Algebras of analytic operator valued functions
Author:
Kenneth O. Leland
Journal:
Trans. Amer. Math. Soc. 194 (1974), 223239
MSC:
Primary 46J25; Secondary 30A96
MathSciNet review:
0377522
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Abstract: This paper proves and generalizes the following characterization of the algebra 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 be the algebra of all continuous functions on open subsets of K into K, and F a subalgebra of with nonconstant elements such that range is closed under uniform convergence on compact sets and domain transformations of the form . Then F is or or . In the general case conditions on B are studied that insure that either F contains an embedment of 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 such that for , there exist complexifications of E and of range f, such that with respect to and the elements of are complex differentiable.
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 [2]
 , A characterization of analyticity. II, Proc. Amer. Math. Soc. 19 (1968), 519527. MR 38 #3395. MR 0235083 (38:3395)
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 , A characterization of analyticity. III, J. Math. Mech. 18 (1968/69), 109123. MR 38 #3396. MR 0235084 (38:3396)
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 , Maximum modulus theorems for algebras of operator valued functions, Pacific J. Math. 39 (1971), 121138. MR 0310652 (46:9750)
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 , Characterizations of solutions of , Studia Math. 29 (1967/68), 125132. MR 36 #5365. MR 0222313 (36:5365)
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 M. A. Naĭmark, Normed rings, GITTL, Moscow, 1956; English transl., Noordhoff, Groningen, 1959. MR 19, #870; 22 #1824.
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 G. T. Whyburn, Topological analysis, 2nd rev. ed., Princeton Math. Series, no. 23, Princeton Univ. Press, Princeton, N.J., 1964. MR 29 #2758. MR 0165476 (29:2758)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197403775224
PII:
S 00029947(1974)03775224
Keywords:
Operator valued functions,
function algebras,
complex differentiability,
analyticity,
Banach algebras,
complexification
Article copyright:
© Copyright 1974 American Mathematical Society
