The centre of the second conjugate algebra of the Fourier algebra for infinite products of groups.

*(English)*Zbl 1067.22004Let \(G\) 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.

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.

Reviewer: Eberhard Kaniuth (Paderborn)

##### 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 |