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## State spaces of operator algebras: Basic theory, orientations, and C*-products, by

Citations: | 10 - 4 self |

### Citations

1277 |
Operator algebras and quantum statistical mechanics
- Bratteli, Robinson
- 1979
(Show Context)
Citation Context ...ly realized on different Hilbert spaces. Formally, this means that one considers multiple representations of the same operator algebra. Specifically, this happens (1) in quantum statistical mechanics =-=[4]-=- where different values of temperature typically give rise to inequivalent representations of a single operator algebra and (2) in quantum field theory on curved spacetimes [18] where the vacuum repre... |

688 | Quantum Field Theory on Curved Spacetime and Black Hole Thermodynamics.
- Wald
- 1994
(Show Context)
Citation Context ...m statistical mechanics [4] where different values of temperature typically give rise to inequivalent representations of a single operator algebra and (2) in quantum field theory on curved spacetimes =-=[18]-=- where the vacuum representations according to different observers are generally inequivalent. It seems to be fairly widely believed that Jordan algebras, more specifically JBalgebras, are really the ... |

335 |
Equivariant KK -theory and the Novikov conjecture,
- Kasparov
- 1988
(Show Context)
Citation Context ... has a naturally associated operator algebra! Moreover, operator algebra techniques have paid off handsomely with, for example, major applications to group representations [7], the Novikov conjecture =-=[12]-=-, Connes’ index theorem for foliations [5], and Jones’ work in knot theory [10]. In order to appreciate Alfsen and Shultz’s contribution, one needs to know a little about states and order. Every concr... |

249 |
A class of C∗-algebras and topological Markov chains
- Cuntz, Krieger
(Show Context)
Citation Context ...sentation on the Hilbert space L2 (G) generates both a C*-algebra C∗ (G) and a von Neumann algebra W ∗ (G). There are operator algebras naturally associated to foliated manifolds [5], directed graphs =-=[6]-=-, Euclidean Bruhat-Tits buildings [13], and Poisson manifolds [15]. It sometimes seems that almost every mathematical object has a naturally associated operator algebra! Moreover, operator algebra tec... |

237 | The Octonions.
- Baez
- 2002
(Show Context)
Citation Context ...vincing, and Jordan algebras turned out to be no help at all in dealing with field theoretic issues. Still, hopes of finding a profitable use of Jordan algebras in quantum mechanics persist (see e.g. =-=[3]-=-). The exposition in these two volumes is excellent, and the work they describe is certainly a tour de force, but the ultimate results characterizing state spaces seem difficult to apply because one i... |

148 |
Algebraic Methods in Statistical Mechanics and Quantum Field Theory ,
- Emch
- 1972
(Show Context)
Citation Context ...ems to be fairly widely believed that Jordan algebras, more specifically JBalgebras, are really the proper tool in this arena, although very few people actually use them. (A rare partial exception is =-=[8]-=-.) Alfsen and Shultz raise this point and emphasize the connection with physics repeatedly throughout their two books. Indeed, Jordan algebras were originally conceived as a model for the bounded obse... |

49 |
A survey of foliations and operator algebras”, Operator algebras and applications
- Connes
- 1982
(Show Context)
Citation Context ..., then its left representation on the Hilbert space L2 (G) generates both a C*-algebra C∗ (G) and a von Neumann algebra W ∗ (G). There are operator algebras naturally associated to foliated manifolds =-=[5]-=-, directed graphs [6], Euclidean Bruhat-Tits buildings [13], and Poisson manifolds [15]. It sometimes seems that almost every mathematical object has a naturally associated operator algebra! Moreover,... |

41 |
A representation theory for commutative topological algebra.
- Kadison
- 1951
(Show Context)
Citation Context ...eak* compact convex subset of the dual Banach space A ′, and for each self-adjoint element x ∈Awe have a continuous affine function ˆx : S(A) → R defined by ˆx(ρ) =ρ(x). Now an old theorem of Kadison =-=[11]-=- states that the map x ↦→ ˆx is an isometric isomorphism from Asa, the self-adjoint part of A, onto the space of all continuous affine functions from S(A) intoR. For von Neumann algebras one considers... |

38 |
Groups acting on buildings, operator K-theory, and Novikov's conjecture,
- KASPAROV, SKANDALIS
- 1991
(Show Context)
Citation Context ... generates both a C*-algebra C∗ (G) and a von Neumann algebra W ∗ (G). There are operator algebras naturally associated to foliated manifolds [5], directed graphs [6], Euclidean Bruhat-Tits buildings =-=[13]-=-, and Poisson manifolds [15]. It sometimes seems that almost every mathematical object has a naturally associated operator algebra! Moreover, operator algebra techniques have paid off handsomely with,... |

8 |
State spaces of Jordan algebras
- Alfsen, Shultz
- 1978
(Show Context)
Citation Context ...st important auxiliary objects associated with an operator algebra is its state space. The two books under review describe the authors’ solutions, obtained together with H. Hanche-Olsen and B. Iochum =-=[1]-=-, [2], [9], to the problems: What data must be added to a state space so that the operator algebra can be recovered? and Which convex sets can arise as state spaces? As that work is now around twenty ... |

6 |
Quantization as a functor, Quantization, Poisson brackets and beyond
- Landsman
- 2002
(Show Context)
Citation Context ... C∗ (G) and a von Neumann algebra W ∗ (G). There are operator algebras naturally associated to foliated manifolds [5], directed graphs [6], Euclidean Bruhat-Tits buildings [13], and Poisson manifolds =-=[15]-=-. It sometimes seems that almost every mathematical object has a naturally associated operator algebra! Moreover, operator algebra techniques have paid off handsomely with, for example, major applicat... |

6 |
Continuous-trace C*-algebras not isomorphic to their opposite algebras
- Phillips
- 2001
(Show Context)
Citation Context ... in general. Indeed, granting that there exist C*-algebras which are not isomorphic to their “opposite” algebra obtained by reversing the order of the product (a surprisingly difficult fact; see e.g. =-=[16]-=-), it follows that one cannot hope to entirely recover A from S(A), since S(A) ∼ = S(Aop )alwaysholds. The extra structure that needs to be added to S(A) in order to fully determine the C*-algebra A i... |

4 |
State spaces of C-algebras
- Alfsen, Hanche-Olsen, et al.
- 1980
(Show Context)
Citation Context ...portant auxiliary objects associated with an operator algebra is its state space. The two books under review describe the authors’ solutions, obtained together with H. Hanche-Olsen and B. Iochum [1], =-=[2]-=-, [9], to the problems: What data must be added to a state space so that the operator algebra can be recovered? and Which convex sets can arise as state spaces? As that work is now around twenty years... |

2 |
Anneaux d’opérateurs et représentations des groupes
- Dixmier
- 1995
(Show Context)
Citation Context ...st every mathematical object has a naturally associated operator algebra! Moreover, operator algebra techniques have paid off handsomely with, for example, major applications to group representations =-=[7]-=-, the Novikov conjecture [12], Connes’ index theorem for foliations [5], and Jones’ work in knot theory [10]. In order to appreciate Alfsen and Shultz’s contribution, one needs to know a little about ... |

2 |
Normal state spaces of Jordan and von Neumann algebras
- Iochum, Shultz
- 1983
(Show Context)
Citation Context ...nt auxiliary objects associated with an operator algebra is its state space. The two books under review describe the authors’ solutions, obtained together with H. Hanche-Olsen and B. Iochum [1], [2], =-=[9]-=-, to the problems: What data must be added to a state space so that the operator algebra can be recovered? and Which convex sets can arise as state spaces? As that work is now around twenty years old,... |

2 |
structures in analysis, in: “Jordan algebras” (Oberwolfach
- Rodŕıguez-Palacios, Jordan
- 1992
(Show Context)
Citation Context ...z et al. is the best example of an application of JB-algebras to the theory of operator algebras, and they are also useful in the study of infinite-dimensional complex domains [14] and in other areas =-=[17]-=-. I mentioned earlier that operator algebras have important connections to physics; let me come back to that now. In quantum mechanics a physical system is modelled by a Hilbert space, with states of ... |

1 |
algebras and holomorphy, Functional Analysis, Holomorphy, and Approximation Theory
- Kaup, Jordan
- 1978
(Show Context)
Citation Context ...work of Alfsen and Shultz et al. is the best example of an application of JB-algebras to the theory of operator algebras, and they are also useful in the study of infinite-dimensional complex domains =-=[14]-=- and in other areas [17]. I mentioned earlier that operator algebras have important connections to physics; let me come back to that now. In quantum mechanics a physical system is modelled by a Hilber... |