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Transactions of the American Mathematical Society

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

Author: Cornel Pasnicu
Journal: Trans. Amer. Math. Soc. 310 (1988), 703-714
MSC: Primary 46L05; Secondary 46M10
MathSciNet review: 929238
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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.

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