Operator space structure and amenability for Figà-Talamanca-Herz algebras |
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Authors: | Anselm Lambert Matthias Neufang Volker Runde |
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Institution: | a Fachrichtung 6.1 Mathematik, Universität des Saarlandes, Postfach 151150, 66041 Saarbrücken, Germany b School of Mathematics and Statistics, 4364 Herzberg Laboratories, Carleton University, Ottawa, Ontario, Canada K1S 5B6 c Department of Mathematical and Statistical Sciences, University of Alberta, CAB 632, Edmonton, Alberta, Canada T6G 2G1 |
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Abstract: | Column and row operator spaces—which we denote by COL and ROW, respectively—over arbitrary Banach spaces were introduced by the first-named author; for Hilbert spaces, these definitions coincide with the usual ones. Given a locally compact group G and p,p′∈(1,∞) with , we use the operator space structure on to equip the Figà-Talamanca-Herz algebra Ap(G) with an operator space structure, turning it into a quantized Banach algebra. Moreover, we show that, for p?q?2 or 2?q?p and amenable G, the canonical inclusion Aq(G)⊂Ap(G) is completely bounded (with cb-norm at most , where is Grothendieck's constant). As an application, we show that G is amenable if and only if Ap(G) is operator amenable for all—and equivalently for one—p∈(1,∞); this extends a theorem by Ruan. |
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Keywords: | primary 47L25 secondary 43A15 43A30 46B70 46J99 46L07 47L50 |
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