Boundaries of the cone of positive linear maps and its subcones in matrix algebras

BOUNDARIES OF THE CONE OF POSITIVE LINEAR MAPS AND ITS SUBCONES IN MATRIX ALGEBRAS

SEUNG-HYEOK KYE

1. Introduction

Let M

be the C*-algebra of all n x n matrices over the complex field, and JP'[Mm, Mn] the convex cone of all positive linear maps from Mm into M

that is, the maps which send the set of positive semi- definite matrices in Mm into the set of positive semi-definite matrices in M n. The convex structures of JP'[Mm, M n] are highly complicated even in low dimensions, and several authors [CL, KK, LW, 0, R, S, W] have considered the possibility of decomposition ofJP'[M

M

into subcones.

A linear map

A

B between C*-algebras is said to be s-positive if the linear map

ids : A

Ms

B

Ms is positive, and com- pletely positive if it is s-positive for each natural number s

1,2, ... , where ids : Ms

Ms is the identity map. We denote by JP's[A, B]

(respectively JP'

[A, BD the convex cone of all s-positive (respectively completely positive) linear maps from A into B. Choi [Cl] gave an example in JP'n-dMn,Mn] \ JP'n[Mn, M n], and showed that JP'n[A,MnJ and JP'm[Mm, BJ coincide with the cone of all completely positive linear maps. In order to understand the boundary structures of JP'l[Mn, M n], the author has characterized the maximal faces OfJP'l[M

M

in [KI].

Using the methods in [SW], he [K2] has also found all maximal faces of JP'

[Mm , M n], and investigated how JP'

[Mm , M n] is sitting down in JP'l[M

M

In this note, we continue the same work for the cone JP's [Mm, M n].

Motivated by [SW] and [K2], we define for each m x n matrix V a

Received December 1, 1995. Revised February 26, 1996.

1991 AMS Subject Classification: 46L05, 15A30.

Key words: Positive linear maps, matrix algebras, maximal faces.

Partially supported by MOE and GARC.

linear functional (1.1)

on the space of all linear maps from Mm into M n, in Section 2.

turns out that every maximal face of lfs[Mm,Mn] is exposed one obtained from this functional associated with a matrix V with rank :S s. With this information, we see in Section 3 how the cones JPs[Mm,Mn] and JPt[M

M

are related for different s and t.

Throughout this note, we fix natural numbers m and n, and denote by just JP s for the convex cone of all s-positive linear maps from Mm

into M

We also denote by m

n the minimum of m and n, then JP

= JP

as was mentioned above. The vector space Mm,n of all m x n. complex matrices is endowed withthe inner product arising from the usual trace of M n. Then Mm,n is isometrically isomorphic to the space cm l8l cn. Every vector in C n will be considered as an

x 1 matrix. By the interior int C of a convex set C in RV, we mean the relative interior of C with respect to the affine manifold generated by C. The boundary ac of C is the set difference C \ int C:

2. Maximal faces of JP

For an m x n matrix V with rank V = r, choose an orthonormal basis {1J},'." 1Jr} of (Ker V)

Then we have V = Ej=l ei1Jj with ei

V1Ji for

1, ... , r. Denote by

r

Pv

L ^{eie; l8l} E ii

^{Mm l8l} M

r

1Jv = L ^{1Ji l8l ei}

^C

^{l8l cr,}

i==l

where {Eii } ^and {ei} are the usual matrix units for M r and orthonor- mal basis of cr. For a linear map </>: Mm

M n, we define the scalar number

(2.1)

Then we have

r

=

L

i,j=l r

= L

i,j=l

where

M n

Mm is given by

r} is an another orthononnal basis of (Ker V)

and

r

Wi

= L

j=l

i=l, ... ,r,

with an r x r unitary matrix

then we calculate

L

= L L

L

i,j=l

Therefore, we see that the value 8q,(V) does not depend on the choice

of

1]r}. For each s = 1, ... , m

n, we denote

Ms

{V E Mm,n : IIVII

1, rank V ::; s}.

Then Ms is a compact subset of Mm,n.

PROPOSITION

2.1. For

linear map <p : Mm

M

and ana number s

1, ... , m /\ n, the following are equivalent:

(i) <p E

(ii) 6.p(V)

0 for each V EMs.

Proof. If

E JP's and rank V

r

s then

® idr is a positive li map, and so we have (i)

(ii). For the converse, put

s

(2.2) P = L ^{eie;®Eij E Mm®M}

i,j=1 Then we have

s

71

L71j0

j E e

0

j=1

which is nonnegative by the assumption (ii), because Ej=1 ej71;

in Ms with normalization. Since every vector of en ^{® Cs is of}

form 71 in (2.2), we see that (<p ® ids)(P) is positive semi-definite.

conclude that

® id

is a positive linear map, because every pos semi-definite matrix in Mm®M

is the nonnegative linear combina of matrices of the form P in (2.2).

For each s

1, ... ,m /\ n and V

Ms, we define (2.3) Fs[V]

{<p E JP's : 6.p(V)

O}.

If s

1 and V

e71* E Mh then F

[V] is nothing but the maximal F[e,71] of JP'I, as was introduced in [Kt]. For a family V = ^{{Vk :}

1, ... , q} of m x n matrices, we define the completely positive li:

map <Pv: Mm

M

by

s

<Pv : X

L Vk * XVk,

k=1

Recall [C2] that every element in the cone JP'

= JP'mAn arise

this form. If V

E;=I(V71j)71; E M

with orthonormal vec

{ 7]I , . . . ,

7]r}, then we have

(2.4)

q r

8.pv (V)

L L (Vt V 7]i7]jV*Vk7]j, 7]i) k=l i,j=l

q r

=

L L (7]jV*Vk7]j, 7]7V*Vk7]i) k=l i,j=l

~ 1;. (t.(V'voq;,q;)) (t. (V'VO~i'~i))

q q

=

L ITr(V*Vk )1

L I(Vk, V)1

k=l k=l

Therefore, we see that (2.5)

is nothing but the maximal face F[V] of ClF, as was shown in [K2].

Now, we fix an integer s

n.

V

Ms then we see that Fs[V] is an exposed face of lFs by Proposition 2.1. Because Fmf\n[V]

is nonempty by (2.5) and FmAn[V] C Fs[V] , we see that Fs[V] is nonempty. Actually, we have seen in Theorem 3.2 of [K2] that ev- ery F

[V] contains a unital linear map. We know that the trace map T : Mm

M n given by T( X)

Tr (X)I is an interior point of IF s' In order to characterize the interior of IF

we need to calculate 8

(V) as follows:

r

8

(V)

L (Tr [(V7]i)(V

)*]7]j, 7]i) i,j=l

r r

= LTr [(V7]i)(V7]i)*] = L(V7]i' V7]i)

i=l 1=1

r

=

L(V*V7]i,7]i)

Tr(V*V)

IIVII

i=l

PROPOSITION

2.2. Let c/J

lP

where s

1, ... ,

n is

fixed integer. Then the following are equivalent:

(i) c/J is an interior point of lP

(ii) 9</>(V) > 0 for each V EMs.

Proof. If c/J E int lP

then c/J

(1 - t)r + t'lj; with t < 1 and 'Ij; E lP

Therefore, we have

9</>(V) = (1 - t)9

(V) + ^t9.,p(V) = (1 - t) + ^t9.,p(V) > 0

whenever V

Ms. For the converse, we note that the map V

9</>(V) is a continuous function on the compact set Ms. We take c with 0 <

c < 1 such that

V

Ms ==> 9",(V) 2:: c,

and put 'Ij; = (1 - ^_1_)

^{+ _I_c/J>} Then we have

1-e 1-e

9.,p(V)

(1 -_1_) _1-e ⁹

^{(V) +} _1_ 9",(V) _1-e 2:: 0,

for each V E Ms, and so 'Ij; E lPs. Since _1_ > 1, we have c/J E int lP

0 1-e

Now, we are ready to characterize maximal faces of the convex cone lPs for s = 1, ... , m

n.

THEOREM

2.3. Let s

1, ... ,

n be a fixed integer. H V EMs then the set F

[V] is a maximal face of the convex cone lP

Conversely, every maximal face of lP

is of the form F

[V] with an V EMs.

Proof. Take a family V of m x n matrices such that span V

Applying Theorem 2.5 (vii) of [KI], it suffices to show that

Assume that there is

E Ms such that

O. Then we have

The relation (2.4) and Oq,v(W) = 0 shows that W E (span V)..L, and so W is a scalar multiple of V. Hence, we see that Oq,(W) > 0 since

</>

rt. Fs[V]. This contradiction says that Oq,(W) > 0 for each W EMs,

and

E int lP

by Proposition 2.2.

For the converse, let F be a maximal face of lPs with an interior point

Then there is V E Ms such that Oq,(V) =

by Proposition

2.2. Then

E Fs[V] by definition, and so it follows that F C Fs[V]

since Fs[V] contains an interior point of F. The conclusion follows from the maximality of F. 0

3. Relations between lPs and lPt

By the discussion in the previous section, we see that the boundary of lPs is determined by the linear functional

with V

Ms.

5

increase then more linear functionals are needed to determine the boundary of lP

THEOREM

3.1. Let

= 1,2, ... ,

n be given with s <

Then

we have the following:

(i) H V E Ms then Ft[V]

Fs[V] n alPt c alP s.

(ii) HV _{E M t \} Ms then intFt[V]

intlPs.

Proof. Assume that V E Ms. Then by definition (2.3), we have

Fs[V] n lPt

Ft[V], and so Fs[V] n alPt c Ft[V]. Because Ft[V] lies on the boundary of lPt the required relation hold.

For the second part,' assume that there is

E int Ft [V] \ int lP s.

Then

lies on the boundary of lP s and so

E Fs[W] ^{for aWE} Ms.

Since

E Ft [V] E alPt, we have

E Ft[W] by the first part. But,

</>

E int Ft[V] implies that Ft[V] C Ft[W], and we have Ft[W]

Ft[V]

by the maximality. Considering completely positive linear maps in these two faces, we see that {V, W} is linearly dependent. Therefore, we have V EMs. 0

COROLLARY

3.2. Let s, t = 1,2, ... ,m

n be given with s < t, and

V

Mt. Then we have

In [K2], we have seen that every maximal face of P

contains a unital completely positive linear map. We denote by Ps[I] the convex compact set of all s-positive unitallinear maps from Mm into M

and put

(3.1)

for V E Ms. Then every results in this note hold when Ps and Fs[V]

are replaced by Ps[I] and Fs[V,I].

We conclude this note with comments on copositive linear maps.

We denote by tps : Ms

Ms the transpose maps. A linear map

</> :

A

B between C*-algebras is said to be s-copositive if the linear

map

tps : A

Ms

B

Ms is positive, and completely copositive

if it is s-copositive for each natural number s = ^{1,2, ....} We denote by

B) (respectively

BD the convex cone of all s-copositive (respectively completely copositive) linear maps from A into B.

In order to characterize the boundary of ps, we modify (2.1) to define

(3.2)

r

= 2:

i,j=l

The proofs of the following theorem about the convex cone ps are same as before, and will be omitted.

PROPOSITION

3.3. For a fixed natural number s = 1, ... , m /\ n, we have the following:

(i) A linear map

Mm

M

lies in ps if and only if Ot/>(V)

0 for each V EMs.

(ii)

E ps is

interior point of ps if and only if Ot/>(V) > 0 for

each V EMs.

(iii) For each V E Ms, the set

(3.3) FS[V)

E ps : Ot/>(V)

O}

is a maximal face of the convex cone ps. Conversely, every maximal face of ps is of the form FS[V) with

V EMs.

(iv) Let s,

1,2, ... ,

n be given with s < t. HV E Ms then

Ft [V) = FS [V] n apt. H V E M t \ Ms then int Ft [V] ^C int ps.

References

[Cl] M.-D. Choi, Positive linear maps on C*-algebras, Canad. J. Math.24(1972), 520-529.

[C2] , Completely positive linear maps on complex matrices, Linear Alg.

Appl.10(1975), 285-290.

[CL] M.-D. Choi and T.-T. Lam, Extremal positive semidefinite forms, Math. Ann.

231 (1977), 1-18.

[KK] H.-J. Kim and S.-H. Kye, Indecomposable extreme positive linear maps in matrix algebras, Bull. London Math. Soc. 26 (1994), 575-58l.

[K1] S.-H. Kye, Facial structures for positive linear maps between matrix algebras, Canad. Math. Bull. 39 (1996), 74-82.

[K2] , On the convex set of all completely positive linear maps in matrix algebras, Math. Proc. Cambridge Philos. Soc. (to appear).

[LW] C.-K. Li and H.-J. Woerdeman, Special classes of positive and completely positive maps,preprint.

[0] H. Osaka, A class of extremal positive maps in 3 x 3 matrix algebras, Publ.

RIMS, Kyoto Univ. 28 (1992), 747-756.

[R] A. G. Robertson, Positive projections on C*-algebras and extremal positive maps,J. London Math. Soc. (2) 32 (1985), 133-140.

[SW] R. R. Smith and J. D. Ward, The geometric structure of generalized state spaces,J. Funet. Anal.40 (1981), 170-184.

[S] E. Stl1lrmer, Decomposable positive maps on C*-algebras, Proc. Amer. Math.

Soc. 86 (1982), 402-404.

[W] S. 1. Woronowicz, Positive maps of low dimensional matrix algebras, Rep.

Math. Phys. 10 (1976), 165-183.

Department of Mathematics Seoul National University Seoul 151-742, KOREA

E-mail: [email protected]