Algebras of analytic operator valued functions

Leland, Kenneth O.

doi:10.1090/S0002-9947-1974-0377522-4

Algebras of analytic operator valued functions
HTML articles powered by AMS MathViewer

by Kenneth O. Leland PDF

Trans. Amer. Math. Soc. 194 (1974), 223-239 Request permission

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.

References

Kenneth O. Leland, A characterization of analyticity, Duke Math. J. 33 (1966), 551–565. MR 197751
Kenneth O. Leland, A characterization of analyticity. II, Proc. Amer. Math. Soc. 19 (1968), 519–527. MR 235083, DOI 10.1090/S0002-9939-1968-0235083-7
Kenneth O. Leland, A characterization of analyticity. III, J. Math. Mech. 18 (1968/1969), 109–123. MR 0235084, DOI 10.1512/iumj.1969.18.18011
Kenneth O. Leland, Algebras of integrable functions. II, Rocky Mountain J. Math. 2 (1972), no. 2, 207–224. MR 500103, DOI 10.1216/RMJ-1972-2-2-207
Kenneth O. Leland, Maximum modulus theorems for algebras of operator valued functions, Pacific J. Math. 40 (1972), 121–138. MR 310652, DOI 10.2140/pjm.1972.40.121
Kenneth O. Leland, Characterizations of solutions of $\Delta f=cf$, Studia Math. 29 (1967/68), 125–132. MR 222313, DOI 10.4064/sm-29-2-125-132

Normed rings

19

22

Gordon Thomas Whyburn, Topological analysis, Second, revised edition, Princeton Mathematical Series, No. 23, Princeton University Press, Princeton, N.J., 1964. MR 0165476, DOI 10.1515/9781400879335