Linear Maps betweenC*-Algebras Whose Adjoints Preserve Extreme Points of the Dual Ball

Abstract

We give a structural characterization of linear operators from one C*-algebra into another whose adjoints map extreme points of the dual ball onto extreme points. We show that up to a V-isomorphism, such a map admits of a decomposition into a degenerate and a non-degenerate part, the non-degenerate part of which appears as a Jordan V-morphism followed by a``rotation'' and then a reduction. In the case of maps whose adjoints preserve pure states, the degenerate part does not appear, and the``rotation'' is but the identity. In this context the results concerning such pure state preserving maps depend on and complement those of Sto% rmer [1963, Acta Math. 110, 233 278, 5.6 and 5.7]. In conclusion we consider the action of maps with``extreme point preserving'' adjoints on some specific C*-algebras. 1998Academic Press

Nice surjections on spaces of operators

“…Our results imply automatic continuity of these maps with respect to other topologies on spaces of operators. We also formulate the corresponding result for L (X,Y ) thereby proving an analogue of the result from [9] for L p (1 < p = 2 < ∞) spaces. We also formulate results when nice operators are not of the canonical form, extending and correcting the results from [8].…”

“…Motivated by a description due to Labuschagne and Mascioni [9] of such maps for the space of compact operators on a Hilbert space, in this article we consider a description of nice surjections on K (X,Y ) for Banach spaces X,Y . We give necessary and sufficient conditions when nice surjections are given by composition operators.…”

“…A linear map T : X → Y is said to be nice if y * • T is an extreme point of the unit ball of X * for every extreme point y * of the unit ball of Y * . It is easy to see that such an operator is of norm one (see [9,16]). Such operators between Banach spaces have been well studied in the literature (see [1,15] and the references therein, and also the more recent article [11]).…”

“…Here Λ ⊗ y * denotes the functional (Λ ⊗ y * )(T ) = Λ(T * (y * )). The main aim of this paper is to formulate and prove an abstract analogue of part A of Theorem 16 of [9] that completely describes nice operators between spaces of compact operators on Hilbert spaces, for nice surjections. Our results are valid for a class of Banach spaces that include the L p -spaces for 1 < p = 2 < ∞.…”

“…In §3 which has the main result, we deal with an analogue of part B of Theorem 16 [9]. We first formulate conditions similar to operators preserving ultra-weakly continuous extreme points on L (H) and show that for certain reflexive Banach spaces X, Y , weak * -continuous-extreme point-preserving surjections on L (X,Y ) are given by composition operators.…”