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Pedersen ideal and group algebras


Author: Klaus Hartmann
Journal: Proc. Amer. Math. Soc. 83 (1981), 183-188
MSC: Primary 46L05; Secondary 43A20
DOI: https://doi.org/10.1090/S0002-9939-1981-0620009-0
MathSciNet review: 620009
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Abstract: For a locally compact $ {T_2}$ group $ G$ which has an open subgroup of polynomial growth (e.g., $ G$ a group that has a compact neighbourhood invariant under inner automorphisms or $ G$ a compact extension of a locally compact nilpotent group) the intersection of the Pedersen ideal of the group $ {C^ * }$-algebra with $ {L^1}(G)$ is dense in $ {L^1}(G)$ (Theorem 1). For groups with small invariant neighbourhoods this intersection is the smallest dense ideal of $ {L^1}(G)$, and it consists exactly of those $ f \in {L^1}(G)$ whose "Fourier transform" vanishes outside some (closed) quasicompact subset of $ \hat G$ (Theorem 3); the Pedersen ideal of $ {C^ * }(G)$ is described as the set of all $ a \in {C^ * }(G)$ for which $ \left\{ {\pi \in \hat G:\pi (a) \ne 0} \right\}$ is contained in some (closed) quasicompact subset of $ \hat G$ (Theorem 2).


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1981-0620009-0
Keywords: Group algebra, group $ {C^ * }$-algebra, Pedersen ideal, polynomial growth, functional calculus, SIN-group
Article copyright: © Copyright 1981 American Mathematical Society

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