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Let AH be the real linear space of all Hermitian elements in A . Now AH = A+−A+; in a natural way AH is an ordered topological vector space with positive cone A+ (see [3], [6], [8]). The cone of all positive linear forms on AH will be denoted by P . Restriction of a positive linear form on A to AH gives an element ∗Work done as a student at the University of Groningen (the Netherlands). Received by the editors September 1995. Communicated by J. Schmets. 1991 Mathematics Subject Classification : 46L05, 46A40.

A quick and simple proof of Sherman's theorem on order in {$C\sp *$}-algebras