Nonseparability and Uniformities in Topological Groups

Bouziad, Ahmed ; Troallic, Jean-Pierre

HAL, hal-00372943 / Harvested from HAL

Résumé

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 UL(G) (respectively UR(G)) denote the real Banach space of all left (respectively right) uniformly continuous bounded real-valued functions on G, and let $U(G) = UL(G)\cap UR(G)$. If UL(G) = UR(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 UR(G)/U(G) is nonseparable. Moreover, we prove that for a large class of topological groups G, if G is not FSIN then UR(G)/U(G) contains a linear isometric copy of $l^\infty$. 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.

Détail

Publié le : 2004-07-05
Classification:  Left (right) uniformly discrete subset,  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.,  22A05;54E15; 22A10,  [MATH.MATH-GN]Mathematics [math]/General Topology [math.GN],  [MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA]