Linear Maps Between Certain Nonseparable C ∗ -Algebras

Abstract

There exists a noninjective commutative C*-algebra A such that every bounded linear map of any C*-algebra into A is decomposed as a linear combination of positive linear maps. 1. Introduction. A bounded linear map : A —> B between C*-algebras is said to be completely positive if every multiplicity map (h®idn: A®Mn —> B®Mn is positive, and completely bounded if supn|| idn|| < oo. Recently, Wittstock (14, Satz 4.5) proved that the linear span of completely positive maps of a unital C*-algebra into an injective C*-algebra is identical with the set of all completely bounded maps (see also (6)). We showed in (3) (resp. (2)) that given a separable C*-algebra B, if every bounded linear (resp. completely bounded) map of any C*- algebra into B is a linear combination of positive linear (resp. completely positive) maps, then B is finite-dimensional, namely injective. The purpose of this paper is to study the positive decomposability for bounded linear maps in the absence of the separability assumption. The main result is that there exists a nonstonean compact Hausdorff space T such that every bounded linear map of any C*-algebra into C(T), the C*-algebra of all continuous complex functions on T, is decomposed as a linear combination of positive linear maps. This result is used to answer negatively Smith's conjecture (8). In addition, we show that every bounded linear map of any partially ordered Banach space with normal positive cone into Cr(T), the Banach space of all continuous real functions on the space T, is decomposed as the difference of two positive bounded linear maps.