In this paper we investigate when various Banach spaces associated to a locally compact group G have the fixed point property for nonex-pansive 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 A has the fixed point property then for any normal element a of A, the spectrum σ(a) is countable and that the group C *-algebra C * (G) has weak normal structure if and only if G is finite.

In this paper we investigated when the dual of a certain function space defined on a locally compact group has certain geometric properties. More particularly, we asked when weak ∗ ^{*} compact convex subsets in these spaces have normal structure, and when the norm of these spaces satisfies one of several types of Kadec-Klee property. As samples of the results we have obtained, we have proved, among other things, the following two results: (1) The measure algebra of a locally compact group has weak ∗ ^{*} -normal structure iff it has property SUKK ∗ ^{*} iff it has property SKK ∗ ^{*} iff the group is discrete; (2) Among amenable locally compact groups, the Fourier-Stieltjes algebra has property SUKK ∗ ^{*} iff it has property SKK ∗ ^{*} iff the group is compact. Consequently the Fourier-Stieltjes algebra has weak ∗ ^{*} -normal structure when the group is compact.

Let E be a dual Banach space. E is said to have quasi-weak*-normal structure if for each weak* compact convex subset K of E there exists x e K such that ||* -y\\ < άiam(K) for all y e K. E is said to satisfy Lim's condition if whenever { x a } is a bounded net in E converging to 0 in the weak* topology and lim ||x α || = s then lim α \\x a + y\\ = s + \\y\\ for any y e E. Lim's condition implies (quasi) weak*-normal structure. Let H be a Hilbert space. In this paper, we prove that &~(H), the space of trace class operators on //, always has quasi-weak*-normal structure for any H; tΓ(H) satisfies Lim's condition if and only if H is finite dimensional. We also prove that the space of bounded linear operator on H has quasi-weak*-normal structure if and only if H is finite dimensional; the space of compact operators on // has quasi-weak-normal structure if and only if H is separable. Finally we prove that if X is a locally compact Hausdorff space, then C 0 (X)* satisfies Lim's condition if and only if C 0 (X)* is isometrically isomorphic to /i(Γ) for some non-empty set Γ.

A convexity space is a non‐empty set X together with a family ℓ of subsets of X that contains X and the empty set Ø, and is closed under arbitrary intersections. We give necessary and sufficient conditions, in terms of ℓ and real‐valued convexity‐preserving functions on X, for the existence of a real linear structure for X with respect to which ℓ is precisely the family of all convex sets of the linear space X.

Abstract. A theory of summability is developed in amenable semigroups. We give necessary and (or) sufficient conditions for matrices to be almost regular, almost Schur, strongly regular, and almost strongly regular. In particular, when the amenable semigroup is the additive positive integers, our theorems yield those results of J. P. King, P. Schaefer and G. G. Lorentz for some of the matrices mentioned above.