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Quasi-states and quasi-measures. (English) Zbl 0744.46052
The subject of the paper is the nonlinearity of the quasi-states on commutative $$C^*$$-algebras $$C(X)$$. A complex-valued function $$\rho$$ on a unital $$C^*$$-algebra $$A$$ is called a quasi-state if it is a state on each unital $$C^*$$-subalgebra generated by a selfadjoint element and if $$\rho(a+ib)=\rho(a)+i\rho(b)$$ for every self-adjoint elements $$a$$ and $$b$$.
The author first establishes the correspondence between the quasi-states on $$C(X)$$ and the so-called quasi-measures on $$X$$ with total mass 1, where a quasi-measure is defined on the class of compact sets and that of open sets in $$X$$. For linear states, this correspondence reduces to the familiar Riesz representation theorem. Therefore, to construct a nonlinear quasi-state on $$C(X)$$ amounts to constructing a quasi-measure which is not (the restriction of) a regular Borel measur on $$X$$. Indeed, such a quasi-measure is constructed given that $$X$$ is the unit square in the plane.
Reviewer: C.-h.Chu (London)

##### MSC:
 46L30 States of selfadjoint operator algebras 46J05 General theory of commutative topological algebras 46L05 General theory of $$C^*$$-algebras
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