On two methods of studying the invertibility of operators in $C^*$-algebras induced by dynamical systems

Operators of the form
$$ bu(x)=\sum a_k(x)u(\alpha_k^{-1}(x)) $$
are studied in $L_2(X,\mu)$, where the $a_k$ are given functions and the $\alpha_k\colon X\to X$ are given bijective mappings. A class of $C^*$-algebras including the algebras generated by these operators is also considered. It is proved that these algebras are isomorphic; as a consequence, the spectrum is invariant under rotations and independent of the space the operators act in, and the set of Fredholm operators in the above class coincides with the set of invertible operators. Two methods of studying the invertibility of operators belonging to the above-mentioned algebras are described. The first method relies on establishing the relation between the invertibility of an operator $b$ and the hyperbolicity of the associated linear extension $\beta$. The second method is based on constructing, from the operator $b$, a family of operators $\pi_x(b)$ in the algebra generated by the classical weighted shift operators in $l_2$, such that $b$ is invertible if and only if all the $\pi_x(b)$ are invertible.
Bibliography: 47 titles.