STRONGLY TRANSLATION CONTINUOUS FUNCTIONALS ON THE SPACE LUC(G) FOR AN AMENABLE TOPOLOGICAL GROUP G
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As is well known (thanks to Graham, Lau, and Leinert), for a locally compact Hausdorff group G and for X 2 fM(G)¤ LUC(G)¤ L1(G)¤, VN(G) VN(G)¤g (and for some other function spaces), the characterization of the set Xc of elements in X for which the natural translation operation is continuous with respect to the topology on G and the norm topology on X is a difficult task. In particular, if Xc is sufficiently small, then G is often compact, or even finite, depending on the case. Let G be an amenable locally compact group and let N be the closed subspace of LUC(G) formed by functions for which all invariant means on LUC(G) vanish on the modulus of the function. A total family of strongly translation continuous functionals on the quotient space LUC(G)=N is constructed.
Abstract
As is well known (thanks to Graham, Lau, and Leinert), for a locally compact Hausdorff group G and for X 2 fM(G)¤ LUC(G)¤ L1(G)¤, VN(G) VN(G)¤g (and for some other function spaces), the characterization of the set Xc of elements in X for which the natural translation operation is continuous with respect to the topology on G and the norm topology on X is a difficult task. In particular, if Xc is sufficiently small, then G is often compact, or even finite, depending on the case. Let G be an amenable locally compact group and let N be the closed subspace of LUC(G) formed by functions for which all invariant means on LUC(G) vanish on the modulus of the function. A total family of strongly translation continuous functionals on the quotient space LUC(G)=N is constructed.
참고문헌 (6)
- C. C. Graham, "Separable translation-invariant subspaces of M(G) and other dual spaces on locally compact groups", Colloq. Math, 55(1), pp.131~145 (1988)
- C. C. Graham, "Continuity of translation in the dual of L1(G) and related spaces", Trans. Amer. Math. Soc, 328(2), pp.589~618 (1991)
- E. Hewitt, "Abstract Harmonic Analysis. Vol. I. Structure of Topological Groups, Integration Theory, Group Representations, Grundlehren der Mathematischen Wissenschaften, 115" (1979), Springer-Verlag
- F. P. Greenleaf, "Invariant Means on Topological Groups and Their Applications" (1969), Van Nostrand
- A. L. T. Paterson, "Amenability", Amer. Math. Soc., Providence, RI (1988)
- A. I. Shtern, "Almost convergence and its applications to the Fourier–Stieltjes localization", Russ. J. Math. Phys, 1(1), pp.115~125 (1993)