Journal of Operator Theory

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

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

Authors: Yuri Karlovich (1) and Ivan Loreto-Hernandez (2)
Author institution:(1) Facultad de Ciencias, Universidad Autonoma del Estado de Morelos, Cuernavaca, 62209, Mexico
(2) Facultad de Ciencias, Universidad Autonoma del Estado de Morelos, Cuernavaca, 62209, Mexico

Summary: 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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