Algebras of analytic operator valued functions

Author:
Kenneth O. Leland

Journal:
Trans. Amer. Math. Soc. **194** (1974), 223-239

MSC:
Primary 46J25; Secondary 30A96

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

MathSciNet review:
0377522

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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]**K. O. Leland,*A characterization of analyticity*, Duke Math. J.**33**(1966), 551-565. MR**33**#5914. MR**0197751 (33:5914)****[2]**-,*A characterization of analyticity*. II, Proc. Amer. Math. Soc.**19**(1968), 519-527. MR**38**#3395. MR**0235083 (38:3395)****[3]**-,*A characterization of analyticity*. III, J. Math. Mech.**18**(1968/69), 109-123. MR**38**#3396. MR**0235084 (38:3396)****[4]**-,*Algebras of integrable functions*. II, Rocky Mountain J. Math.**2**(1972), 207-224. MR**0500103 (58:17809)****[5]**-,*Maximum modulus theorems for algebras of operator valued functions*, Pacific J. Math.**39**(1971), 121-138. MR**0310652 (46:9750)****[6]**-,*Characterizations of solutions of*, Studia Math.**29**(1967/68), 125-132. MR**36**#5365. MR**0222313 (36:5365)****[7]**M. A. Naĭmark,*Normed rings*, GITTL, Moscow, 1956; English transl., Noordhoff, Groningen, 1959. MR**19**, #870;**22**#1824.**[8]**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)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
46J25,
30A96

Retrieve articles in all journals with MSC: 46J25, 30A96

Additional Information

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