Quasi-states and quasi-measures.

*(English)*Zbl 0744.46052The 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.

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 |

##### Keywords:

nonlinearity of the quasi-states on commutative \(C^*\)-algebras; self- adjoint elements; quasi-measures; Riesz representation theorem
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##### References:

[1] | Aarnes, J.F, Physical states on a C∗-algebra, Acta math., 122, 161-172, (1969) · Zbl 0183.14203 |

[2] | Aarnes, J.F, Quasi-states on C∗-algebras, Trans. amer. math. soc., 149, 601-625, (1970) · Zbl 0212.15403 |

[3] | Akemann, C.A; Newberger, S.M, Physical states on a C∗-algebra, (), 500 · Zbl 0272.46037 |

[4] | Bunce, L.J; Wright, J.D.Maitland, Quantum measures and states on Jordan algebras, Comm. math. phys., 98, 187-202, (1985) · Zbl 0579.46049 |

[5] | Christensen, E, Measures on projections and physical states, Comm. math. phys., 86, 529-538, (1982) · Zbl 0507.46052 |

[6] | Christenson, C.O; Voxman, W.L, Aspects of topology, (1977), Dekker New York · Zbl 0347.54001 |

[7] | Dieudonné, J, Foundations of modern analysis, (1960), Academic Press New York/London · Zbl 0100.04201 |

[8] | Gleason, A.M, Measures on closed subspaces of a Hilbert space, J. math. mech., 6, 885-893, (1957) · Zbl 0078.28803 |

[9] | Kadison, R.V, Transformation of states in operator theory and dynamics, Topology, 3, 177-198, (1965) · Zbl 0129.08705 |

[10] | Mackey, G.W, Quantum mechanics and Hilbert space, Amer. math. monthly, 64, 45-57, (1957) · Zbl 0137.23805 |

[11] | Mackey, G.W, The mathematical foundations of quantum mechanics, (1963), Benjamin New York · Zbl 0114.44002 |

[12] | Rudin, W, Real and complex analysis, (1987), McGraw-Hill New York · Zbl 0925.00005 |

[13] | Yeadon, F.J, Measures on projections in W∗-algebras of type II, Bull. London math. soc., 15, 139-145, (1983) · Zbl 0522.46045 |

[14] | Yeadon, F.J, Finitely additive measures on projections in finite W∗-algebras, Bull. London math. soc., 16, 145-150, (1984) · Zbl 0574.46048 |

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