Journal of Operator Theory

Volume 73, Issue 1, Summer 2015 pp. 211-242.

$C^*$-algebra of nonlocal convolution type operators with piecewise slowly oscillating data

Authors: (1) Yuri Karlovich, (2) Iván Loreto-Hernández
Author institution: (1) Facultad de Ciencias, Universidad Autónoma del Estado de Morelos, Cuernavaca, 62209, México
(2) Facultad de Ciencias, Universidad Autónoma del Estado de Morelos, Cuernavaca, 62209, México

Summary: $ \newcommand{\fB}{\mathfrak{B}} \newcommand{\R}{\mathbb{R}} \newcommand{\cK}{\mathcal{K}} $ The $C^*$-subalgebra $\fB$ of all bounded linear operators on the space $L^2(\R)$, which is generated by all multiplication operators by piecewise slowly oscillating functions, by all convolution operators with piecewise slowly oscillating symbols and by the range of a unitary representation of the group of all translations on $\R$, is studied. A faithful representation of the quotient $C^*$-algebra $\fB^\pi=\fB/\cK$ in a Hilbert space, where $\cK$ is the ideal of compact operators on $L^2(\R)$, is constructed by applying a local-trajectory method and appropriate spectral measures. This gives a Fredholm symbol calculus for the $C^*$-algebra $\fB$ and a Fredholm criterion for the operators $B\in\fB$.

DOI: http://dx.doi.org/10.7900/jot.2013nov11.2039
Keywords: Convolution type operator, piecewise slowly oscillating function, local-trajectory method, spectral measure, $C^*$-algebra, faithful representation, Fredholmness.

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