Weak separation properties for closed subgroups of locally compact groups

Zsolt Tanko¹

¹ Department of Pure Mathematics University of Waterloo Waterloo, Ontario, N2L 3G1, Canada

Studia Mathematica, Tome 238 (2017) no. 2, pp. 177-200

Résumé

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 $\mathrm {VN}(G)\rightarrow \mathrm {VN}_{H}(G)$, and (3) the approximability of the characteristic function $\chi _{H}$ by functions in $M_{cb}A(G)$ with respect to the weak$^{*}$ topology of $M_{cb}A(G_{d})$. 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 the fact 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 deduce (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 $H$.

Détail

DOI : 10.4064/sm8723-3-2017

Keywords: three separation properties closed subgroup locally compact group studied existence bounded approximate indicator existence completely bounded invariant projection mathrm rightarrow mathrm approximability characteristic function chi functions respect weak * topology h separation property kaniuth lau characterized existence certain bounded approximate indicators discretized analogue h separation property equivalent nbsp moreover related characterization amenability terms group containing closed subgroup weak amenability satisfies approximation property combination existence natural projection sense lau lger shown suffice deduce nbsp several consequences involving cb multiplier completion given finally convolution technique averaging closed subgroup developed weaken condition existence bounded approximate indicator nbsp