# Journal of Operator Theory

Volume 73, Issue 2, Spring 2015 pp. 533-546.

On the $C^*$-algebra generated by Toeplitz operators and Fourier multipliers on the Hardy space of a locally compact group

**Authors**:
Ugur Gul

**Author institution:**Hacettepe University, Department of Mathematics, 06800, Beytepe,
Ankara, Turkey

**Summary: **Let $G$ be a locally compact abelian Hausdorff topological group
which is non-compact and whose Pontryagin dual $\Gamma$ is
partially ordered. Let $\Gamma^{+}\subset\Gamma$ be the semigroup
of positive elements in $\Gamma$. The Hardy space $H^{2}(G)$ is
the closed subspace of $L^{2}(G)$ consisting of functions whose
Fourier transforms are supported on $\Gamma^{+}$. In this paper we
consider the $C^*$-algebra $C^{*}(\mathcal{T}(G)\cup
F(C(\dot{\Gamma^{+}})))$ generated by Toeplitz operators with
continuous symbols on $G$ which vanish at infinity and Fourier
multipliers with symbols which are continuous on one point
compactification of $\Gamma^{+}$ on the Hilbert--Hardy space\break
$H^{2}(G)$. We characterize the character space of this $C^*$-algebra
using a theorem of Power.

**DOI: **http://dx.doi.org/10.7900/jot.2014mar12.2055

**Keywords: **$C^*$-algebras, Toeplitz operators, Hardy space of a locally compact group

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