DUALITY FOR POSITIVE LINEAR MAPS IN MATRIX ALGEBRAS

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1 MATH. SCAND. 86 (2000), 130^142 DUALITY FOR POSITIVE LINEAR MAPS IN MATRIX ALGEBRAS MYOUNG-HOE EOM AND SEUNG-HYEOK KYE Abstract We characterize extreme rays of the dual cone of the cone consisting of all s-positive (respectively t-copositive) linear maps between matrix algebras. This gives us a characterization of positive linear maps which are the sums of s-positive linear maps and t-copositive linear maps, which generalizes Strmer's characterization of decomposable positive linear maps in matrix algebras. With this duality, it is also easy to describe maximal faces of the cone consisting of all s-positive (respectively t-copositive) linear maps between matrix algebras. 1. Introduction The structures of the convex cone of positive linear maps between C -algebras are turned out to be extremely complicated even when the domain and the range are low dimensional matrix algebras M n. Several authors including [2], [4], [9], [10], [12], [14] and [15] have tried to decompose the cone into smaller cones consisting of more well-behaved positive linear maps such as completely positive and completely copositive linear maps. We denote by b h and T h the space of all bounded linear operators and trace class operators on a Hilbert space h, respectively. One of the methods to examine the possibility of decomposition is to use the duality between the space b A; b h of all bounded linear operators from a C -algebra A into b h and the projective tensor product T h ^A given by hx y;iˆtr y x t ; x 2T h ; y2 A; 2 b A; b h ; where Tr and t denote the usual trace and the transpose, respectively. Using this duality, Woronowicz [15] has shown that every positive linear map from the matrix algebra M 2 into M n is the sum of a completely positive linear map and a completely copositive linear map if and only if n 3. The above duality is also useful to study extendibility of positive linear maps as was con- Partially supported by BSRI-MOE and GARC-KOSEF. Received July 21, 1997

2 DUALITY FOR POSITIVE LINEAR MAPS IN MATRIX ALGEBRAS 131 sidered by Strmer [13]. We denote by P s A; BŠ (respectively P s A; BŠ) the convex cone of all s-positive (respectively s-copositive) linear maps from a C -algebra A into a C -algebra B. WealsodenotebyP 1 A; BŠ (respectively P 1 A; BŠ) the cone of all completely positive (respectively completely copositive) linear maps. The predual cones of P s A; b h Š and P s A; b h Š with respect to the above pairing has been determined by Itoh [3]. If we restrict ourselves to the cases of matrix algebras, then the above pairing may be expressed " by! X # m 1 ha;iˆtr e ij e ij A t ˆ Xm h e ij ; a ij i; for A ˆ Pm a ij e ij 2 M n M m and a linear map : M m! M n, where fe ij g isthematrixunitsofm m and the bilinear form in the right-side is given by hx; Yi ˆTr YX t for X; Y 2 M n. Then (1) defines a bilinear pairing between the space M n M m ˆ M nm of all nm nm matrices over the complex field and the space L M m ; M n of all linear maps from M m into M n. In this note, we show that the predual cone of P s M m ; M n Š with respect to the pairing (1) is generated by rank one matrices in M nm whose range vectors in C nm correspond to m n matrices of ranks s. The predual cone of P s M m ; M n Š is obtained by block-transposing that of P s M m ; M n Š.Withthis information, it is easy to characterize the predual cone of P s M m ; M n Š P t M m ; M n Š. As an application, we extend Strmer's result [12] to give a characterization of linear maps which are sums of s-positive linear maps and t-copositive linear maps. We also show that Choi's examples [1] of non-decomposable positive linear maps are not the sum of 3-positive linear maps and 2-copositive linear maps. The second author [6], [7], [8] has modified the method in [11] to characterize maximal faces of the cones P s M m ; M n Š and P s M m ; M n Š, and all faces of the cones P 1 M m ; M n Š and P 1 M m ; M n Š.Generally,itturnsoutthateverymaximalfaceofaconvexconeinafinitedimensional space corresponds to an extreme ray of the predual cone, whenever every extreme ray of the predual cone is exposed with respect to the pairing. This enables us to describe maximal faces of the cones P s M m ; M n Š and P s M m ; M n Š simultaneously. Compare with [7]. We develop in Section 2 some general aspects of dual cones how maximal faces of a cone correspond to extreme rays of the dual cone, and characterize extreme rays of the predual cones of P s M m ; M n Š and P s M m ; M n Š in Section 3. We also examine in Section 4 the Choi's example mentioned above. Throughout this note, we fix natural numbers m and n, and denote by just P s (respectively P s ) for the cone P s M m ; M n Š (respectively P s M m ; M n Š). Note that P 1 ˆ P m^n and P 1 ˆ P m^n in these notations, where m ^ n denotes the minimum of fm; ng.

3 132 MYOUNG-HOE EOM AND SEUNG-HYEOK KYE This note has been completed during the second author was staying at the University of Illinois at Urbana-Champaign. He is grateful for the kind hospitality of Professor Zhong-Jin Ruan and the financial support from the Yonam Foundation. 2. Duality of convex cones. Let X and Y be finite dimensional normed space, which are dual each other with respect to a bilinear pairing h ; i. For a subset C of X (respectively D of Y), we define the dual cone C (respectively D )bythesetofally2y (respectively x 2 X) such that hx; yi 0foreachx2C (respectively y 2 D). It is clear that C is the closed convex cone of X generated by C. Itisalso easy to see that the dual cone of the intersection C 1 \ C 2 of two cones C 1 and C 2 is nothing but the closed cone generated by C1 and C 2. In other words, we have the identities: 2 C 1 \ C 2 ˆ C1 [ C 2 ; C 1 [ C 2 ˆ C1 \ C 2 ; whenever C 1 and C 2 are closed convex cones of X. Indeed, we have C i ˆ Ci C1 [ C 2 for i ˆ 1; 2, and so C 1 \ C 2 C1 [ C 2,whichimplies one direction of the first identity. On the other hand, Ci C 1 \ C 2 implies C1 [ C 2 C 1 \ C 2. The second identity follows from the first one. For a face F of a closed convex cone C of X, we define the subset F 0 of C by F 0 ˆfy2C : hx; yi ˆ0 for each x 2 Fg: It is then clear that F 0 is a closed face of C.Ifwetakeaninteriorpointx 0 of F then we see that F 0 ˆfy2C : hx 0 ; yi ˆ0g: Recall that a point x 0 of a convex set C is said to be an interior point if for any x 2 C there is t > 1suchthat 1 t x tx 0 2 C. IfC isaconvexsubset of a finite dimensional space then the set C of all interior points of C is nothing but the relative interior of C with respect to the affine manifold generated by C. It is clear that F F 00 for any face F of C. Therefore, we have F 0 F 000 F 0, and so it follows that F 0 ˆ F 000. We say that a face F of a closed convex cone C is exposed with respect to the pairing h ; i if there exists y 0 2 C such that F ˆfx 2 C : hx; y 0 iˆ0g. IfafaceF is exposed by y 0 2 C then take a face G of C such that y 0 is an interior point of G. Then F ˆ G 0,andsoF 00 ˆ G 000 ˆ G 0 ˆ F. Therefore, we have the following:

4 DUALITY FOR POSITIVE LINEAR MAPS IN MATRIX ALGEBRAS 133 Lemma 2.1. Let F be a closed face of a closed convex cone C. Then F is exposed if and only if F ˆ F 00.ThesetF 00 is the smallest exposed face containing F. If all closed faces of C and C are exposed with respect to the pairing, then it is clear from Lemma 2.1 that the correspondence F 7!F 0 is an order reversing one-to-one mapping from the complete lattice f C of all closed faces of C onto the complete lattice f C. From this, it is easily seen that this map is an order reversing lattice isomorphism. Indeed, it is clear that F1 0 _ F 2 0 F 1 ^ F 2 0 ; F1 0 ^ F 2 0 F 1 _ F 2 0 : Then it follows that F 1 _ F 2 ˆ F1 00 _ F 2 00 F 1 0 ^ F F 1 _ F 2 00 ˆ F 1 _ F 2 ; and so, we have 3 F1 0 _ F 2 0 ˆ F 1 ^ F 2 0 ; F1 0 ^ F 2 0 ˆ F 1 _ F 2 0 : From now on throughout this section, we assume that C is a closed convex cone of X on which the pairing is non-degenerate, thatis, 4 x 2 C; hx; yi ˆ0 for each y 2 C ˆ) x ˆ 0: This assumption guarantees the existence of a point 2 C with the property: 5 x 2 C; x 6ˆ 0 ˆ) hx;i > 0; which is seemingly stronger than (4). Indeed, we take for each x 2 C a neighborhood U x of x and a point y x 2 C such that hz; y x i > 0forz2U x. Put C ˆfx 2 C : kxk ˆg for >0. Then since C 1 is compact, we see that there exist x 1 ;...; x r 2 C 1 such that U x1 [...[ U xr C 1. We may put ˆ y x1 y xr. As an another immediate consequence of (4), we also have 6 F 2 f C ; F 0 ˆ C ˆ) F ˆf0g: Lemma 2.2. Foragivenpointy2 C, the following are equivalent: (i) y is an interior point of C. (ii) hx; yi > 0 for each nonzero x 2 C. (iii) hx; yi > 0 for each x 2 C which generates an extreme ray. Proof. If y is an interior point of C then we may take t < 1andz2C such that y ˆ 1 t tz, where 2 C is a point with the property (5). Then we see that hx; yi ˆ 1 t hx;i thx; zi > 0

5 134 MYOUNG-HOE EOM AND SEUNG-HYEOK KYE for each nonzero x 2 C. It is clear that (ii) and (iii) are equivalent. Now, we assume (ii), and take an arbitrary point z 2 C.ThensinceC 1 is compact, ˆ supfhx; zi : x 2 C 1 g is finite, and we see that hx; zi 1foreachx 2 C 1=. We also take with 0 <<1suchthathx; yi for each x 2 C 1=.Put w ˆ z 1 1 y: Then we see that hx; wi 0foreachx2C 1=,andsow2C.Sincezwas an arbitrary point of C and 1 1 > 1, we see that y is an interior point of C. Lemma 2.3. If F is a maximal face of C then there is an extreme ray L of C such that F ˆ L 0. Proof. Note that F lies in the boundary of C.Ifwetakeaninterior point y 0 of F then there is x 0 2 C which generates an extreme ray L such that hx 0 ; y 0 iˆ0by Lemma 2.2. Since x 0 is an interior point of L, we see that y 0 2 L 0 \ int F, fromwhichweinferthatf L 0. Because L 0 6ˆ C by (6), we have F ˆ L 0. Since L 00 0 ˆ F in the above lemma, we see that every maximal face of C is of the form G 0 for a unique nonzero exposed face G. Note that L 00 need not be minimal even among nonzero exposed faces. Lemma 2.4. If L is an exposed face of C which is minimal among nonzero exposed faces, then L 0 is a maximal face of C. Proof. Assume that L 0 is not maximal, and take a maximal face F of C such that L 0 6ˆ F. Then there exists a nonzero exposed face M of C such that F ˆ M 0,andsoLˆL 00 F 0 ˆ M 00.SinceLis minimal among nonzero exposed faces, we have L ˆ M and L 0 ˆ M 0 ˆ F, which is a contradiction. Now, we summarize as follows: Theorem 2.5. Let X and Y be finite-dimensional normed spaces with a non-degenerate bilinear pairing h ; i on a closed convex cone C in X. Assume that every extreme ray of C is exposed with respect to the pairing. Then L 0 is a maximal face of C for each extreme ray L of C. Conversely, every maximal face of C is of the form L 0 for a unique extreme ray L of C. We will see that there is an extreme ray in P 1 M 3 ; M 3 Š which is not exposed, while every extreme ray of the dual cone of P s (respectively P s )isexposed.

6 DUALITY FOR POSITIVE LINEAR MAPS IN MATRIX ALGEBRAS Positive linear maps. In this section, every vector in the space C r will be considered as an r 1 matrix. The usual orthonormal basis of C r and matrix unit M r will be denoted by fe i : i ˆ 1;...; rg and fe ij : i; j ˆ 1;...rg respectively, regardless of the dimension r. ForamatrixAˆ P x ij e ij 2 M n M m,wedenoteby A the block-transpose P m x ji e ij of A. Every vector z 2 C n C m may be written in a unique way as z ˆ Pm iˆ1 z i e i with z i 2 C n for i ˆ 1; 2;...; m. We say that z is an s-simple vector in C n C m if the linear span of fz 1 ;...; z m g has the dimension s. For an s-simple vector z ˆ Pm iˆ1 z i e i 2 C n C m, take a generator fu 1 ; u 2 ;...; u s g of the linear span of fz 1 ; z 2 ;...; z m g in C n, and define a ik 2 C, a k 2 C m, u 2 C n C s and w 2 C m C s by z i ˆ Xs a ik u k 2 C n ; i ˆ 1; 2;...; m; 7 kˆ1 a k ˆ Xm iˆ1 u ˆ Xs kˆ1 w ˆ Xs kˆ1 a ik e i 2 C m ; k ˆ 1; 2;...; s; u k e k 2 C n C s ; a k e k 2 C m C s ; where u k denotes the vector whose entries are conjugates of those of the vector u k.thenzz ˆ Pm z iz j e ij 2 M n M m, z i z j ˆ Ps k;`ˆ1 a ika j`u k u ` 2 M n,and so we have hzz ;iˆxm h e ij ; z i z j i ˆ Xm X s k;`ˆ1 ˆ Xm X s k;`ˆ1 ˆ Xm X s k;`ˆ1 a ik a j`h e ij ; u k u `i a ik a j`h e ij u`; u k i C n a ik a j`h e ij e k` u; ui C n C s; for a linear map : M m! M n. We also have ww ˆ Ps k;`ˆ1 a ka ` e k` 2 M m M s,and

7 136 MYOUNG-HOE EOM AND SEUNG-HYEOK KYE id s ww ˆ Xs k;`ˆ1 a k a ` e k` ˆ Xs X m k;`ˆ1 Therefore, it follows that 8 hzz ;iˆh id s ww u; ui C n C s; where id s denotes the identity map of M s. With the exactly same calculation as above, we also have 9 h zz ;iˆh tp s ww u; ui C n C s; a ik a j` e ij e k`: for an s-simple vector z 2 C n C m and a linear map : M m! M n, where tp s denotes the transpose map of M s. Theorem 3.1. For a linear map : M m! M n, we have the following: (i) The map is s-positive if and only if hzz ;i0 foreachs-simplevector z 2 C n C m. (ii) The map is s-copositive if and only if h zz ;i0 for each s-simple vector z 2 C n C m. Proof. Assume that is s-positive and take an s-simple vector z ˆ Pm iˆ1 z i e i 2 C n C m. Then the identity (8) shows that hzz ;i0. For the converse, assume that hzz ;i0foreachs-simple vector z 2 C n C m. For each w 2 C m C s and u 2 C n C s,wetakea k 2 C m and z i 2 C n as in the relations (7). Then we see that id s ww is positive semidefinite by (8), and so id s is a positive linear map. The exactly same argument may be applied for the second statement if we use the identity (9). For s ˆ 1; 2;...; m ^ n, we define convex cones V s and V s in M n M m by V s M n M m ˆfzz : z is an s-simple vector in C n C m g ; V s M n M m ˆf zz : z is an s simple vector in C n C m g : Then Theorem 3.1 and the identity (2) say that the following pairs 10 V s ; P s ; V t ; P t ; V s \ V t ; P s P t are dual each other, for s; t ˆ 1; 2;...; m ^ n. We note that V m^n M n M m is nothing but the cone M n M m of all positive semi-definite matrices in M n M m. Corollary 3.2. Alinearmap : M m! M n is the sum of an s-positive linear map and a t-copositive linear map if and only if ha;i0 for each A 2 V s \ V t. Strmer [12] characterized the decomposable positive maps among linear maps from a C -algebra into b h. Foralinearmap : M m! M n,this

8 DUALITY FOR POSITIVE LINEAR MAPS IN MATRIX ALGEBRAS 137 tells us that is the sum of a completely positive linear map and a completely copositive linear map if and only if the following 11 id p V p \ V p M m M p M n M p : holds for p ˆ 1; 2;... In order to generalize this result for the sums of s-positive and t-copositive linear maps, we use block-wise Hadamard product. For two block matrices X ˆ Pp k;`ˆ1 x k` e k` 2 M n M p and Y ˆ Pp k;`ˆ1 y k` e k` 2 M m M p,wedefinetheblock-wise Hadamard product by X Y ˆ Xp x k` y k` 2 M n M m : k;`ˆ1 Then for every linear map : M m! M n, we see that the following identity h id p Y ; Xi ˆ Xp h y k` ; x k`i 12 k;`ˆ1 ˆ Xm ˆ * + e ij ; Xp hy k`; e ij ix k` Xm X p k;`ˆ1 k;`ˆ1 * + hy k`; e ij ix k` e ij ; ˆhX Y;i holds, using the relation y k` ˆ Pm hy k`; e ij ie ij.fora2m n M m,wedenote by A 2 M m M n the shuffle of A, thatis, xy ˆ y x. Thenitis easy to see that A 2 V s M n M m ()A 2 V s M m M n ; 13 A 2 V t M n M m ()A 2 V t M m M n : Let y ˆ Pp kˆ1 y k e k 2 C m C p be an s-simple vector with y k ˆ Ps ˆ1 b ku 2 C m for k ˆ 1;2;...;p. Then we have Xs y k y ` ˆ b k b` u u ;ˆ1 ˆ Xs X m ;ˆ1 ˆ Xs X m ;ˆ1 b k b` hu u ; e ijie ij b k b` hu ; e i ihu ; e j ie ij :

9 138 MYOUNG-HOE EOM AND SEUNG-HYEOK KYE For an arbitrary given x ˆ Pp kˆ1 x k e k 2 C n C p, put z ˆ Xp b k x k 2 C n ; ˆ 1; 2;...; s; Then we have kˆ1 w i ˆ Xs ˆ1 w ˆ Xm iˆ1 xx yy ˆ Xp hu ; e i iz 2 C n ; i ˆ 1; 2;...; m; w i e i 2 C n C m : k;`ˆ1 ˆ Xp X s x k x ` y ky ` X m k;`ˆ1 ;ˆ1 ˆ Xm w i w j e ij ˆ ww ; b k b` hu ; e i ihu ; e j ix k x ` e ij which belongs to V s M n M m,sincew is an s-simple vector of C n C m. Therefore, we see that 14 X 2 M n M p ; Y 2 V s M m M p ˆ)X Y 2 V s M n M m : By the same argument, we also have 15 X 2 M n M p ; Y 2 V t M m M p ˆ)X Y 2 V t M n M m : Theorem 3.3. For a linear map : M m! M n, we have the following: (i) is s-positive if and only if id n V s M m M n M n M n. (ii) is t-copositive if and only if id n V t M m M n M n M n. (iii) is the sum of an s-positive linear map and a t-copositive linear map if and only if id n V s \ V t M m M n M n M n. Proof. If is s-positive and Y 2 V s M m M p with p ˆ 1; 2;...,thenwe have h id p Y ; Xi ˆhX Y;i0 for each X 2 M n M p by (14) and the duality between V s and P s. Therefore, id p Y 2 M n M p. For the converse, note that every A 2 M n M m is written by A ˆ A J m ˆ J n A ;

10 where J r ˆ Pr e ij e ij 2 M r M r for r ˆ 1; 2;... Therefore, for each A 2 V s M n M m, we have ha;iˆhj n A ;iˆh id n A ; J n i0 by(13).thisproves(i).theexactlysameargumentalsoproves(ii)and(iii) if we use (15) and (13). We note that the trace map X 7!Tr X I (respectively the identity matrix) is a typical interior point of the cones P m^n and P m^n (respectively V 1 ). It is also easy to see that these play the roª les of in (5) for any pairs of dual cones in (10). We also note that every face of V m^n is exposed with respect to the pairing. To see this, take a face F of V m^n and an interior point A of F. Then F consists of all positive semi-definite matrices whose range spaces are contained in the range space of A. If we take a positive semi-definite matrix B whose range space is orthogonal to that of A and a linear map : M n! M m such that P m e ij e ij ˆ B, then we see that is completely positive, and F is exposed by. In this way, we see that every face of P m^n corresponds to a face of V m^n, which is determined by the range space of an interior point. Therefore, every face of P m^n corresponds to a subspace of C n C m, in the orderreversing way. For an order preserving lattice isomorphism between faces of P m^n and subspaces of C n C m ˆ M m;n, we refer to [8]. Since every extreme ray of V s is an extreme ray of V m^n and P s is larger than P m^n, it follows that every extreme ray of V s is exposed with respect to the pairing. The same argument holds for the pair V s ; P s, because the block transpose map A 7!A is linear. Therefore, we may apply Theorem 2.5 to get the following: Theorem 3.4. Let P s (respectively P s be the convex cone of all s-positive (respectively s-copositive) linear maps from M m into M n. For each s-simple vector z 2 C n C m,theset f 2 P s : hzz ;iˆ0g respectively f 2 P s : h zz ;iˆ0g is a maximal face of P s (respectively P s ). Conversely, every maximal face of P s (respectively P s ) arises in this form for an s-simple vector z 2 C n C m. 4. Examples DUALITY FOR POSITIVE LINEAR MAPS IN MATRIX ALGEBRAS 139 The first example of an indecomposable positive linear map between M 3 was given by Choi [1] by considering a positive semi-definite biquadratic form which is not the sum of the squares of bilinear forms. This example : M 3! M 3 is defined by

11 140 MYOUNG-HOE EOM AND SEUNG-HYEOK KYE x 11 x 12 x 13 x 11 x 12 x 13 x x 21 x 22 x 23 A x 21 x 22 x 23 0 x 11 0 A; x 31 x 32 x 33 x 31 x 32 x x 22 where 1. Later, Strmer [12] showed that the above map is not decomposable by the condition (11). In order to apply Corollary 3.2, we modify the matrix in [12] to define 0 A ˆ@ e 11 2 e 22 e 33 e 12 e 13 e 21 e 11 e 22 2 e 33 e 23 e 31 e 32 2 e 11 e 22 e 33 1 A2 M 3 M 3 ; where is a nonnegative real number. It is easy to see that A is positive semi-definite whenever 0, and A 2 V 3.Ifweput z 1 ˆ e 2 e 4 ; z 2 ˆ e 6 e 8 ; z 3 ˆ e 7 e 3 then we see that A ˆ X3 iˆ1 z i z i e 1 e 1 e 5e 5 e 9e 9 ; and so A 2 V 2, whenever 0. A direct calculation shows that ha;iˆ3 1 ; which is negative if ˆ 1=2 for example. Therefore, we see that the map is not the sum of a 3-positive linear map and a 2-copositive linear map. The authors were not able to determine whether the above matrix A belongs to V 2. If this is the case then we may conclude that isnotthesumofa2-positive linear map and a 2-copositive linear map. See [14] for the case of ˆ 1. Actually, we could not find an explicit example of matrix which lies in V 2 \ V 2 n V 1, although we know that this set is nonempty since there are examples of positive linear maps between M 3 which are not the sums of 2-positive linear maps and 2-copositive linear maps. See [5], [9] and [14]. The following proposition says that we must consider matrices whose ranks are at least two, in order to find examples in V 2 \ V 2 n V 1. Proposition 4.1. Let x 2 C n C m. Then the rank one matrix xx 2 M n M m lies in V m^n \ V m^n ifandonlyifitliesinv 1. Proof. Put x ˆ Pm iˆ1 x i e i 2 C n C m.ifxx 2 V 1 then x is a 1-simple vector, and so we may write x ˆ Pm iˆ1 iy e i 2 C n C m. If we put ^x ˆ Pm iˆ1 iy e i then we have xx ˆ ^x^x, and so xx 2 V m^n. For the converse, we may assume that x 1 6ˆ 0 without loss of generality. For a ˆ Pm iˆ1 a i e i 2 C n C m, note that

12 DUALITY FOR POSITIVE LINEAR MAPS IN MATRIX ALGEBRAS 141 a xx a ˆ Xm a i x jx i a j ˆ Xm hx i ; a j ihx j ; a i i: If we take a ˆ x k e 1 x 1 e k,fork ˆ 2; 3;...; m, then a xx a ˆ 2 jhx 1 ; x k ij 2 kx 1 k 2 kx k k 2 ; which should be nonnegative. Therefore, we see that x k is a scalar multiple of x 1 for each k ˆ 2; 3;...; m, andsox is a 1-simple vector. Note that the map with ˆ 1 generates an extreme ray as was shown in [2]. We remark that this ray is not exposed. To see this, first note that if 2 C 3 C 3 is a 1-simple vector then h ;iˆtr t Šˆh ; i: So, if h ;iˆ0 then by a direct calculation we see that the pair ; is one of the following: 16 e 1 ; e 3 ; e 2 ; e 1 ; e 3 ; e 2 ; ; ; where ˆ e ia ; e ib ; e ic, ˆ e ia ; e ib ; e ic and ˆ a; b; c runs through R 3. Therefore, if we denote by L the extreme ray generated by then we have L 0 ˆfxx 2 V 1 M 3 M 3 : x ˆ e 1 e 2 ; e 2 e 3 ; e 3 e 1 ; 2 R 3 g : By the arguments in Section 5 of [6], we see that L 6ˆ L00. Added in proof. It was shown in the paper [16] by Kil-Chan Ha that the map in section 4 is not the sum of a 2-positive map and a 2-copositive map. REFERENCES 1. M.-D. Choi, Positive semidefinite biquadratic forms, Linear Algebra Appl. 12 (1975), 95^ M.-D. Choi and T.-T. Lam, Extremal positive semidefinite forms, Math. Ann. 231 (1977), 1^ T. Itoh, Positive maps and cones in C -algebras, Math. Japon 31 (1986), 607^ H.-J. Kim and S.-H. Kye, Indecomposable extreme positive linear maps in matrix algebras, Bull. London Math. Soc. 26 (1994), 575^ S.-H. Kye, A class of atomic positive maps in 3-dimensional matrix algebras, Elementary operators and applications (Blaubeuren, 1991), pp. 205^209, World-Scientific, S.-H. Kye, Facial structures for positive linear maps between matrix algebras, Canad. Math. Bull. 39 (1996), S.-H. Kye, Boundaries of the cone of positive linear maps and its subcones in matrix algebras, J. Korean Math. Soc. 33 (1996),

13 142 MYOUNG-HOE EOM AND SEUNG-HYEOK KYE 8. S.-H. Kye, On the convex set of all completely positive linear maps in matrix algebras, Math. Proc. Cambridge Philos. Soc., 122 (1997), 45^ H. Osaka, A class of extremal positive maps in 3 3 matrix algebras, Publ. Res. Inst. math, Sci. 28 (1992), A.G.Robertson,Positive projections on C -algebras and extremal positive maps, J. London Math. Soc. (2) 32 (1985), 133^ R. R. SmithandJ. D. Ward, The geometric structure of generalized state spaces, J. Funct. Anal. 40 (1981), 170^ E. Strmer, Decomposable positive maps on C -algebras, Proc. Amer. Math. Soc. 86 (1982), E. Strmer, Extension of positive maps into B h, J. Funct. Anal. 66 (1986), K. Tanahashi and J. Tomiyama, Indecomposable positive maps in matrix algebras, Canad. Math. Bull. 31 (1988), S. L. Woronowicz, Positive maps of low dimensional matrix algebras, Rep. Math. Phys. 10 (1976), 165^ K.-C. Ha, Atomic positive linear maps in matrix algebras, Publ. RIMS, Kyoto Univ. 34 (1998), 591^599. DEPARTMENT OF MATHEMATICS SEOUL NATIONAL UNIVERSITY SEOUL KOREA