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On the \(C^*\)-algebra generated by Toeplitz operators and Fourier multipliers on the Hardy space of a locally compact group. (English) Zbl 1389.47082

Let \(G\) be a non-compact, locally compact commutative Hausdorff topological group whose Pontryagin dual group \(\Gamma\) is partially ordered, and let \(\Gamma^+=\{\gamma\in\Gamma:\gamma\geq e\}\) be the semigroup of positive elements of \(\Gamma\), where \(e\) is the unit of \(G\). Let \(P\) stand for the orthogonal projection of the Hilbert space \(L^2(G)\) onto the Hardy space \(H^2(G)=\{f\in L^2(G): \widehat{f}(\gamma)=0\) for all \(\gamma\notin\Gamma^+\}\), where \(\widehat{f}={\mathcal F}f\) and \({\mathcal F}: L^2(G)\to L^2(\Gamma)\) is the Fourier transform. Let \(\dot{G}=G\cup\{\infty\}\) and \(\dot{\Gamma^+}= \Gamma^+\cup\{\infty\}\) denote the one point compactifications of \(G\) and \(\Gamma^+\), respectively, let \(C_0(G)\) be the set of continuous functions on \(G\) that vanish at \(\infty\), let \(C(\dot{\Gamma^+})\) be the set of continuous functions on \(\dot{\Gamma^+}\), and let \(M_\phi\) and \(M_\theta\) denote the multiplication operators by \(\phi\in C_0(G)\) and \(\theta\in C(\dot{\Gamma^+})\), respectively. The \(C^*\)-algebra \(\Psi(C_0(G),C(\dot{\Gamma^+}))\) generated by all Toeplitz operators \(T_\phi=PM_\phi:H^2(G)\to H^2(G)\) with \(\phi\in C_0(G)\) and by all operators (Fourier multipliers) \(D_\theta={\mathcal F}^{-1}M_\theta{\mathcal F} :H^2(G)\to H^2(G)\) with \(\theta\in C(\dot{\Gamma^+})\) is studied. The character space \(M(\Psi)\) of the \(C^*\)-algebra \(\Psi(C_0(G),C(\dot{\Gamma^+}))\) is described provided that \(\Gamma^+\) separates the points of \(G\), which means that for any \(t_1,t_2\in G\) with \(t_1\neq t_2\) there is a \(\gamma\in\Gamma^+\) such that \(\gamma_(t_1)\neq\gamma(t_2)\). Namely, under all conditions indicated above, it is proved that \(M(\Psi)\cong(\dot{G}\times\{\infty\}) \cup(\{\infty\}\times\dot{\Gamma^+})\).

MSC:

47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
47L80 Algebras of specific types of operators (Toeplitz, integral, pseudodifferential, etc.)
47C15 Linear operators in \(C^*\)- or von Neumann algebras
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