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BOUNDARIES FOR SPACES OF HOLOMORPHIC FUNCTIONS ON M-IDEALS IN THEIR BIDUALS
 

Summary: BOUNDARIES FOR SPACES OF HOLOMORPHIC FUNCTIONS ON
M-IDEALS IN THEIR BIDUALS
MARŽIA D. ACOSTA, RICHARD. M ARON, AND LUIZA A. MORAES
Abstract. For a complex Banach space X, let Au(BX) be the Banach algebra of
all complex valued functions defined on BX that are uniformly continuous on BX and
holomorphic on the interior of BX, and let Awu(BX ) be the Banach subalgebra consisting
of those functions in Au(BX) that are uniformly weakly continuous on BX. In this
paper we study a generalization of the notion of boundary for these algebras, originally
introduced by Globevnik. In particular, we characterize the boundaries of Awu(BX )
when the dual of X is separable. We exhibit some natural examples of Banach spaces
where this characterization provides concrete criteria for the boundary. We also show
that every non-reflexive Banach space X which is an M-ideal in its bidual cannot have
a minimal closed boundary for Au(BX).
1. Introduction
A classical result of Silov ([30], [31, Theorem 7.4] or [17, Theorem I.4.2]) states that if
K is a compact Hausdorff topological space and A is a unital and separating subalgebra
of C(K) then there is a minimal closed subset M K such that f = maxmM |f(m)|
for every f A. This set is known as the Silov boundary for A.
Five years after Silov's paper, Bishop ([11, Theorem 1]) proved that if A is a separating
Banach algebra of continuous functions on a compact metrizable space K, then K has a

  

Source: Aron, Richard M. - Department of Mathematics, Kent State University

 

Collections: Mathematics