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Convex and linear effect algebras. (English) Zbl 0956.46002
In the paper it is shown that convex effect algebras arise naturally in the description of physical statistical system. It is proved that a convex effect algebra \(P\) posseses a separating (resp. an order determining) set of states if and only if the corrsponding ordered linear space \((V,K)\) has no nonzero \(u\)-infinitesimals (resp. \((V,K)\) is an order unit space). Using these assertions it is shown that an effect algebra \(P\) is imbeddable in an interval \([\theta,u]\) of an order unit space if and only if the state space of \(P\) is order determining. An element \(a\) of effect algebra is sharp (resp. extremal) if \(a\wedge a'=0\) (resp. \(a=\lambda b\oplus (1-\lambda)c\) for \(\lambda\in (0,1)\) implies that \(b=c\)). Some properties of sharp and extremal elements are considered. One of the basic result is the following Theorem.
Suppose that the state space of \([\theta,u]\) is sharply determining.
(i) Every sharp element of \([\theta,u]\) is extremal and hence the sharp and extreme elements of \([\theta,u]\) coincide.
(ii) If \(a,b\in [\theta,u]\) are sharp and \(a\perp b\) then \(a\oplus b\) is sharp.
An alternative definition of a convex effect algebra called a \(CE\)-algebra is considered. It is shown that a convex effect algebra \(P\) is an \(MV\)-algebra if and only if \(P\) is lattice ordered.

MSC:
46A40 Ordered topological linear spaces, vector lattices
46B40 Ordered normed spaces
46N50 Applications of functional analysis in quantum physics
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