VN}}(G)\} $, the space of almost periodic operators in $ {\text{VN}}(G)$. Let $ C_\delta ^*(G)$ be the $ {C^*}$-algebra generated by $ \{ \lambda (x):x \in G\} $. Then $ C_\delta ^*(G) \subset {\text{AP}}(\hat G)$. For a compact $ G$, let $ E$ be the rank one operator on $ {L^2}(G)$ that sends $ h \in {L^2}(G)$ to the constant function $ \int {h(x)dx} $. We have the following results: (1) There exists a compact group $ G$ such that $ E \in$   AP$ (\hat G)\backslash C_\delta ^*(G)$. (2) For a compact Lie group $ G$, $ E \in {\text{AP(}}\hat G{\text{)}} \Leftrightarrow E \in C_\delta ^*(G) \Leftrightarrow {L^\infty }(G)$ has a unique left invariant mean $ \Leftrightarrow G$ is semisimple. (3) If $ G$ is an extension of a locally compact abelian group by an amenable discrete group then $ {\text{AP}}(\hat G) = C_\delta ^*(G)$. (4) Let $ G = {{\mathbf{F}}_r}$, the free group with $ r$ generators, $ 1 < r < \infty $. If $ T \in {\text{VN}}(G)$ and $ u \mapsto u \cdot T$ is a compact operator from $ B(G)$ into $ {\text{VN}}(G)$ then $ T \in C_\delta ^*(G)$.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1990-0943301-9
PII: S 0002-9947(1990)0943301-9
Keywords: Locally compact groups, von Neumann algebras, left regular representation, Fourier algebras, Fourier-Stieltjes algebras, $ {C^*}$-algebras, left translation operators, compact groups, amenable groups, free groups, Lorentz groups, approximate identities, almost periodic operators, weakly almost periodic operators
Article copyright: © Copyright 1990 American Mathematical Society