## The change-making problem (1975)

Venue: | J. Assoc. Comput. Mach |

Citations: | 15 - 0 self |

### BibTeX

@ARTICLE{Bunce75thechange-making,

author = {L. J. Bunce and J. D. Maitl},

title = {The change-making problem},

journal = {J. Assoc. Comput. Mach},

year = {1975},

pages = {125--128}

}

### OpenURL

### Abstract

Abstract. Let A be a von Neumann algebra with no direct summand of Type I2, and let P(A) be its lattice of projections. Let X be a Banach space. Let m: P(A) → X be a bounded function such that m(p + q) = m(p) + m(q) whenever p and q are orthogonal projections. The main theorem states that m has a unique extension to a bounded linear operator from A to X. In particular, each bounded complex-valued finitely additive quantum measure on P(A) has a unique extension to a bounded linear functional on A. Physical background In von Neumann’s approach to the mathematical foundations of quantum mechanics, the bounded observables of a physical system are identified with a real linear space, L, of bounded selfadjoint operators on a Hilbert space H. It is reasonable to assume that L is closed in the weak operator topology and that whenever x ∈ L then x 2 ∈ L. (Thus L is a Jordan algebra and contains spectral projections.) Then the projections in L form a complete orthomodular lattice, P, otherwise known as the lattice of “questions ” or the quantum logic of the physical system. A quantum measure is a map µ: P → R such that whenever p and q are orthogonal projections µ(p + q) = µ(p) + µ(q). In Mackey’s formulation of quantum mechanics [11] his Axiom VII makes the assumption that L = L(H)sa. Mackey states, that in contrast to his other axioms, Axiom VII has no physical justification; it is adopted for mathematical convenience. One of the technical advantages of this axiom was that, by Gleason’s Theorem, a completely additive positive quantum measure on the projections of L(H) is the restriction of a bounded linear functional (provided H is not two-dimensional). In order to weaken Axiom VII it was desirable to strengthen Gleason’s Theorem.

### Citations

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165 |
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(Show Context)
Citation Context ...eorem B was established by Christensen [7] for properly infinite algebras and algebras of Type In and by Yeadon [15, 16] for algebras of finite type. The first major progress had been made by Gleason =-=[9]-=-, who, by using an ingenious geometric argument, settled the question for positive completely additive measures on the projections of L(H). (See [8] for an elementary proof of this deep result.) Aarne... |

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(Show Context)
Citation Context ...he quantum logic of the physical system. A quantum measure is a map µ: P → R such that whenever p and q are orthogonal projections µ(p + q) = µ(p) + µ(q). In Mackey’s formulation of quantum mechanics =-=[11]-=- his Axiom VII makes the assumption that L = L(H)sa. Mackey states, that in contrast to his other axioms, Axiom VII has no physical justification; it is adopted for mathematical convenience. One of th... |

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(Show Context)
Citation Context .... The first major progress had been made by Gleason [9], who, by using an ingenious geometric argument, settled the question for positive completely additive measures on the projections of L(H). (See =-=[8]-=- for an elementary proof of this deep result.) Aarnes [1] and Gunson [10] made important contributions, especially concerning continuity properties. Paszkiewicz [13], working independently of Christen... |

13 |
Probability measures on projections in von Neumann algebras
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- 1989
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Citation Context ...to extend his results to nonfactorial von Neumann algebras, he requires µ to be positive and countably additive. A lucid and meticulous exposition of Theorem B for positive measures is given by Maeda =-=[12]-=-. 1. Vector measures The following short argument shows that Theorem A is a consequence of Theorem B. Lemma 1.1. Let A be a von Neumann algebra such that each finitely additive (complex) measure on P(... |

6 |
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(Show Context)
Citation Context ...rem. Theorem B answers a natural question first posed by G. W. Mackey some thirty years ago. When µ is positive, that is, when µ(e) ≥ 0 for each projection e. Theorem B was established by Christensen =-=[7]-=- for properly infinite algebras and algebras of Type In and by Yeadon [15, 16] for algebras of finite type. The first major progress had been made by Gleason [9], who, by using an ingenious geometric ... |

2 |
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Citation Context ...e thirty years ago. When µ is positive, that is, when µ(e) ≥ 0 for each projection e. Theorem B was established by Christensen [7] for properly infinite algebras and algebras of Type In and by Yeadon =-=[15, 16]-=- for algebras of finite type. The first major progress had been made by Gleason [9], who, by using an ingenious geometric argument, settled the question for positive completely additive measures on th... |

1 |
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Citation Context ...who, by using an ingenious geometric argument, settled the question for positive completely additive measures on the projections of L(H). (See [8] for an elementary proof of this deep result.) Aarnes =-=[1]-=- and Gunson [10] made important contributions, especially concerning continuity properties. Paszkiewicz [13], working independently of Christensen and Yeadon, established Theorem B for σ-finite factor... |

1 |
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(Show Context)
Citation Context ...D, then either B ⊂ C ⊂ A, where C ≈ M4(C) or B ⊂ C ⊕ D, where C ≃ M4(C) and D ⊂ E ⊂ A with E ≈ M3(C). Hence µ is linear on B. By “patching” together Type I2 factorial subalgebras of A it can be shown =-=[2]-=-, building on techniques of Christensen [7], that µ is uniformly continuous on P(A) and µ is linear on each Type In subalgebra of A. In particular, µ is linear on W(1, p, q), where W(1, p, q) is the W... |

1 |
and linear extensions of quantum measures on Jordan operator algebras
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(Show Context)
Citation Context ...such that ∑ ∞ m+1 2−n < ε. Let e be a projection such that (1) |µ(p) − µ(epe) − µ((1 − e)p(1 − e))| < ε/m for each p ∈ P(A) ; (2) |µ(a + b) − µ(a) − µ(b)| < ε/m whenever a ≥ 0, b ≥ 0, and a + b ≤ e ; =-=(3)-=- |µ(c + d) − µ(c) − µ(d)| < ε/m whenever c ≥ 0, d ≥ 0, and c + d ≤ 1 − e. Then, whenever x ≥ 0, y ≥ 0, and x + y ≤ 1, |µ(x + y) − µ(x) − µ(y)| < 20ε. We shall now sketch a proof of Theorem B for A a p... |

1 | logic, state space geometry and operator algebras - Quantum - 1984 |

1 | measures and states on Jordan algebras - Quantum - 1985 |

1 |
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- 1972
(Show Context)
Citation Context ...n ingenious geometric argument, settled the question for positive completely additive measures on the projections of L(H). (See [8] for an elementary proof of this deep result.) Aarnes [1] and Gunson =-=[10]-=- made important contributions, especially concerning continuity properties. Paszkiewicz [13], working independently of Christensen and Yeadon, established Theorem B for σ-finite factors, however, to e... |

1 |
Measures on projections in
- Paszkiewicz
- 1985
(Show Context)
Citation Context ...es on the projections of L(H). (See [8] for an elementary proof of this deep result.) Aarnes [1] and Gunson [10] made important contributions, especially concerning continuity properties. Paszkiewicz =-=[13]-=-, working independently of Christensen and Yeadon, established Theorem B for σ-finite factors, however, to extend his results to nonfactorial von Neumann algebras, he requires µ to be positive and cou... |

1 |
on projections in W ∗ -algebras of Type II1
- Measures
- 1983
(Show Context)
Citation Context ...e thirty years ago. When µ is positive, that is, when µ(e) ≥ 0 for each projection e. Theorem B was established by Christensen [7] for properly infinite algebras and algebras of Type In and by Yeadon =-=[15, 16]-=- for algebras of finite type. The first major progress had been made by Gleason [9], who, by using an ingenious geometric argument, settled the question for positive completely additive measures on th... |