Topological methods in solvability theory of multidimensional pair integral operators with homogeneous kernels of compact type (Q744275)

In the paper, the author ``calculate[s] the homotopy equivalence classes of Fredholm and invertible operators in the \(C^\ast\)-algebra of pair operators with homogeneous kernels of compact type.'' A homogeneous measurable kernel \(k\) of degree \(-n\) belongs to the class \(\mathcal M_n\) if \(\int_{\mathbb R^n} |k(x,s)||x|^{-n/2}\,dx < \infty\) and \(\int_{\mathbb R^n} |k(s,y)||y|^{-n/2}\,dy < \infty\). In the corresponding \(C^\ast\)-algebra \(\text{Op}(\mathcal M_n)\), consider the closed subalgebra \(\mathfrak V_n\) generated by integral operators of the form \[ (K_{(k)}f)(x) = \int_{\mathbb R^n} k(x,y)f(y)\,dy \] with homogeneous (\(k(\alpha x,\alpha y) = \alpha^{-n}k(x,y)\) for all \(\alpha > 0\), \(x,y\in \mathbb R^n\)) kernels \(k(x,y)\) in \(\mathcal{C}_n\) (see Section 1.3 of the paper for the definition of \(\mathcal{C}_n\)). Let \(\mathfrak W_n\) be the \(C^\ast\)-algebra generated by the operators of form \(\lambda I + A_-M_{\chi_{\mathbb R_-}}+ A_+M_{\chi_{\mathbb R_+}} + T\), where \(\lambda\in \mathbb C\), \(A_\pm\) is in the closed subalgebra \(\mathfrak V_n\) and \(T \in \mathcal K(\mathcal H)\) is a compact operator, \(M_\varphi\) is the operator of multiplication with a function \(\varphi\). The main result of the paper is Theorem 3.2, stating a short exact hexagon of abelian groups \[ \begin{tikzcd} \mathbb Z \rar["0"] & K_0(\mathfrak W_n) \rar["z_{[0]}"] & \mathbb Z\dar\\ \mathbb Z \oplus \mathbb Z \uar["p_1"] & K_1(\mathfrak W_n) \lar["z_{[1]}"] & 0\lar\end{tikzcd} \] from which one deduces the homotopy type of the operator \(C^\ast\)-algebra \(\mathfrak W\) and its index theory.