Topology Proceedings 28 No. 2 (2004), pp. 343-359

Nonseparability and uniformities in topological groups

A. Bouziad and J.-P. Troallic

Let G be a Hausdorff topological group. If the left and right uniform structures L_G and R_G on G coincide, then G is said to be balanced, or a SIN-group. Let U_L(G) (respectively U_R(G)) denote the real Banach space of all left (respectively right) uniformly continuous bounded real-valued functions on G, and let U(G) = U_L(G)ÇU_R(G). If U_L(G) = U_R(G), then G is said to be functionally balanced, or to be an FSIN-group. We prove that if G is not an FSIN-group, then the quotient Banach space U_R(G)/U(G) is nonseparable. Moreover, we prove that for a large class of topological groups G, if G is not FSIN then U_R(G)/U(G) contains a linear isometric copy of l^¥. We also establish the equivalence between SIN and FSIN properties in various cases. In particular, we show that for any topological group G strongly functionally generated by its right precompact subsets, SIN and FSIN properties are equivalent.

Keywords: Topological group, Left (right) uniform structure, Left (right) uniformly continuous bounded real-valued function, SIN-group, FSIN-group, Left (right) thin subset, Left (right) neutral subset, Left (right) uniformly discrete subset

Mathematics Subject Classification: 22A05 54E15 (22A10)

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Topology Proceedings, Volume 28, No. 2 (2004)
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