Representations of locally compact groups on QSL_p-spaces and a p-analog of the Fourier–Stieltjes algebra

Volker Runde

Vol. 221 (2005), No. 2, 379–397

DOI: 10.2140/pjm.2005.221.379

Abstract

For a locally compact group G and p ∈ (1,∞), we define B_p(G) to be the space of all coefficient functions of isometric representations of G on quotients of subspaces of L_p spaces. For p = 2, this is the usual Fourier–Stieltjes algebra. We show that B_p(G) is a commutative Banach algebra that contractively (isometrically, if G is amenable) contains the Figà-Talamanca–Herz algebra A_p(G). If 2 ≤ q ≤ p or p ≤ q ≤ 2, we have a contractive inclusion B_q(G) ⊂ B_p(G). We also show that B_p(G) embeds contractively into the multiplier algebra of A_p(G) and is a dual space. For amenable G, this multiplier algebra and B_p(G) are isometrically isomorphic.

Keywords

locally compact groups, representations, coefficient functions, QSL_p-spaces, Figà-Talamanca–Herz algebras, multiplier algebra, amenability

Mathematical Subject Classification 2000

Primary: 46J99

Secondary: 22D12, 22D35, 43A07, 43A15, 43A65, 46J99

Milestones

Received: 13 February 2004

Revised: 17 October 2004

Published: 1 October 2005