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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Algebras of analytic operator valued functions


Author: Kenneth O. Leland
Journal: Trans. Amer. Math. Soc. 194 (1974), 223-239
MSC: Primary 46J25; Secondary 30A96
MathSciNet review: 0377522
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Abstract: This paper proves and generalizes the following characterization of the algebra $ A(K,K)$ of complex analytic functions on open subsets of the complex plane K into K to the case of algebras of functions on a real Euclidean space E into a real Banach algebra B.

Theorem. Let $ F(K,K)$ be the algebra of all continuous functions on open subsets of K into K, and F a subalgebra of $ F(K,K)$ with nonconstant elements such that $ { \cup _{f \in F}}$ range $ f = K,F$ is closed under uniform convergence on compact sets and domain transformations of the form $ z \to {z_0} + z\sigma ,z,{z_0},\sigma \in K$. Then F is $ F(K,K)$ or $ A(K,K)$ or $ \bar A(K,K) = \{ \bar f;f \in A(K,K)\} $.

In the general case conditions on B are studied that insure that either F contains an embedment of $ F(R,R)$ and thus contains quite arbitrary continuous functions or that the elements of F are analytic and F can be expressed as a direct sum of algebras $ {F_1}, \ldots ,{F_n}$ such that for $ i = 1, \ldots ,n$, there exist complexifications $ {M_i}$ of E and $ {N_i}$ of $ {\cup _{f \in {F_i}}}$ range f, such that with respect to $ {M_i}$ and $ {N_i}$ the elements of $ {F_i}$ are complex differentiable.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1974-0377522-4
PII: S 0002-9947(1974)0377522-4
Keywords: Operator valued functions, function algebras, complex differentiability, analyticity, Banach algebras, complexification
Article copyright: © Copyright 1974 American Mathematical Society