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 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.

**[1]**Kenneth O. Leland,*A characterization of analyticity*, Duke Math. J.**33**(1966), 551–565. MR**0197751****[2]**Kenneth O. Leland,*A characterization of analyticity. II*, Proc. Amer. Math. Soc.**19**(1968), 519–527. MR**0235083**, 10.1090/S0002-9939-1968-0235083-7**[3]**Kenneth O. Leland,*A characterization of analyticity. III*, J. Math. Mech.**18**(1968/1969), 109–123. MR**0235084****[4]**Kenneth O. Leland,*Algebras of integrable functions. II*, Rocky Mountain J. Math.**2**(1972), no. 2, 207–224. MR**0500103****[5]**Kenneth O. Leland,*Maximum modulus theorems for algebras of operator valued functions*, Pacific J. Math.**40**(1972), 121–138. MR**0310652****[6]**Kenneth O. Leland,*Characterizations of solutions of Δ𝑓=𝑐𝑓*, Studia Math.**29**(1967/1968), 125–132. MR**0222313****[7]**M. A. Naĭmark,*Normed rings*, GITTL, Moscow, 1956; English transl., Noordhoff, Groningen, 1959. MR**19**, #870;**22**#1824.**[8]**Gordon Thomas Whyburn,*Topological analysis*, Second, revised edition. Princeton Mathematical Series, No. 23, Princeton University Press, Princeton, N.J., 1964. MR**0165476**

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DOI:
https://doi.org/10.1090/S0002-9947-1974-0377522-4

Keywords:
Operator valued functions,
function algebras,
complex differentiability,
analyticity,
Banach algebras,
complexification

Article copyright:
© Copyright 1974
American Mathematical Society