## ON EXTREMAL POSITIVE MAPS ACTING BETWEEN TYPE I FACTORS (812)

### BibTeX

@MISC{812onextremal,

author = {},

title = {ON EXTREMAL POSITIVE MAPS ACTING BETWEEN TYPE I FACTORS},

year = {812}

}

### OpenURL

### Abstract

Abstract. The paper is devoted to the problem of classification of extremal positive maps acting between B(K) and B(H) where K and H are Hilbert spaces. It is shown that every positive map with the property that rank φ(P) ≤ 1 for any one-dimensional projection P is a rank 1 preserver. It allows to characterize all decomposable extremal maps as those which satisfy the above condition. Further, we prove that every extremal positive map which is 2-positive turns out to automatically completely positive. Finally we get the same conclusion for such extremal positive maps that rank φ(P) ≤ 1 for some one-dimensional projection P and satisfy the condition of local complete positivity. It allows us to give a negative answer for Robertson’s problem in some special cases. 1.

### Citations

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Operator Algebras and Quantum Statistical Mechanics I
- Bratteli, Robinson
- 1979
(Show Context)
Citation Context ...] = ∑ [ k i=1 ξi + ∑n j=k+1 αijξj, ] ηi = 0. Now, condition (iii) implies that φ is continuous on the subspace Bf(K) on K with respect to the strong topology. But Bf(K) is strongly dense in B(K) (cf. =-=[4]-=-), so φ can be uniquely extended to the whole B(K). □16 M. MARCINIAK 5. Structural results As it was mentioned if φ is a non-zero map then always there exists such a triple ξ, x and λ that the condit... |

34 |
Positive maps of low dimensional matrix algebras
- Woronowicz
- 1976
(Show Context)
Citation Context ... in cones P(B(C 2 ), C 2 ), P(B(C 2 ), C 3 ) and P(B(C 3 ), C 2 ) there are no other than the mentioned above extremal elements. This is a consequence of the results of Størmer ([25]) and Woronowicz (=-=[31]-=-) that these cones contain only decomposable maps. However, there are known some other examples of extremal positive maps between matrix algebras for greater dimensions (see [7, 10, 12]). Obviously, t... |

29 | Linear preserver problems
- Li, Pierce
- 2001
(Show Context)
Citation Context ...such that rankφ(P) = 1 for some one-dimensional projection P. The last statement before the above remark tell us to draw our attention to the theory of the so called linear preservers (for survey see =-=[3]-=-). In particular, we are interested in the problem of rank 1 preservers i.e. linear maps T : B(K) → B(H) such that rankφ(X) = 1 whenever rankX = 1 for X ∈ B(K). They are well described. By the result ... |

27 |
Positive linear maps of operator algebras
- Størmer
- 1963
(Show Context)
Citation Context ...erization of extremal elements in the cone of all positive maps. The explicit form of extremal positive unital maps is described fully only for the simplest non-trivial case of 2 ×2 complex matrices (=-=[25]-=-). Let us warn that in our paper we consider a larger class of all positive (i.e. not necessarily unital) maps. These both classes have a little bit different structures. Positive unital maps form a c... |

20 | A family of indecomposable positive linear maps based on entangled quantum states
- Terhal
(Show Context)
Citation Context ... + a11 −a23 −a31 −a32 a33 + a22 ⎤ ⎦ . (1.1) It was the first known example of a nondecomposable map. Let us mention also that in the literature several examples of nondecomposable maps are described (=-=[5, 8, 10, 12, 22, 23, 26, 29, 30]-=-). Although some conditions equivalent to decomposability are known ([27]), proving that a positive map is nondecomposable is a very difficultEXTREMAL POSITIVE MAPS 3 task. But it seems that providin... |

18 |
State spaces of operator algebras
- Alfsen, Shultz
- 2001
(Show Context)
Citation Context ...ly unital) maps. These both classes have a little bit different structures. Positive unital maps form a convex subset of the cone of all unital maps but it is not a base for this cone in the sense of =-=[1]-=-. As it was shown in [19] even in the case of 2×2 matrices the structure of extremal positive unital maps differs from the structure of extremal elements in the cone of all positive maps. On the other... |

17 |
Positive semidefinite biquadratic forms
- Choi
- 1975
(Show Context)
Citation Context ...[25]) and Woronowicz ([31]) that these cones contain only decomposable maps. However, there are known some other examples of extremal positive maps between matrix algebras for greater dimensions (see =-=[7, 10, 12]-=-). Obviously, they are necessarily nondecomposable. The most famous example is that which belongs to P(B(C 3 ), C 3 ) given by Choi in [7] ⎛⎡ φ⎝ ⎣ a11 a12 a13 a21 a22 a23 a31 a32 a33 ⎤⎞ ⎦⎠ = ⎡ ⎣ a11 +... |

17 | On a characterization of positive maps
- Majewski, Marciniak
(Show Context)
Citation Context ... and linear preservers theory. Our main motivation is to give answers for questions contained in Remarks 1.2 and 1.3. However, if it comes to our methods we will use the technique presented in papers =-=[14, 17, 18, 19, 20]-=-. The paper is organized as follows. In Section 2 we give an ’almost’ positive answer for the question from Remark 1.3 (Theorem 2.2). It will allow us to characterize decomposable extremal maps as tho... |

16 |
On inner products in linear metric spaces
- Jordan, Neumann
- 1935
(Show Context)
Citation Context ...parallelogram identity such that for any ξK µ(−ξ) = µ(ξ), µ(iξ) = µ(ξ) (4.2) Then there is a positive operator M on K such that for any ξ ∈ K µ(ξ) = 〈ξ, Mξ〉. (4.3) Proof. We apply main arguments from =-=[11]-=-. For the readers convenience we give the full proof. Firstly, define for ξ, η ∈ K (ξ, η)R = 1 (µ(ξ + η) − µ(η − ξ)) . (4.4) 4 It follows from (4.2) that (ξ, η)R = (η, ξ)R (4.5)14 M. MARCINIAK for an... |

16 |
Decomposable positive maps on C ∗ -algebras
- Størmer
- 1982
(Show Context)
Citation Context ...t us mention also that in the literature several examples of nondecomposable maps are described ([5, 8, 10, 12, 22, 23, 26, 29, 30]). Although some conditions equivalent to decomposability are known (=-=[27]-=-), proving that a positive map is nondecomposable is a very difficultEXTREMAL POSITIVE MAPS 3 task. But it seems that providing new examples of extremal maps is of extremal difficulty. Apart from the... |

14 |
Completely positive maps on complex matrices
- Choi
- 1975
(Show Context)
Citation Context ...from the structure of extremal elements in the cone of all positive maps. On the other hand, let us remind that all extremal elements of the cone of completely positive maps are fully recognized (see =-=[6, 2]-=-). If we consider maps from B(K) into B(H) where K and H are finite dimensional Hilbert spaces, then a map φ is extremal in the cone of completely positive maps if and only if φ(X) = AXA ∗ , X ∈ B(K),... |

11 |
Atomic positive linear maps in matrix algebras
- Ha
- 1998
(Show Context)
Citation Context ...[25]) and Woronowicz ([31]) that these cones contain only decomposable maps. However, there are known some other examples of extremal positive maps between matrix algebras for greater dimensions (see =-=[7, 10, 12]-=-). Obviously, they are necessarily nondecomposable. The most famous example is that which belongs to P(B(C 3 ), C 3 ) given by Choi in [7] ⎛⎡ φ⎝ ⎣ a11 a12 a13 a21 a22 a23 a31 a32 a33 ⎤⎞ ⎦⎠ = ⎡ ⎣ a11 +... |

11 |
Facial structures for positive linear maps between matrix algebras
- Kye
- 1996
(Show Context)
Citation Context ...S 19 For any unit vectors η ∈ K and y ∈ H let us define Fη,y = {φ ∈ P(B(K), H) : φ(ηη ∗ )y = 0} (5.17) One can easily check that it is a face of the cone of all positive maps. Let us recall that Kye (=-=[13]-=-) showed that each maximal face in the cone P(B(K), H) is of the above form for some η and y provided that K and H are finite dimensional. Observe that for any ξ and x we have Gξ,x = ⋂ y⊥x Fξ,y. Moreo... |

10 |
Maps on matrix spaces
- unknown authors
(Show Context)
Citation Context ...s B, C : K → H by Bη = λ −1/2 φ(ηξ ∗ )x, (3.2) Cη = λ −1/2 φ(ξη ∗ )x (3.3) where η ∈ K and let ψ and χ be maps from B(K) into B(H) determined by ψ(X) = BXB ∗ , (3.4) χ(X) = CX T C ∗ for X ∈ B(K) (cf. =-=[24]-=-). Then we have the following (3.5) Proposition 3.1. Assume that φ(X) = AXA ∗ (resp. φ(X) = AX T A ∗ ) for X ∈ B(K) where A ∈ B(K, H) is some non-zero operator. Let ξ, x and λ fulfil (3.1). Take the o... |

9 | On the structure of entanglement witnesses and new class of positive indecomposable maps, Open Sys
- Chru´sciński, Kossakowski
(Show Context)
Citation Context ... + a11 −a23 −a31 −a32 a33 + a22 ⎤ ⎦ . (1.1) It was the first known example of a nondecomposable map. Let us mention also that in the literature several examples of nondecomposable maps are described (=-=[5, 8, 10, 12, 22, 23, 26, 29, 30]-=-). Although some conditions equivalent to decomposability are known ([27]), proving that a positive map is nondecomposable is a very difficultEXTREMAL POSITIVE MAPS 3 task. But it seems that providin... |

8 |
Linear rank and corank preserving maps on B(H) and an application to *-semigroup isomorphisms of operator ideals, Linear Algebra Appl
- Győry, Molnár, et al.
(Show Context)
Citation Context ...rm φ(X) = AXA ∗ or φ(X) = AX T A ∗ then rankφ(P) ≤ 1 for every one-dimensional projection P on K. Motivated by the considerations from the above paragraph we can ask whether the converse is true (cf. =-=[9]-=- and references therein). The aim of this paper is to present a new approach to the problem of classification of extremal maps in P(B(K), H) which is based on point of view coming from the convex anal... |

8 |
Moyls Transformations on tensor product spaces
- Marcus, B
- 1959
(Show Context)
Citation Context ...are interested in the problem of rank 1 preservers i.e. linear maps T : B(K) → B(H) such that rankφ(X) = 1 whenever rankX = 1 for X ∈ B(K). They are well described. By the result of Marcus and Moyls (=-=[21]-=-) we know that each injective rank 1 preserver is of the form T(X) = MXN or T(X) = MX T N, X ∈ P(K), for some M ∈ B(K, H) and N ∈ B(H, K). Lim ([15]) proved that the similar4 M. MARCINIAK form follow... |

8 |
On positive linear maps between matrix algebras
- Tang
- 1986
(Show Context)
Citation Context ... + a11 −a23 −a31 −a32 a33 + a22 ⎤ ⎦ . (1.1) It was the first known example of a nondecomposable map. Let us mention also that in the literature several examples of nondecomposable maps are described (=-=[5, 8, 10, 12, 22, 23, 26, 29, 30]-=-). Although some conditions equivalent to decomposability are known ([27]), proving that a positive map is nondecomposable is a very difficultEXTREMAL POSITIVE MAPS 3 task. But it seems that providin... |

7 |
Indecomposable positive maps in low-dimensional matrix algebras
- Osaka
- 1991
(Show Context)
Citation Context |

5 |
k -Decomposability of positive maps, Quantum Probability and infinite dimensional analysis vol
- W, Majewski
(Show Context)
Citation Context ... and linear preservers theory. Our main motivation is to give answers for questions contained in Remarks 1.2 and 1.3. However, if it comes to our methods we will use the technique presented in papers =-=[14, 17, 18, 19, 20]-=-. The paper is organized as follows. In Section 2 we give an ’almost’ positive answer for the question from Remark 1.3 (Theorem 2.2). It will allow us to characterize decomposable extremal maps as tho... |

5 | Decomposability of extremal positive maps on M2(C
- Majewski, Marciniak
(Show Context)
Citation Context ...oth classes have a little bit different structures. Positive unital maps form a convex subset of the cone of all unital maps but it is not a base for this cone in the sense of [1]. As it was shown in =-=[19]-=- even in the case of 2×2 matrices the structure of extremal positive unital maps differs from the structure of extremal elements in the cone of all positive maps. On the other hand, let us remind that... |

4 |
Positive linear maps between matrix algebras which fix diagonals
- Kye
- 1995
(Show Context)
Citation Context ...[25]) and Woronowicz ([31]) that these cones contain only decomposable maps. However, there are known some other examples of extremal positive maps between matrix algebras for greater dimensions (see =-=[7, 10, 12]-=-). Obviously, they are necessarily nondecomposable. The most famous example is that which belongs to P(B(C 3 ), C 3 ) given by Choi in [7] ⎛⎡ φ⎝ ⎣ a11 a12 a13 a21 a22 a23 a31 a32 a33 ⎤⎞ ⎦⎠ = ⎡ ⎣ a11 +... |

4 |
Linear transformations of tensor spaces preserving decomposable vectors
- Lim
- 1975
(Show Context)
Citation Context ... well described. By the result of Marcus and Moyls ([21]) we know that each injective rank 1 preserver is of the form T(X) = MXN or T(X) = MX T N, X ∈ P(K), for some M ∈ B(K, H) and N ∈ B(H, K). Lim (=-=[15]-=-) proved that the similar4 M. MARCINIAK form follows from a weaker assumption. Namely, it is enough to assume that rankφ(X) ≤ 1 for every X ∈ B(K) such that rankX = 1. Remark 1.3. Observe that if a m... |

4 |
Indecomposable cones
- Loewy, Schneider
- 1975
(Show Context)
Citation Context ...chneider which goes in this direction. To this end we recall that a cone K is indecomposable if there are no non-empty subsets K1, K2 ⊂ K such that K = K1 + K2 and spanK1 ∩ spanK2 = {0}. Theorem 1.1 (=-=[16]-=-). Let V + be a cone in V and assume V + = hull(ExtV + ). Then the following conditions are equivalent: (1) V + is indecomposable. (2) If T ∈ L(V, V ) is such that kerT = {0} and T(ExtV + ) ⊆ ExtV + ,... |

4 |
Schwarz inequalities and the decomposition of positive maps on C ∗ -algebras
- Robertson
- 1983
(Show Context)
Citation Context |

4 |
Decomposition of positive projections on C*-algebras
- Størmer
- 1980
(Show Context)
Citation Context |

4 |
Extreme positive operators on convex cones
- Tam
- 1995
(Show Context)
Citation Context ...s of extremal difficulty. Apart from the mentioned above results there is another line of research in the mathematical literature which deals with similar problems. It comes from convex analysis (see =-=[28]-=- and references therein). The main object in this framework is an ordered linear space, i.e. a pair (V, V + ) where V is a finite dimensional linear space while V + is a pointed cone in V . Having two... |

3 |
Generalized Choi maps in three-dimensional matrix algebra, Linear Algebra Appl
- Cho, Kye, et al.
- 1992
(Show Context)
Citation Context |

3 | On the structure of positive maps between matrix algebras
- Majewski, Marciniak
(Show Context)
Citation Context ... and linear preservers theory. Our main motivation is to give answers for questions contained in Remarks 1.2 and 1.3. However, if it comes to our methods we will use the technique presented in papers =-=[14, 17, 18, 19, 20]-=-. The paper is organized as follows. In Section 2 we give an ’almost’ positive answer for the question from Remark 1.3 (Theorem 2.2). It will allow us to characterize decomposable extremal maps as tho... |

3 |
Extremals and exposed faces of the cone of positive maps
- Yopp, Hill
(Show Context)
Citation Context ...position of the element X. Consequently, the cone of all decomposable maps is the hull of all maps which have one of the previously mentioned two forms. Coming back to positive maps, it was proved in =-=[32]-=- that maps of the above two forms are extremal also in the cone P(B(K), H) of all positive maps. Further, let us note that in cones P(B(C 2 ), C 2 ), P(B(C 2 ), C 3 ) and P(B(C 3 ), C 2 ) there are no... |