Geometry and analytic boundaries of Marcinkiewicz sequence spaces

Abstract

We investigate the geometric structure of the unit ball of the Marcinkiewicz sequence space Graphic, giving characterizations of its real and complex extreme points and of the exposed points in terms of the symbol Ψ. Using our knowledge of the geometry of Graphic we then give necessary and sufficient conditions for a subset of Graphic to be a boundary for Graphic, the algebra of functions which are uniformly continuous on Graphic and holomorphic on the interior of Graphic. We show that it is possible for the set of peak points of Graphic to be a boundary for Graphic yet for Graphic not to have a Šilov boundary in the sense of Globevnik.