$\ast $-Homomorphisms of matrix algebras over pseudo-solenoids that are approximated by $\ast $-isomorphisms

A pseudo-solenoid is a compact connected metrizable space that is an inverse limit of circles and has a characteristic feature, called hereditary indecomposability. The class of pseudo-solenoids has a topological rigidity in that two pseudo-solenoids $X$ and $Y$ are homeomorphic if and only if their first integral Čech cohomology groups are isomorphic. We show that the class of matrix algebras over pseudo-solenoids has a similar rigidity: two matrix algebras $M_{n}(C(X))$ and $M_{n}(C(Y))$ are isomorphic as $C^\ast $-algebras if and only if they have isomorphic $K_1$-groups.