Premium
Measurable Schur Multipliers and Completely Bounded Multipliers of the Fourier Algebras
Author(s) -
Spronk Nico
Publication year - 2004
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/s0024611504014650
Subject(s) - mathematics , bounded function , multiplier (economics) , invariant (physics) , pure mathematics , bimodule , locally compact space , discrete mathematics , combinatorics , algebra over a field , mathematical analysis , economics , mathematical physics , macroeconomics
Let G be a locally compact group, L p ( G ) be the usual L p ‐space for 1 ⩽ p ⩽ ∞, and A( G ) be the Fourier algebra of G . Our goal is to study, in a new abstract context, the completely bounded multipliers of A( G ), which we denote M cb A( G ). We show that M cb A( G ) can be characterised as the ‘invariant part’ of the space of (completely) bounded normal L ∞ ( G )‐bimodule maps on B(L 2 ( G )), the space of bounded operators on L 2 ( G ). In doing this we develop a function‐theoretic description of the normal L ∞ ( X , μ)‐bimodule maps on B(L 2 ( X , μ)), which we denote by V ∞ ( X , μ), and name the measurable Schur multipliers of ( X , μ). Our approach leads to many new results, some of which generalise results hitherto known only for certain classes of groups. Those results which we develop here are a uniform approach to obtaining the functorial properties of M cb A( G ), and a concrete description of a standard predual of M cb A( G ). 2000 Mathematics Subject Classification 46L07, 43A30, 43A15, 46A32, 22D10, 22D12 (primary), 22D25, 43A20, 43A07 (secondary).