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## Characterisations of the weak expectation property (2013)

Citations: | 1 - 0 self |

### Citations

362 |
Operator Spaces
- Effros, Ruan
- 2000
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Citation Context ...called the operator system structure of S. Every complex ∗-vector space equipped with a family of matricial cones and an order unit satisfying natural axioms can, by virtue of the Choi-Effros Theorem =-=[5]-=-, be represented faithfully as an operator system acting on some Hilbert space. When a particular embedding is not specified, the order unit of an operator system will be denoted by 1. A map φ : S → T... |

239 |
Completely bounded maps and operator algebras, Cambridge
- Paulsen
- 2002
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Citation Context ... bijection φ : S → T CHARACTERISATIONS OF THE WEAK EXPECTATION PROPERTY 3 of operator systems S and T is a complete order isomorphism if both φ and φ−1 are completely positive. We refer the reader to =-=[25]-=- for further properties of operator systems and completely positive maps. An operator system tensor product S⊗τ T of operator systems S and T is an operator system structure on the algebraic tensor pr... |

210 |
Introduction to Operator Space Theory
- Pisier
- 2003
(Show Context)
Citation Context ... Proof. The fact that C∗(Fn) and C ∗(SL2(Z)) have the lifting property (LP) for all n ∈ N ∪ {∞} was established by Kirchberg [19]. There are alternate proofs for the assertion that C∗(Fn) has LP: see =-=[23, 28]-=-, for example. Suppose that φ : C∗(Z2) → B/J is a unital completely positive map, where B is a unital C*-algebra and J ⊆ B is a closed ideal. Let b ∈ B be a selfadjoint contractive lifting of φ(h), wh... |

129 |
On non-semisplit extensions, tensor products and exactness of group C∗- algebras
- Kirchberg
- 1993
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Citation Context ...=1Z2), where ∗ n j=1Z2 is the n-fold free product of n copies of Z2, n ∈ N. Proof. The fact that C∗(Fn) and C ∗(SL2(Z)) have the lifting property (LP) for all n ∈ N ∪ {∞} was established by Kirchberg =-=[19]-=-. There are alternate proofs for the assertion that C∗(Fn) has LP: see [23, 28], for example. Suppose that φ : C∗(Z2) → B/J is a unital completely positive map, where B is a unital C*-algebra and J ⊆ ... |

42 | Tensor products of operator algebras - Effros, Lance - 1977 |

42 | About the QWEP conjecture
- Ozawa
(Show Context)
Citation Context ... Proof. The fact that C∗(Fn) and C ∗(SL2(Z)) have the lifting property (LP) for all n ∈ N ∪ {∞} was established by Kirchberg [19]. There are alternate proofs for the assertion that C∗(Fn) has LP: see =-=[23, 28]-=-, for example. Suppose that φ : C∗(Z2) → B/J is a unital completely positive map, where B is a unital C*-algebra and J ⊆ B is a closed ideal. Let b ∈ B be a selfadjoint contractive lifting of φ(h), wh... |

39 |
On nuclear C∗-algebras
- Lance
- 1973
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Citation Context ... on WEP, these results give new formulations of Connes’ embedding problem. 1. Introduction In this paper we deduce various characterisations of Lance’s weak expectation property (WEP) for C*-algebras =-=[21]-=-. Lance’s original definition of WEP requires that every faithful representation of the C*-algebra possesses a so-called weak expectation or, equivalently, that the universal representation, which is ... |

32 | Commutants of unitaries in UHF algebras and functorial properties of exactness - Kirchberg - 1994 |

29 |
The full C∗-algebra of the free group on two generators
- Choi
- 1980
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Citation Context ...sition property in B. We will next formulate the Connes Embedding Problem in terms of the Riesz decomposition property. Since C∗(F2) is a residually finite dimensional (for brevity, RFD) C*-algebra =-=[3]-=-, there is a C*-algebraic embedding C∗(F2) →֒ ∞∏ k=1 Mn(k), for some sequence (n(k))k∈N of natural numbers. Theorem 6.9. Connes’ embedding problem has an affirmative solution if and only if C∗(F2) has... |

23 | C ∗ -algebras by Example, Fields Institute Monographs - Davidson - 1996 |

21 |
Free products of completely positive maps and spectral sets
- Boca
- 1991
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Citation Context ...tal C*-algebras, we denote by A1∗A2 the free product C*-algebra, amalgamated over the unit. The same notation is used for free products of groups. The following result, which combines results of Boca =-=[2]-=- and Pisier [28, Theorem 1.11], will be useful for us in the sequel. Theorem 3.1. Let A1, . . . ,An be unital C*-algebras and ϕi : Ai → B(H) be unital completely positive maps, i = 1, . . . , n. Then ... |

20 | A simple proof of a theorem of Kirchberg and related results on - Pisier - 1996 |

18 | On Gromov–Hausdorff convergence for operator metric spaces
- Kerr, Li
(Show Context)
Citation Context ... these two representations to give two more characterisations of WEP. We first recall some basic facts about coproducts of operator systems. Coproducts in this category were used by D. Kerr and H. Li =-=[18]-=-, where the authors described the amalgamation process over a joint operator subsystem. T. Fritz demonstrated some applications of this concept in quantum information theory [11]. A categorical treatm... |

16 | Nuclear C∗-algebras and injectivity: the general case - Choi, Effros - 1977 |

16 | Operator system structures on ordered spaces
- Paulsen, Todorov, et al.
(Show Context)
Citation Context ... system arising from the natural inclusion of S ⊗ T into B(H⊗K). (b) The maximal tensor product max. For each n ∈ N, let Dn = {A ∗(P ⊗ Q)A : A ∈Mn,km(C), P ∈Mk(S)+, Q ∈Mm(T )+}. The Archimedanisation =-=[26]-=- of the family (Dn)n∈N of cones is an operator system structure on S⊗T ; the corresponding operator system is denoted by S ⊗max T . (c) The commuting tensor product c. By definition, an element X ∈ Mn... |

15 |
Operator system quotients of matrix algebras and their tensor products
- Farenick, Paulsen
- 2012
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Citation Context ...C(n)⊗min NC(m) = NC(n)⊗max NC(m). We conclude this section with another realisation of NC(n) as a quotient of a matrix operator system, which leads to a different characterisation of WEP. Following =-=[10]-=-, let Wn = span{uiu ∗ j : i, j = 0, 1, . . . , n} ⊆ C ∗(Fn), where we have set u0 = 1. Let β : Mn+1 →Wn be the linear map given by β(Ei,j) = 1 n+1u ∗ i uj, i, j = 0, . . . , n. It follows from [10] th... |

15 |
quelques notions fondamentales dans la théorie générale des opérations linéaires
- Riesz, Sur
- 1940
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Citation Context ...rdered function space theory, or in general, in ordered topological lattice theory, the vast use of Riesz interpolation and decomposition properties dates back to F. Riesz’s studies in the late ’30’s =-=[29]-=-. The reader may refer to [1] and the bibliography therein for broad applications of this concept. We also refer to [15] for a noncommutative Riesz interpolation property that characterises WEP. In th... |

13 | Tensor Products of Operator Systems - Kavruk, Paulsen, et al. - 2011 |

12 | Tsirelson’s problem and Kirchberg’s conjecture.
- Fritz
- 2012
(Show Context)
Citation Context ... and C ∗(∗nj=1Z2) have the lifting property, which is stronger than the local lifting property. Moreover, SL2(Z) contains a copy of F2 (see, e.g., [19, p.486]) as does ∗ n j=1Z2 for n ≥ 3 (see, e.g., =-=[12]-=-). Thus, Proposition 3.5 applies to each of these groups. Remark. That the free product of n copies of Z2 detects WEP was also established by T. Fritz in [12] using a different method. Since SL3(Z) ... |

10 | Quotients, exactness and nuclearity in the operator system category, preprint
- Kavruk, Paulsen, et al.
(Show Context)
Citation Context ...uch that J = ker φ. If J ⊆ S is kernel, then one may endow the ∗-vector space S/J with an operator system structure such that the canonical quotient map qJ : S → S/J is unital and completely positive =-=[17]-=-. An element (xi,j+J ) is positive in Mn(S/J ) if and only if for every ǫ > 0, there exist elements yi,j ∈ J such that (xi,j + yi,j) + ǫ1n ∈ Mn(S)+. Moreover, if J ⊆ kerφ for some completely positive ... |

9 | Operator system structures on the unital direct sum of C*-algebras, arXiv:1011.1247
- Fritz
- 2010
(Show Context)
Citation Context ... by D. Kerr and H. Li [18], where the authors described the amalgamation process over a joint operator subsystem. T. Fritz demonstrated some applications of this concept in quantum information theory =-=[11]-=-. A categorical treatment and further results can be found in the thesis of the CHARACTERISATIONS OF THE WEAK EXPECTATION PROPERTY 17 second author [14]. We next extend the results from [11] and [14] ... |

7 |
Nuclearity related properties in operator systems, preprint arXiv:1107.2133
- Kavruk
- 2011
(Show Context)
Citation Context ... (xi,j + yi,j) + ǫ1n ∈ Mn(S)+. Moreover, if J ⊆ kerφ for some completely positive map φ : S → T , then there exists a completely positive map φ̇ : S/J → T such that φ = φ̇ ◦ qJ . A null subspace of S =-=[14]-=- is a subspace J which does not contain positive elements other than 0. It was shown in [14] that every null subspace is a kernel. Definition 2.2. A unital completely positive map φ : S → T is called ... |

6 | C∗-algebras with the weak expectation property and a multivariable analogue of Ando’s theorem on the numerical radius
- Farenick, Kavruk, et al.
(Show Context)
Citation Context ...epresentation. Thus, one is free to choose a preferred faithful representation of the C*-algebra to attempt to determine if it has WEP. These results expand on earlier work of the first three authors =-=[8]-=- and of the second author [15] that also obtained such representation-free characterisations of WEP. One major motivation for the desire to obtain such a plethora of characterisations of WEP are the r... |

6 | The weak expectation property and Riesz interpolation, arXiv 1201.5414
- Kavruk
(Show Context)
Citation Context ...ree to choose a preferred faithful representation of the C*-algebra to attempt to determine if it has WEP. These results expand on earlier work of the first three authors [8] and of the second author =-=[15]-=- that also obtained such representation-free characterisations of WEP. One major motivation for the desire to obtain such a plethora of characterisations of WEP are the results of Kirchberg, who prove... |

6 |
Tensor Products and Nuclear C∗-algebras
- Lance
- 1982
(Show Context)
Citation Context ...(ii) A has the complete 2-Riesz decomposition property in B(H); (iii) A has the complete n-Riesz decomposition property in B(H) for every n ∈ N. In contrast to the original definition of WEP given in =-=[22]-=-, the characterisation given in Theorem 6.4 only makes reference to a single concrete representation of A. Moreover, we shall see below that B(H) can be replaced by an arbitrary C*-algebra having WEP.... |

4 |
Operator systems from discrete groups, preprint
- Farenick, Kavruk, et al.
- 1209
(Show Context)
Citation Context ...llowing [17], we let Sn = span{1, ui, u ∗ i : 1 = 1, . . . , n} ⊆ C ∗(Fn), where u1, . . . , un are the generators of Fn viewed as elements of C ∗(Fn) and u−1i = u ∗ i , i = 1, . . . , n. We also let =-=[9]-=- NC(n) be the operator system NC(n) = span{1, hi : i = 1, . . . , n} ⊆ C ∗(∗nj=1Z2), where h1, . . . , hn are the canonical generators of ∗ n j=1Z2 (that is, hi is the non-trivial element of the ith c... |

4 | On maximal tensor products and quotient maps of operator systems - Han |

2 |
Cones and duality, volume 84
- Aliprantis, Tourky
- 2007
(Show Context)
Citation Context ...or in general, in ordered topological lattice theory, the vast use of Riesz interpolation and decomposition properties dates back to F. Riesz’s studies in the late ’30’s [29]. The reader may refer to =-=[1]-=- and the bibliography therein for broad applications of this concept. We also refer to [15] for a noncommutative Riesz interpolation property that characterises WEP. In this paper we give a characteri... |

2 |
About the Connes embedding conjecture
- Ozawa
(Show Context)
Citation Context ... system. We next provide a multivariable version of [9, Proposition 5.11]; that latter result identified NC(2) with a dual of a diagonal matrix operator system. This result can also be found in Ozawa =-=[24]-=-. Theorem 5.9. Let Rn,k = {(a 1 1, . . . , a 1 k, . . . , a n 1 , . . . , a n k) ∈ C nk : k∑ i=1 ali = k∑ i=1 ami , for all l,m}. Then R∗n,k ∼= ∐ni=1C k unitally and completely order isomorphically. I... |