# 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)
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 $B\in\fB$.
Keywords: convolution type operator, piecewise slowly oscillating function, local-trajectory method, spectral measure, $C^*$-algebra, faithful representation, Fredholmness