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Weak amenability of Fourier algebras and local synthesis of the anti-diagonal
Lee, H.H.,Ludwig, J.,Samei, E.,Spronk, N. Academic Press ; Elsevier Science B.V. Amsterdam 2016 Advances in mathematics Vol.292 No.-
<P>We show that for a connected Lie group G, its Fourier algebra A(G) is weakly amenable only if G is abelian. Our main new idea is to show that weak amenability of A(G) implies that the anti-diagonal, Delta G = {(g, g(-1)) : g is an element of G} is a set of local synthesis for A(G x G). We then show that this cannot happen if G is non-abelian. We conclude for a locally compact group G, that A(G) can be weakly amenable only if it contains no closed connected non-abelian Lie subgroups. In particular, for a Lie group G, A(G) is weakly amenable if and only if its connected component of the identity G(e) is abelian. (C) 2016 Elsevier Inc. All rights reserved.</P>
Some Weighted Group Algebras are Operator Algebras
Lee, Hun Hee,Samei, Ebrahim,Spronk, Nico Cambridge University Press 2015 Proceedings of the Edinburgh Mathematical Society Vol.58 No.2
<B>Abstract</B><P>Let <I>G</I> be a finitely generated group with polynomial growth, and let <I>ω</I> be a weight, i.e. a sub-multiplicative function on <I>G</I> with positive values. We study when the weighted group algebra <I>ℓ</I><SUP>1</SUP> (<I>G, ω</I>) is isomorphic to an operator algebra. We show that <I>ℓ</I><SUP>1</SUP> (<I>G, ω</I>) is isomorphic to an operator algebra if <I>ω</I> is a polynomial weight with large enough degree or an exponential weight of order 0 < <I>α</I> < 1. We demonstrate that the order of growth of <I>G</I> plays an important role in this problem. Moreover, the algebraic centre of <I>ℓ</I><SUP>1</SUP> (<I>G, ω</I>) is isomorphic to a <I>Q</I>-algebra, and hence satisfies a multi-variable von Neumann inequality. We also present a more detailed study of our results when <I>G</I> consists of the <I>d</I>-dimensional integers ℤ<SUP><I>d</I></SUP> or the three-dimensional discrete Heisenberg group ℍ3(ℤ). The case of the free group with two generators is considered as a counter-example of groups with exponential growth.</P>