The MackeyGleason problem
Authors:
L. J. Bunce and J. D. Maitland Wright
Journal:
Bull. Amer. Math. Soc. 26 (1992), 288293
MSC (2000):
Primary 46L50
MathSciNet review:
1121569
Fulltext PDF
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Let A be a von Neumann algebra with no direct summand of Type , and let be its lattice of projections. Let X be a Banach space. Let be a bounded function such that 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 complexvalued finitely additive quantum measure on has a unique extension to a bounded linear functional on A.
 [1]
Johan
F. Aarnes, Quasistates on
𝐶*algebras, Trans. Amer. Math.
Soc. 149 (1970),
601–625. MR 0282602
(43 #8311), http://dx.doi.org/10.1090/S00029947197002826024
 [2]
L.
J. Bunce and J.
D. Maitland Wright, Complex measures on projections in von Neumann
algebras, J. London Math. Soc. (2) 46 (1992),
no. 2, 269–279. MR 1182483
(93j:46069), http://dx.doi.org/10.1112/jlms/s246.2.269
 [3]
L.
J. Bunce and J.
D. Maitland Wright, Continuity and linear extensions of quantum
measures on Jordan operator algebras, Math. Scand. 64
(1989), no. 2, 300–306. MR 1037464
(91f:46096)
 [4]
, The MackeyGleason problem for vector measures on projections in a von Neumann algebra, submitted.
 [5]
L.
J. Bunce and J.
D. Maitland Wright, Quantum logic, state space geometry and
operator algebras, Comm. Math. Phys. 96 (1984),
no. 3, 345–348. MR 769351
(86e:81017)
 [6]
L.
J. Bunce and J.
D. Maitland Wright, Quantum measures and states on Jordan
algebras, Comm. Math. Phys. 98 (1985), no. 2,
187–202. MR
786572 (86k:46101)
 [7]
Erik
Christensen, Measures on projections and physical states,
Comm. Math. Phys. 86 (1982), no. 4, 529–538. MR 679201
(85b:46072)
 [8]
Roger
Cooke, Michael
Keane, and William
Moran, An elementary proof of Gleason’s theorem, Math.
Proc. Cambridge Philos. Soc. 98 (1985), no. 1,
117–128. MR
789726 (86h:46098), http://dx.doi.org/10.1017/S0305004100063313
 [9]
Andrew
M. Gleason, Measures on the closed subspaces of a Hilbert
space, J. Math. Mech. 6 (1957), 885–893. MR 0096113
(20 #2609)
 [10]
J.
Gunson, Physical states on quantum logics. I, Ann. Inst. H.
Poincaré Sect. A (N.S.) 17 (1972), 295–311
(English, with French summary). MR 0336364
(49 #1139)
 [11]
G. W. Mackey, The mathematical foundations of quantum mechanics, Benjamin, 1963.
 [12]
Shûichirô
Maeda, Probability measures on projections in von Neumann
algebras, Rev. Math. Phys. 1 (1989), no. 23,
235–290. MR 1070091
(92m:46100), http://dx.doi.org/10.1142/S0129055X89000122
 [13]
Adam
Paszkiewicz, Measures on projections in 𝑊*factors, J.
Funct. Anal. 62 (1985), no. 1, 87–117. MR 790772
(86m:46060), http://dx.doi.org/10.1016/00221236(85)900217
 [14]
Masamichi
Takesaki, Theory of operator algebras. I, SpringerVerlag, New
YorkHeidelberg, 1979. MR 548728
(81e:46038)
 [15]
F.
J. Yeadon, Finitely additive measures on projections in finite
𝑊*algebras, Bull. London Math. Soc. 16
(1984), no. 2, 145–150. MR 737242
(85i:46087), http://dx.doi.org/10.1112/blms/16.2.145
 [16]
F.
J. Yeadon, Measures on projections in 𝑊*algebras of type
𝐼𝐼₁, Bull. London Math. Soc.
15 (1983), no. 2, 139–145. MR 689246
(84g:46089), http://dx.doi.org/10.1112/blms/15.2.139
 [1]
 J. F. Aarnes, Quasistates on algebras, Trans. Amer. Math. Soc. 149 (1970), 601625. MR 0282602 (43:8311)
 [2]
 L. J. Bunce and J. D. M. Wright, Complex measures on projections in von Neumann algebras, J. London Math. Soc. (2) (to appear). MR 1182483 (93j:46069)
 [3]
 , Continuity and linear extensions of quantum measures on Jordan operator algebras, Math. Scand. 64 (1989), 300306). MR 1037464 (91f:46096)
 [4]
 , The MackeyGleason problem for vector measures on projections in a von Neumann algebra, submitted.
 [5]
 , Quantum logic, state space geometry and operator algebras, Comm. Math. Phys. 96 (1984), 345348. MR 769351 (86e:81017)
 [6]
 , Quantum measures and states on Jordan algebras, Comm. Maths. Phys. 98 (1985), 187202. MR 786572 (86k:46101)
 [7]
 E. Christensen, Measures on projections and physical states, Comm. Math. Phys. 86 (1982), 529538. MR 679201 (85b:46072)
 [8]
 R. Cooke, M. Keane, and W. Moran, An elementary proof of Gleason's Theorem, Math. Proc. Cambridge Philos. Soc. 98 (1985), 117128. MR 789726 (86h:46098)
 [9]
 A. M. Gleason, Measures on the closed subspaces of a Hilbert space, J. Math. Mech. 6 (1957), 885893. MR 0096113 (20:2609)
 [10]
 J. Gunson, Physical states on quantum logics I, Ann. Inst. H. Poincaré 17 (1972), 295311. MR 0336364 (49:1139)
 [11]
 G. W. Mackey, The mathematical foundations of quantum mechanics, Benjamin, 1963.
 [12]
 S. Maeda, Probability measures on projections in von Neumann algebras, Reviews in Mathematical Physics 1 (1990), 235290. MR 1070091 (92m:46100)
 [13]
 A. Paszkiewicz, Measures on projections in factors, J. Funct. Anal. 62 (1985), 87117. MR 790772 (86m:46060)
 [14]
 M. Takesaki, Theory of operator algebras, Springer, 1979. MR 548728 (81e:46038)
 [15]
 F. J. Yeadon, Finitely additive measures on projections in finite algebras, Bull. London Math. Soc. 16 (1984), 145150. MR 737242 (85i:46087)
 [16]
 , Measures on projections in algebras of Type , Bull. London Math. Soc. 15 (1983), 139145. MR 689246 (84g:46089)
Similar Articles
Retrieve articles in Bulletin of the American Mathematical Society
with MSC (2000):
46L50
Retrieve articles in all journals
with MSC (2000):
46L50
Additional Information
DOI:
http://dx.doi.org/10.1090/S027309791992002744
PII:
S 02730979(1992)002744
Article copyright:
© Copyright 1992
American Mathematical Society
