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Transactions of the American Mathematical Society

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Topological invariant means on the von Neumann algebra $ {\rm VN}(G)$


Author: Ching Chou
Journal: Trans. Amer. Math. Soc. 273 (1982), 207-229
MSC: Primary 22D25; Secondary 43A07, 46L10
DOI: https://doi.org/10.1090/S0002-9947-1982-0664039-7
MathSciNet review: 664039
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Abstract: Let $ VN(G)$ be the von Neumann algebra generated by the left regular representation of a locally compact group $ G$, $ A(G)$ the Fourier algebra of $ G$ and $ TIM(\hat G)$ the set of topological invariant means on $ VN(G)$. Let $ {\mathcal{F}_1} = \{ \mathcal{O} \in {({l^\infty })^ \ast }\} :\mathcal{O} \geqslant 0,\;\vert\vert\mathcal{O}\vert\vert = 1$ and $ \mathcal{O}(f) = 0\;{\text{if}}\;f \in {l^\infty }$ and $ f(n) \to 0\} $. We show that if $ G$ is nondiscrete then there exists a linear isometry $ \Lambda $ of $ {({l^\infty })^ \ast }$ into $ VN{(G)^ \ast }$ such that $ \Lambda ({\mathcal{F}_1}) \subset TIM(\hat G)$. When $ G$ is further assumed to be second countable then $ {\mathcal{F}_1}$ can be embedded into some predescribed subsets of $ TIM(\hat G)$. To prove these embedding theorems for second countable groups we need the existence of a sequence of means $ \{ {u_n}\} $ in $ A(G)$ such that their supports in $ VN(G)$ are mutually orthogonal and $ \vert\vert u{u_n} - {u_n}\vert\vert \to 0\;{\text{if}}\;u$ is a mean in $ A(G)$.

Let $ F(\hat G)$ be the space of all $ T \in VN(G)$ such that $ m(T)$ is a constant as $ m$ runs through $ TIM(\hat G)$ and let $ W(\hat G)$ be the space of weakly almost periodic elements in $ VN(G)$. We show that the following conditions are equivalent: (i) $ G$ is discrete, (ii) $ F(\hat G)$ is an algebra and (iii) $ (A(G) \cdot VN(G)) \cap F(\hat G) \subset W(\hat G)$.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1982-0664039-7
Keywords: Locally compact groups, von Neumann algebra of a group, Fourier algebra, topological invariant means, supports of normal states, topological almost convergent functionals, amenable groups, isometric linear embeddings
Article copyright: © Copyright 1982 American Mathematical Society

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