Weak factorizations of operators in the group von Neumann algebras of certain amenable groups and applications (Q652245)

Let \(G\) be a locally compact group and \(A(G)\) be the Fourier algebra of \(G\). Let \(A(G)^{**}\) denote the second dual of \(A(G)\) with the first Arens multiplication ``\(\square\)'' defined by \(\langle m\square n,\,f\rangle= \langle m,n\cdot f\rangle\), where \(\langle n \cdot f, a\rangle = \langle n,f\cdot a\rangle\), \(m,n\in A(G)^{**}\), \(f\in A^*\) and \(a\in A(G)\). Then \(A(G)^{**}\) is a Banach algebra extending the multiplication of \(A(G)\). Let \(Z(A(G)^{**})\) denote the algebraic centre of \(A(G)\). Since \(A(G)\) is commutative, \(A(G)^{**}\supseteq Z(A(G)^{**})\supseteq A(G)\). \(A(G)\) is called Arens regular if \( Z(A(G)^{**})= A(G)^{**}\); \(A(G)\) is called strongly Arens irregular if \(Z(A(G)^{**})= A(G)\). It was shown in [\textit{A. T.-M. Lau} and \textit{J. C. S. Wong}, Proc. Am. Math. Soc. 107, No. 4, 1031--1036 (1989; Zbl 0696.43001)] that if \(G\) is amenable, then \(A(G)\) is Arens regular if and only if \(G\) is finite. However, it was announced in the lecture of \textit{V. Losert} given at the meeting ``Harmonic Analysis, Operator Algebra and Representations'' at CIRM, Luminy in 2008 that if \(G\) is the compact group \(SU(3)\), or the locally compact group \(SL(2,\mathbb{R})\), \(A(G)\) is not strongly Arens irregular, but \(A(G)\) is strongly Arens irregular when \(G\) is the compact group \(SU(2)\). However, as known (see [\textit{A. T.-M. Lau} and \textit{V. Losert}, J. Funct. Anal. 112, No. 1, 1--30 (1993; Zbl 0788.22006)]) for a large class of amenable groups which include discrete amenable groups, \(A(G)\) is strongly Arens irregular. In this beautiful and well written paper, the authors show that if \(G\) is a compact group whose local weight \(b(G)\) has uncountable cofinality, then the group von Neumann algebra \(VN(G)= A(G)^*\) has weak uniform factorization property. In particular, for these groups, the centre of \(A(G)^{**}\) is \(A(G)\), i.e. \(A(G)\) is strongly Arens irregular. This result is then extended to the product group \(G\times H\) where \(G\) is as above and \(H\) is an arbitrary amenable locally compact group.