A note on maximum modulus algebras
Pentti K. Jarvi · Annales Academiae Scientiarum Fennicae Series A I Mathematica · 1981
PENTTT rÄnvr 1.Let X be alocally compact Hausdorff space and let A be an algebra of com- plex-valued continuous functions on X.Then ,4 is called a (local) maximum modulus algebra on X, provided that for each compact subset 1( of X with topological bound- ary 0K, and for each f in,4, we have (x) lf@)l = max {lf(Dlip€\K} for Po(K (see [9], [11]).One of themain results of [9]reads asfollows ([9, Theorem2]). Theorem A. If A is a maximum modulus algebra on a plane domain G, and ifA contqins aftmction which is analytic and not constant in G, then euery member of A is analytic in G.Recently, Bear and Hile ([2, Theorem 4]) gave the following extension of Theorem A.[2]