Fixed point property and the Fourier algebra of a locally compact group

Lau, Anthony; Leinert, Michael

doi:10.1090/S0002-9947-08-04622-9

Fixed point property and the Fourier algebra of a locally compact group
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by Anthony To-Ming Lau and Michael Leinert PDF

Trans. Amer. Math. Soc. 360 (2008), 6389-6402 Request permission

Abstract:

We establish some characterizations of the weak fixed point property (weak fpp) for noncommutative (and commutative) $\mathcal {L}^1$ spaces and use this for the Fourier algebra $A(G)$ of a locally compact group $G.$ In particular we show that if $G$ is an IN-group, then $A(G)$ has the weak fpp if and only if $G$ is compact. We also show that if $G$ is any locally compact group, then $A(G)$ has the fixed point property (fpp) if and only if $G$ is finite. Furthermore if a nonzero closed ideal of $A(G)$ has the fpp, then $G$ must be discrete.

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