Atomic Positive Linear Maps in Matrix Algebras

Transcription

1 Publ RIMS, Kyoto Univ. 34 (1998), Atomic Positive Linear Maps in Matrix Algebras By Kil-Chan HA* Abstract We show that all of the known generalizations of the Choi maps are atomic maps. 1. Introduction Let M n be the C*-algebra of all nxn matrices over the complex field and /* S [MJ (respectively, F s [M n ]) the convex cone of all ^--positive (respectively, s-copositive) linear maps between M n. One of the basic problems about the structures of the positive cone /*i[mj is if this cone can be decomposed as the algebraic sum of some simpler subcones. It is well known [22, 25] that every positive linear map between M 2 is decomposable, that is, it can be written as the sum of a completely positive linear map and a completely copositive linear map. But, this is not the case for higher dimensional matrix algebras. The first example of an indecomposable positive linear map in M 3 was given by Choi [5]. Choi and Lam [7] also gave an example of an indecomposable extremal positive linear map in M 3 (see also [6]). Another examples of indecomposable extremal positive linear maps are found in [9, 16, 21]. These maps are neither 2-positive nor 2-copositive, and so they become atomic maps in the sense in [24], that is, they can not be decomposed into the sums of 2-positive linear maps and 2-copositive linear maps. Several authors [1, 2, 10, 15, 17, 24] considered indecomposable positive linear maps as extensions of the Choi's example. The first two examples [2, 10] are generalizations of the Choi's map [5,6] in M 3 and the other maps i n>fc in [1,15,17, 24] (see Section 2 for the definition) are extensions of the Choi map [7] in Communicated by T. Miwa, June 19, Math. Subject Classifications: 46L05, 15A30. partially supported by GARC-KOSEF * Department of Mathematics, Seoul National University, Seoul , KOREA.

2 592 KIL-CHAN HA higher dimensional matrix algebra M n. Among them, examples in [10] and the map T M (n>3) [17] are known to be atomic maps. But the atomic properties of the other indecomposable maps are not determined. Even decomposabilities of the maps T B>fc are remained open except for some special cases [17]. The usual method to show the atomic property of a positive linear map depends on a tedious matrix manipulation. The purpose of this note is to show that all of the above mentioned examples are atomic maps, using the recent result of Eom and Kye [8]. Generalizing the Woronowicz's argument [25], they considered the duality between the space M n M m ( = M nm ) of all nmxnm matrices over the complex field and the space &(M m, M n ) of all linear maps from M m into M n, which is given by (1.1) 04,0> = Tr (<l>(e^ e^^= <0(e (J U,A Lu=i J u=i for A = 'L j=l a i j e i jem n M m and a linear map $:M m -»M M, where {e itj } is the matrix units of M m and the bilinear form in the right-side is given by <jf, Yy = Tr(YX l ) for X 9 YeM n with the usual trace Tr. For the convenience of readers, we briefly explain the results in [8]. For a matrix A =^Tj= i x t,j e ij em w (x)m m, we denote by A J the block-transpose %Tj=i x j,i e ij f A. We say that a vector z = *L =l z i e i ec n C m is an s-simple if the linear span of {z l5 ---,z m } has the dimension <s, where {e l e m } is the usual orthonormal basis of C m. Let F S [MJ (respectively F S [MJ) denote the convex cone in M n M n generated by zz*em n M n (respectively (zz*) T em n M n ) with all ^-simple vectors zec" C". It turns out that F S [MJ (respectively P[MJ) is the dual cone of jp s [MJ (respectively PTMJ) with respect to the pairing (1.1). With this machinery, the maximal faces of F S [MJ and jf[a/j are characterized in terms of ^-simple vectors (see also [12, 13, 14]). Another consequence is a characterization of the cone F s [MJ+F f [MJ : For a linear map $ : M n > M n, the map < is the sum of an ^--positive linear map and a r-copositive linear map if and only if <^,0>>0 for each AE F S [MJ n F f [MJ. This result provides us a useful method to examine the atomic property for the generalizations of the Choi maps mentioned before. Throughout this note, every vector in the space C r will be considered as an r x 1 matrix. The usual orthonormal basis of C r and matrix units of M r will be denoted by {e i :i=l,-~,r} and {e itj :i,j=l, -~,r} respectively, regardless of the dimension r.

3 ATOMIC POSITIVE LINEAR MAPS The Maps i n>fc Let e be the canonical projection of M n to the diagonal part and S be the rotation matrix in M n which sends e t to e i+l (modri) for i =!,-,«. The map T n fc : M n -> M n is defined by T n,*w = (» -*X*) + Z eos'-re *) - X, Xe M n, i=l for fc = l,2,,/! 1. The map T w0 :M M ->M M is also defined by It is easy to see that T n0 is completely positive and!, _! is completely copositive. The positivity of T W fc is equivalent [1] to a certain cyclic inequality, which was shown by Yamagami [26]. The map T 3a is the Choi and Lam's example mentioned in the introduction. For «>4, Osaka showed that T MM _ 2 is not the sum of a 3-positive linear map and a 3-copositive linear map [15], and T W tl is atomic [17]. In this section, we show that the map t wfc is an atomic map for each n>3 and k =1,2,,«2. For each fixed natural number «=1,2,, let {c^: l<z'<3 w } be the 3"-th roots of unity. Then we have (2.1) I>f = 0, 1< <3"-1. i=l For each k = 1,2, -,«, we define m k ez by w fc = -(3 fc ~ 1 1). Then it is easy to see the following: (2.2) m k m l =^m i m j if and only if (k,i) = (lj) or (kj) = (ij). For any y>0, we define a ik, c r ec n by a ik = CD^mk a tl, n-i i c i =e l +ye 2 + ^ ^k + -^MJ fe=3 7 Cr = 5r-i Ci9 1</<3 M, 2<k<n, 2<r<n.

4 594 KIL-CHAN HA For each r= 1,2,,«, /=1,2,,3 fl and j =1,2, -,«, we define b rij ec n by where is the Schur product of n x 1 matrices c,- and a^. We also define z ri ec n C n and ^rem w M M by ^r = Z Z H Z *» J i=l \<r<n. Then we see that each z ri is a 2-simple vector and so A r ev 2 LM n ]. If we write ^l,. by A r = I, n p>q=l(a r ) pfq e p^ M n M n, then it is easy to see that (2.3) (A r ) M = by (2.1) and (2.2). Now we define AeV 2 {M^ by A=-^nr=,A r = Then, by (2.3), we have n y In order to show that AG F 2 [MJ, define u i9 v i9 a f and js^-ec" by 7 1

5 ATOMIC POSITIVE LINEAR MAPS 595 j8 y = e,.(x) e 4- e t e j, 2<i<(n- 2), (i + 2) </ < n, where suffixes are understood in mod«. A direct calculation shows A J = (iw* 4- v t vf + a^*) + Mi* + I Z M y > i=l j=3 i = 2 j = i + 2 and so, we have Ae F 2 [MJ. Furthermore, we also have (A,i nk y = y 2 1, for each «= 3,4, and fc= 1,2,,«2, and so we see that <^,T Mjfc ><0 for 0<y<l. By the result in [8] mentioned in the introduction, we conclude the following: Theorem 2.1. For n>3 and l<k<n 2, the map T nfc :M M ->M M is an atomic positive linear map. 3. The Generalized Choi Map The other generalization of the Choi map is given in [2]. For nonnegative real numbers a, b, and c, the map O[a, 6, c] : M 3 -» M 3 is defined by. 3 0 a - x 2,, *3,3 +**!.! + C *2,2 for each l r =(.x ij -)EM 3. The map <I>[2,0,//] with /x>l is the example of an indecomposable positive linear map given by Choi [6], and $[2,0,2] and <I>[2,0, 1] are the examples given in [5] and [7] respectively. Choi and Lam [7] showed that [2,0, 1] is an extremal positive linear map using the theory of biquadratic forms. Later, Tanahashi and Tomiyama [24] showed the atomic property of the map <D[2,0, 1] which is same as the map T 3>1. It was shown [2] that the map <J)[a, 6, c] is an indecomposable positive linear map if and only if the following conditions are satisfied:

6 596 KIL-CHAN HA (i) l<a<3, (ii) a + b + c>3, 0_^12^A^^( ifl<a<2 (3.1) (iii) - /3-aV "" C< \ 2 / if2<a<3. In this section we show that these conditions imply that <D[0, b, c] is an atom. Let (co e -:/=l,2,3} be the cube roots of unity and s be any positive real number. Define 0 ife ec 3, z i5 w ec 3 C 3 and 5eF 2 [M 3 ] by, ^) t, <% = ( )«i 2? i=l,2,3 9 5" W^ I/ 3 \ 1 / 2 \ / 4 ^ = ^ Z V-* +-2 I «-tf+* 2 I 3\i=l / S \ i=i / \i=3 It is clear that e F 2 [M 3 ]. w,ec 3 C 3 by To show that 5e F 2 [M 3 ], we define z and e i+i ), i= 1,2,3, i= 1,2,3, where suffixes are understood in mod 3. A direct calculation show that It is also easy to calculate (3.2) <5,<D[fl

7 ATOMIC POSITIVE LINEAR MAPS 597 We proceed to show the conditions in (3.1) imply that the pairing in (3.2) is negative. We first consider the case bc = 0. If 6 = 0 then the pairing (3.2) becomes negative for s> / -. If c = 0 then (3.2) is negative for 3 a Q<s< I -. When fec^o, we take s^i-i, then the pairing (3.2) is \b reduced to which is also negative since *Jbc< - in (3.1). Therefore we have Theorem 3.1 Theorem 3.1. The map $>[_a,b, c] is an indecomposable positive linear map if and only if it is an atomic positive linear map. For the Choi's map O[2,0,/j], the condition (3.1) is reduced to the condition \JL > 1. Therefore, we see that the Choi's map <I>[2, 0, ILL] is atomic whenever ^>l. 4. The Robertson's Map An example of an indecomposable positive linear map on M 4 was given by Robertson [18] by considering an extension of an automorphism on a certain spin factors. To describe this map, let a : M 2 -> M 2 be the symplectic involution defined by /.A /* -f \y 5J \-y a The Robertson's map *F : M 4 -> M 4 is defined by Z W) ±(Z + a(y)) tr(x)i 2 for X, Y, Z, We M 2, where tr is the normalized trace on M 2. The indecomposability of this map was shown in [20] by using the Stormer's characterization [23] of decomposability. It turns out that this map is an extremal positive linear map which is neither 2-positive nor 2-copositive [19,21]. So this map is an atomic map. We

8 598 KIL-CHAN HA provide a simple proof. Define z t ec* C 4 and DeF 2 [M 4 ] by Then we see that and so De i / 2[M 4 ]n F 2 [M 4 ]. Furthermore, we can show that the pairing < >, > = by an easy calculation. Consequently, we conclude that *F is an atomic map. References [ 1 ] Ando, T., Positivity of certain maps, Seminar Notes, [ 2 ] Cho, S. J., Kye, S.-H. and Lee, S. G., Generalized Choi maps in 3-dimensional matrix algebras, Linear Alg. AppL, 171 (1992), [3] Choi, M.-D., Positive linear maps on C*-algebras, Canad. Math. J., 24 (1972), [ 4 ] - 3 Completely positive linear maps on complex matrices, Linear Alg. AppL, 10 (1975), [5] -, Positive semidefinite biquadratic forms, Linear Alg. AppL, 12 (1975), [ 6 ] -, Some assorted inequalities for positive linear maps on C *-algebras, J. Operator Theory, 4 (1980), [ 7 ] Choi, M.-D. and Lam, T. Y., Extremal positive semidefinite forms, Math. Ann., 231 (1977), [ 8 ] Eom, M. H. and Kye, S.-H., Duality for positive linear maps in matrix algebras, Math. Scand., to appear. [ 9 ] Kim, H.-J. and Kye, S.-H., Indecomposable extreme positive linear maps in matrix algebras, Bull. London Math. Soc., 26 (1994), [10] Kye, S.-H., Elementary Operators and Applications, World-Scientific, [11] -, Positive linear maps between Matrix algebras which fix diagonals, Linear Alg. AppL, 216 (1995), [12] -, Facial structures for positive linear maps between matrix algebra, Canad. Math. Bull., 39 (1996), [13] -, Boundaries of the cone of positive linear maps and its subcones in matrix algebras, /. Korean Math. Soc., 33 (1996), [14] - s On the convex set of all completely positive linear maps in matrix algebras, Math. Proc. Comb. Phil. Soc., 122 (1997), [15] Osaka, H., Indecomposable positive maps in low dimensional matrix algebras, Linear Alg. AppL, 153 (1991), [16] -, A class of extremal positive maps in 3 x 3 matrix algebras, Publ. RIMS. Kyoto Univ., 28 (1992),

9 ATOMIC POSITIVE LINEAR MAPS 599 [17] Osaka, H., A series of absolutely positive maps in matrix algebras, Linear Alg. AppL, 186 (1993), [18] Robertson, A. G., Automorphisms of spin factors and the decomposition of positive maps, Quart. J. Math. Oxford (2), 34 (1983), [19] 5 Positive extensions of automorphisms of spin factors, Proc. Roy. Soc. Edinburgh Sect. A, 94 (1983), [20], Schwarz inequalities and the decomposition of positive maps on C*-algebras, Math. Proc. Camb. Phil. Soc., 94 (1983). [21], Positive projections on C*-algebra and an extremal positive map, /. London Math. Soc., 32 (1985), [22] Stormer, E., Positive linear maps of operator algebras, Ada Math., 110 (1963), [23] 9 Decomposable positive maps on C*-algebras, Proc. Arner. Math. Soc., 86 (1982), [24] Tanahashi, K. and Tomiyama, J., Indecomposable positive maps in matrix algebras, Canad. Math. Bull., 31 (1988), [25] Woronowicz, S. L., Positive maps of low dimensional matrix algebras, Rep. Math. Phys., 10 (1976), [26] Yamagami, S., Cyclic inequalities, Proc. Arner. Math. Soc., 118 (1993),

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