On inductive limits of certain $C^ *$-algebras of the form $C(X)\otimes F$

Abstract:

A certain class of ${\ast }$-homomorphisms $C(X) \otimes A \to C(Y) \otimes B$, called compatible with a map defined on $Y$ with values in the set of all closed nonempty subsets of $X$, is studied. A local description of ${\ast }$-homomorphisms $C(X) \otimes A \to C(Y) \otimes B$ is given considering separately the cases $X = {\text {point}}$ and $A = {\mathbf {C}}$; this is done in terms of continuous "quasifields" of ${C^{\ast }}$-algebras. Conditions under which an inductive limit $\underrightarrow {\lim }(C({X_k}) \otimes {A_k}, {\Phi _k})$, where each ${\Phi _k}$ is of the above type, is ${\ast }$-isomorphic with the tensor product of a commutative ${C^{\ast }}$-algebra with an AF algebra are given. For such inductive limits the isomorphism problem is considered.