On invariant subalgebras of the Fourier-Stieltjes algebra of a locally compact group
One of the most instructive Banach algebras attached to a locally compact group \(G\) is the commutative Fourier-Stieltjes algebra \(B(G)\), the dual space of the \(C^*\)-algebra \(C^*(G)\) of \(G\); it is spanned by the set \(P(G)\) of all continuous positive definite functions on \(G\). In [J. Funct. Anal. 11, 17-38 (1972; Zbl 0242.22010)], \textit{M. E. Walter} proved the locally compact groups \(G_ 1\) and \(G_ 2\) to be isomorphic if and only if the algebras \(B(G_ 1)\) and \(B(G_ 2)\) are isometrically isomorphic. The carefully written paper under review describes the structure of a [translation-] invariant *-subalgebra \(A\) of \(B(G)\) in terms of a subgroup in \(G\). A direct proof yields that in case \(A\) separates points of \(G\) and it is also conjugation-invariant with respect to the enveloping von Neumann algebra of \(C^*(G)\), it coincides with the Fourier algebra \(A(G)\) of \(G\). As a consequence, the locally compact group \(G\) is amenable if and only if there exists a bijection between the set of all closed normal subgroups of \(G\) and the set of all weak-*-closed invariant *-subalgebras \(A\) of \(G\) with \(A\neq\{0\}\); the proof uses the fact that in the amenable case, \(f\in P(G)\) is a limit on compacta of functions \(\varphi*\widetilde{\varphi}\), \(\varphi\in L^ 2(G)\), with \(\widetilde{\varphi}(x)= \overline {\varphi(x^{-1})}\), \(x\in G\). Also, if the discrete group \(G_ d\) corresponding to \(G\) is amenable, then \(B(G)\) is weak-*-dense in \(B(G_ d)\). The authors prove that not being the case for the nonamenable group \(SL(3,\mathbb{C})\), as it violates Reiter's condition \((P_ 2^*)\). Other general properties are established: If \(A\) is a norm-closed invariant *-subalgebra of \(A(G)\) and \(A\neq \{0\}\), then \(A=A(G/K)\) for some compact normal subgroup \(K\) of \(G\). The weak-*-closed \({\mathcal C}_ 0(G)\)-invariant *-subalgebra \(A\) of \(M^ 1(G)={\mathcal C}_ 0(G)^*\) [resp. the closed \(L^ \infty(G)\)-invariant *-subalgebra of \(L^ 1(G)\)], \(A\neq \{0\}\), is \(M^ 1(H)\) [resp. \(L^ 1(H)\)] for a closed [resp. open] subgroup \(H\) of \(G\). \textit{M. Takesaki} and \textit{N. Tatsuuma} obtained the very last result by different methods [Ann. Math., II. Ser. 93, 344-364 (1971; Zbl 0201.455)].
scientific article; zbMATH DE number 86202
scientific article