Approximate indicators for closed subgroups of locally compact groups with applications to weakly amenable groups

2015/01/28 by Zsolt Tanko, Tanko, Zsolt
Mathematics · #43A22 (Primary) 46J10 #46L07 (Secondary) #Advanced Banach Space Theory #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA)

paper · pdf · doi:10.48550/arxiv.1501.07245

openalex publication_date 2015/01/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We generalize the notion of an approximate indicator for a closed subgroup H of a locally compact group G introduced by Aristov, Runde, and Spronk and extend their characterization of the existence of such nets in terms of the approximability of χ_H in an appropriate weak^* topology. We find that this equivalent condition is satisfied whenever H is weakly amenable and χ_H, considered as acting on ℓ¹(G) by multiplication, extends to a bounded map on VN(G). This occurs in particular when a natural projection VN(G)arrow I(A(G),H)^⊥ exists. Applications are obtained to the existence (and non-existence) of natural and invariant projections onto I(A(G),H)^⊥ and I(A_cb(G),H)^⊥ and to the existence of (Δ-weak) bounded approximate identities in ideals of A(G) and A_cb(G). In particular, we exhibit a locally compact group without the invariant complementation property.

Citations

• Multipliers of the Fourier Algebras of Some Simple Lie Groups and Their Discrete Subgroups

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