# 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

Contents
Full-Text PDF