Generalized Fourier-Stieltjes algebra

Abstract

Let $\{G_{}\}^n_{i=1}$ be locally compact groups and $\mathscr{H}$ be Hilbert space. We define the n-variable Fourier-Stieltjes algebra $B(\Pi^{n}_{1} G_{i}, \mathrm{B}(\mathscr{H}))$ consists all functions \[ \phi : G_{1} \times \cdot \times G_{n} \rightarrow \mathrm{B}(\mathscr{H}) \] for which there exists unitary representations $\pi_{i} : \mathbb{G}_{i} \rightarrow \mathrm{B}(\mathscr{H}_i)$ and a diagram of bounded operators \[ \mathscr{H} \rightarrow^{V} \mathscr{H}_n \rightarrow^{T_{n-1}} \mathscr{H}_{n-1} \rightarrow^{T_{1}} \mathscr{H_1} \rightarrow^{U} \mathscr{H} \] with \[ \phi(s_{1},..., s_{n}) = U\pi_{1}(s_1)T_{1}\pi_{2})s_{2}) \cdot \pi_{n-1}(s_{n-1})T_{n}\pi_{n}(s_{n}V \] We extend the pointwise product on $B(\Pi^{n}_{1} G_{i}, \mathrm{B}(\mathscr{H}))$ under which it forms a completely contractive commutative unital Banach algebra. A diagram of its subalgebras will be introduced.