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In this paper we investigate when various Banach spaces associated to a locally compact group G have the fixed point property for nonexpansive mappings or normal structure. We give sufficient conditions and some necessary conditions about G for the Fourier and Fourier-Stieltjes algebras to have the fixed point property. We also show that if a C*-algebra Qt has the fixed point property then for any normal element a of 2A, the spectrum o(a) is countable and that the group C*-algebra C* (G) has weak normal structure if and only if G is finite.

Fixed point property and normal structure for Banach spaces associated to locally compact groups