Weak separation properties for closed subgroups of locally compact groups

2017/03/08 by Tanko, Zsolt
#43A15 (Primary) 43A22 #43A30 #46L07 (Secondary) #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.1703.02909

Abstract

Three separation properties for a closed subgroup H of a locally compact group G are studied: (1) the existence of a bounded approximate indicator for H, (2) the existence of a completely bounded invariant projection of VN(G) onto VN_H(G), and (3) the approximability of the characteristic function χ_H by functions in M_cbA(G) with respect to the weak^* topology of M_cbA(G_d). We show that the H-separation property of Kaniuth and Lau is characterized by the existence of certain bounded approximate indicators for H and that a discretized analogue of the H-separation property is equivalent to (3). Moreover, we give a related characterization of amenability of H in terms of any group G containing H as a closed subgroup. The weak amenability of G or that G_d satisfies the approximation property, in combination with the existence of a natural projection (in the sense of Lau and Ülger), are shown to suffice to conclude (3). Several consequences of (2) involving the cb-multiplier completion of A(G) are given. Finally, a convolution technique for averaging over the closed subgroup H is developed and used to weaken a condition for the existence of a bounded approximate indicator for H.

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