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The centre of the second conjugate algebra of the Fourier algebra for infinite products of groups. (English) Zbl 1067.22004
Let $$G$$ be a locally compact group, $$\text{VN}(G)$$ the von Neumann algebra generated by the left regular representation $$\rho$$ of $$G$$ on $$L^2(G)$$ and $$\text{UCB}(\widehat G)$$ the $$C^*$$-subalgebra of $$\text{VN}(G)$$ generated by the operators in $$\text{VN}(G)$$ with compact support. This notation is due to the fact that when $$G$$ is abelian, then $$\text{UCB}(\widehat G)$$ is isomorphic to the $$C^*$$-algebra of complex-valued bounded uniformly continuous functions on the dual group $$\widehat G$$ of $$G$$. The dual space $$\text{UCB}(\widehat G)^*$$ of $$\text{UCB}(\widehat G)$$ is a Banach algebra with multiplication $$\langle m\cdot n, T\rangle=\langle m,n\cdot T\rangle$$ for $$m,n\in\text{UCB}(\widehat G)^*$$ and $$T\in\text{UCB}(\widehat G)$$.
It is an interesting and difficult problem to determine the centre of $$\text{UCB}(\widehat G)^*$$. To describe the approach that is available, let $$A(G)$$ and $$B(G)$$ denote the Fourier and the Fourier-Stieltjes algebra of $$G$$, respectively. Moreover, let $$B_\rho(G)$$ be the $$w^*$$-closed subalgebra of $$B(G)= C^*(G)^*$$ associated with $$\rho$$. Then there is a natural isometric embedding $$\pi$$ of $$B_\rho(\widehat G)$$ into the centre of $$\text{UCB}(G)^*$$ with the property that if $$u\in B_\rho(G)$$ and $$S$$ is an element of $$\text{VN}(G)$$ with compact support, then $$\langle\pi(u), S\rangle= \langle uv,S\rangle$$, where $$v$$ is any element of $$A(G)$$ which has compact support and is identically one on the support of $$S$$. In an earlier paper [A. T. Lau and V. Losert, J. Funct. Anal. 112, No. 1, 1–30 (1993; Zbl 0788.22006)] it was shown that $$\pi$$ is surjective for several classes of locally compact groups, including abelian groups, the Heisenberg group, the $$ax+ b$$-group and the motion group of the plane. In the present paper surjectivity of $$\pi$$ is established for second countable locally compact groups $$G$$ of the form $$G= G_0\times\prod^\infty_{j=1} G_j$$, where $$G_0$$ is arbitrary and each $$G_j$$, $$j\geq 1$$, is a non-trivial compact group (Theorem 4.2).
In the above mentioned paper the authors have shown that if $$G$$ is an amenable (equivalently, $$B_\rho(G)= B(G)$$) locally compact group and if $$\pi$$ is surjective, then the centre of $$\text{VN}(G)^* =A(G)^{**}$$ equals $$A(G)$$. Thus, Theorem 4.2 implies that the centre of $$A(G)^{**}$$ equals $$A(G)$$ if $$G$$ is amenable and either of the above product type structure or discrete. However, this centre problem for $$A(G)^{**}$$ is open even for compact semisimple Lie groups and totally disconnected compact groups.

##### MSC:
 22D25 $$C^*$$-algebras and $$W^*$$-algebras in relation to group representations 43A35 Positive definite functions on groups, semigroups, etc. 43A25 Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups
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