Decomposition of Completely Positive Maps

Math. Nachr. 185 (1997), 227-237

Decomposition of Completely Positive Maps

By MICHAEL PAUL of Dresden

(Ff.eceived April 4, 1995)

(Revised Version November 9, 1995)

Abstract. The paper is concerned with completely positive maps on the algebra of unbounded operatore t + ( D ) and on its completion C(D, D+). A decomposition theorem for continuous positive functionrls is proved in [Tim. Loef.), and [Scholz B l ] contains a generalization to maps into operator algebra on finite dimensional Hilbert apacea No. The aim of the present paper is to construct an analogous decompoeition without the assumption that No is finite dimensional. Moreover, the Krrus - theorem [Kraua] is proved for normal completely pd t ive mappinga on L(P, D+). The paper is o r g a n i d M follows. Section 1 contuns the necessary definitions and notations. In Section 2 we prove the decomposition theorem. Section 3 deal with the rtructure of the normal completely positive mappings.

1. Preliminaries

For a dense linear manifold 2, in a complex Hilbert space 31 the set of linear operators L+(2,) := {a : aV C D, a*V C D} is a *-algebra with respect to the usual operations and the involution a I+ a+ := a*Jz). The identity map I is the unit element of L+(D). Let v(V) denote the space of all projections whose range is contained in D. We shall write S(2,) for the finite rank operators of C+(D). The graph topology t on 'D is defined by the family of seminorms { 11 - Ilo := 11. - 11 : a E t+(D)}. We assume that 2, is an (F) - domain, i. e., D[t] is a FZBchet space.

Let D+ denote the conjugate vector space of the dual V[t]'. We use the notation (9, cp') = (cp', cp) := cp'(cp) for all cp E 2, and cp' E 2,+, then (. , a ) extends the scalar product of 3.1 with respect to the canonical lineax embedding V C 71 C V+.

Let L(D, D+) be the *-vector space of all linear mappings x from 23 into Vt for which exist a, b E t+(D) such that I(zcp, +)I 5 IlacpIIllhbII holds for all cp, 1c, E V. The involution is given by (z+cp, +) := (z+, p)for all cp, 1c, E V. L+(V) and the space of bounded operators !B('H) are contained in L(D, ID+). If x E C(D, D+) and

1991 MaQematia Subjzct CIm@ation. Keyurordd and phroscs.

228 Math. Nachr. 186 (1997)

4 , b E L+(D), we define the partial product 4 o z b E L(D, D+) by

(aozbcp, +) := (zbcp, 4++)

for all cp, $ E 2).

4 2 I. We define We denote by F(D, a+) the finite rank mappings in L(D, D+). Let a E C+(D) and

fa := {z E L(D, D+) : 3 c set- I ( ~ ( P , $)I I cll~lllla+ll ,vc~,O E D} , ua := {z E t ( D , D+) : I ( Z C P , $)I 5 II~llll~tlrII 9 VV,+ E 2)) 9

za(2) := inf{c : l(zcp, $11 5 c l l ~ l l 1I~Oll , '% $ E 2))

~b(z) := inf {c : ~(zcp, c p ) ~ I c IIWII~, vv E 2) ) .

It is easy to verify that Za(z) 5 42h(z) I 4 2 a ( ~ ) .

and

Let H a denote the Hilbert space D(E) with inner product (cpl +), := ( Zcpl E$). For each z E La the seaquilinear form (z. - ) is continuous on 1-1, x Ha and, hence, there exists a bounded operator 2 E % ( H a ) such that (zcp, $J) = (29, $)a for all cp,$ E 2). By the same equation each 5 E % ( H a ) defines an element z of La. Moreover, we have I , (%) = ~ ~ 5 ~ ~ B ~ w ~ l . This identification of La and B(3-1,) is used in Section 3.

We furthermore define some topologies on C(D, Do+). The uniform topology TV is given by the family of seminorms

p M ( S ) := SUP I(zcp,$)l 8

* d € M

where M runs over all t- bounded sets of 2). The ultraweak topology ow is determined by the seminorms

where (cpn), ($n) C 2) satisfy

The weak--operator topology on the space C(V, V+) is defined by the family of seminorms z w [(zcp, $) I with cp, $ E D.

The locally convex space L(D, D+)[Tv] is complete for (F)-domains D. Dewte by V(D, D+) be the TQ -closure of F(D, D+) in t ( D , Do+).

Define

B~(D) := { t E ~ ( 3 - 1 ) : t~ c D, t"H c D, atb nuclear, for all 4 , b E Z+(D)}.

For any t E %@) there are sequences (Cpn), (&) C 2, with

Paul, Decomposition of Completely Positive Maps 229

n= 1 n=l

Let z E L(V, Do+). We define the generalized trace by

For more details see [Schmuedgen]. The projective topology rx on 2h(D) is generated by the seminorms va(t) := Tr latal

with a E L+(D), a 2 I. If 2) is an (F)-domain, then ?&(D)[T,] is a R:Bchet space and the dual is &(D, D+).

For any a, -continuous linear mapping N on L(D, Do+) there exists the predual mapping N, : !&(D) -+ !B@) such that

(1.2) tr(N+t)z = t r f (Nz) , --4

for all t E 231(D), for all z E C(D, D+).

Next we define some notions of positive mappings. An element z of L(D, D+) is called positive. (z 2 0) if (z'p, 'p) 2 0 holds for all cp E D, and an (n,n)-matrix (zij) E &(L(D, D+)) is positive ((zij) 2 0) if c' (zij 'pi, 'pi) 2 0 holds for all ~ 1 , . . . ,pn E D. For any set M of operators, we >iZl write M+ for the subset of positive operators.

A mapping T : L(D, D+) + L(D, D+) is said to be positive if z 2 0 implies T z 2 0. Finally, T is said to be completely positive if Tn is positive for all n, where Tn is defined

A mapping N on L(D, Do+) is called normal if N is uw - continuous. A rv -continuous mapping S is said to be singular if S vanish on V ( D , D+). There are (F) - domains 2) such that V(D, D+) and L(D, DO+) coincide, i.e., in these cases nontrivial singular mappings do not exist. There are descriptions for such domains of the form 'D, = nn D(an), where a is a self- adjoint operator ([Tim. Lass 76, Tim. Laef 86 b]).

We sometimes identify the space Mn(L(D, D+)) with L(Dn, D:), where D, denotes the direct sum 2, @ - - - @ D (n times). Of course, Dn is considered as a subspace of the Hilbert direct sum 3-1, := 3-1 @ * * @ 3-1.

on W(L(D9 '0')) by Tn(z i j ) t j= l := (Tzij):j=l*

2. Decomposition theorem

In this section we will prove the following:

Theorem 2.1. Assume that D is an (F) -domain. Let T be a rv -continuous completely positive mapping on C(D, V+). Then there art unique rv -continuous completely positive mappings N, S such that T = N + S, N is normal and S is singular.

230 Math. Nachr. 186 (1997)

The assertion is also true for mappings T : C(D1, '0:) + C(D2, D;), where D1 , D2 are (F) - domains. We only prove the Theorem 2.1 in the particular case to simplify notations.

For Dz = 6 , respectively D2 finite dimensional, we get the decomposition theorem of (Tim.Loef.1, respectively (Scholz 91). It is not essential that the mappings of [Tim. Loef.], [Scholz 911 are defined on C+(D), because any rp -continuous mapping extends continuously to C(D, D+).

The proof of Theorem 2.1 is divided into several lemmas. First we give a criterion for rp -continuity.

Lemma 2.2. Suppose that N : C(D, D+) -$ C(D, D+) is positive and ow - wntinu-

1) N, is rx - continuous ; 2) N is rp - continuous.

Proof. We have already noted that for a a,,, -continuous mapping N the predual mapping N, : Bl(D) -+ !&(D) exists. Fix a E C+(D), a 2 I with N I E L a . For 2 = x+ E CI we have

ow. Then:

-Zf(s)I 5 2 5 Zf(z)I.

The positivity of N yields

(2.1) - If(z) N I 5 N Z 5 Zf(z)NI,

finally, NCI C C,. We prove next the inequality

(2.2) vI(N,t) 5 cv,(t) , for all t E !&(D).

Let t E B1(D). For any z E B(3.1) with 1 1 ~ 1 1 = 1, we have Nz E Ca and by [Schmued- gen, Proposition 3.2.31 it follows that there is a yo E B(3.1) such that

Nz = a o y Z a , IlgzII 5 c := 8Zi(NI).

From this we see

T r ( N , t ) z = t r f (Nz) = tr i(aoy,a) = Tr(ata)y , .

Now we can estimate

which proves (2.2).


F'rom (2.2) we conclude that N, is T~ -closable. For this, suppose t , r? 0 and Net, r? t , where t E ?&(D). By (2.2), we have

0 5 VI(N*tn) 5 CVa(tn) - 0 ,

therefore, v I ( t ) = 0 and t = 0. Because T,, is an (F)- topology, (i) is proved. Assumption (ii) is a direct consequence of [Schaefer: TVS, IV, 7.41. 0

Our next Lemma is concerned with the closure of sets of finite - dimensional operators in the positive part of the rp - bounded sets U,.

Lemma 2.3. Let a E .C+(D), a 2 I . Then S(D)+ n U, is both ow -dense and

Proof. By [Schmuedgen, Proposition 3.5.31 the uw - topology and the weak-operator topology coincide on Ua. We prove the assertion for the weak - operator topology. Each closure is taken with respect to L(D, D+).

Let z E Ua, z 2 0. There is y E B(7-f) with 0 5 y 5 I and 5 = a o ya by [Schmuedgen, Proposition 3.2.31.

We start with the rv - topology. Here [Schmuedgen, Theorem 3.4.11 yields

weak - openator dense in (U,)+.

By the rv - continuity of z t-) a o za on t ( V , D+) it follows

2 = a o ya E {a o (eye)a : e E Y(D))'".

Since eye and a are both in L+(D), we can replace the partial product a o (eye)a by the usual product a(eye)a. Because the weak-operator topology is weaker than the rp -topology, it follows

z E {a(eye)a : e E P(v) )WO. Next we prove

W O

aeyea E S(D)+ nu, . It is enough to show that a weak-operator neighbourhood of the form

I n

u = { z E f ( D , D+) : c I ( z p i , $ i ) - (=yeapi, $ i > l < 7

i-I

with pi, +i E D, c > 0, contains an element of S(D)+ n Ua. For this, consider the finite-dimensional space lin {eacp,, ea$, : i = 1, . . . , n} and

denote by p the projection on it. Define z := aepypea. It is easy to check that J is an element of U. Because y 2 0, we have z 2 0. Since p is a finite-dimensional projection, it follows z E $(Do). Finally, z E U, is an immediate consequence of

0 0 5 ewpe 5 I .

Lemma 2.4. 1. For any projection p E B(X) there is an incmsing net of finite- dimensional projections pa with p = sup, p a .

232 Math. Nachr. 186 (1997)

2. For any positive y f B('H)+ there is an increasing sequence of positive opemtors I n E b('H)+ with y = Supyn and each gn is a finite sum of projections with positive coeficiences.

n

Proof. For (i) consider the net of all finite-dimensional projections in the space 0

We next define the normal mapping N with the help of the predual mapping N,, which is defined by a "restriction" of the dual mapping of T. For this, let t E B@). The functional ut( - ) = tr i(.) is rp -continuous on L(D, D+) by (Schmued- gen, Proposition 5.3.51. Since T is r~-mntinuous by assumption, the functional ut(T. ) is rp -continuous, too. Usually, this functional is not a, -continuous and, therefore, not of the form ua( .) on the whole of t ( D , D+). However, it follows from [Schmuedgen, Corollary 5.3.21 that there is an s E a@) such that

p'H. Assumption (ii) follows by the spectral theorem.

wt(Tz) = trt*Tz = wa(Z) = tr Bz, for all z E 3 ( D , '0').

Now we define N,t := s . Obviously, N, is correctly defined, linear, positive and

(2.3) t r i T z = t r ( K t ) z , for all z E T ( D , D + ) ,

by definition.

Lemma 2.5. N, is rrr -continuous.

Proof. It is enough to show that N, is rrr -closable. Assume tn 2 0 and N*t, 3 t. 0

With respect to the dual pair (!&(D)[rr], L(D, DO+)) the dual mapping N := (N,)* is defined on t ( D , D+). Moreover, N is linear, positive and nw-continuous. By Lemma 2.2 N is rp -continuous.

(2.3) yields t r f z = 0 for all z E T(D, D+). Therefore, t = 0.

Lemma 2.6. N is completely positive.

Proof. From (2.3) it follows that T and N coincide on F(D, D+). Therefore, N is completely positive on T(D, D+).

Fix n E IN. Let ( Z i j ) E &(L(D, D+))+ be a positive matrix. We apply Lemma 2.3 to the space L(Dn, 'of). (We replace a of Lemma 2.3 by the positive matrix ( z i j ) + I ) . Then there exists a net of positive matrices (23) E s('on)+ such that ( ~ 3 ) + (z,j) in the a, -topology on &(Dn, Df). This implies, that the nets x 3 converge to xi, in the a,-topology of t ( D , D+) for all i , j.

By the a, -continuity of N we obtain N z $ % Nx,, for all i , j . From this we have ( N s g ) -t ( N s i j ) in the weak-operator topology of f.(D,, Dz).

Because (s3) E T(D, D+)+, it follows that ( N Z ~ ) 2 0 for all a and, therefore,

Set S := T - N. Obviously, S is linear and rv-continuous. As mentioned in the

( N z i j ) 2 0. 0

preceding proof, T and N coincide on T(D, D+), that is, S is singular.


Lemma 2.7. S is completely positive.

Proof. We have to show Tn 1 Nn for all n E IN. Because S is singular it follows

(2.4) Tnx = N O S , for all x E F(’Dn, D:).

Let u E L+(’D,), u 2 I, and p E B(3-1n) be a projection. By Lemma 2.4 there is an increasing net of finite-dimensional projections (pa) E B(‘Hn) with supremum p. Clearly, a 0 p,a is an increasing net in L(Dn, 23:) with supremum a o pa and a o p,a E T(Dn, ’0,). By positivity of Tn, we have

(2.5) Tn(aop,a) 5 Tn(aoP), for all C Y .

The increasing net (a o p,a) converges to the supremum a o pa in the 0, -topology on L(’Dn, D:). It follows as in the proof of Lemma 2.6 that N ( a o paa) converges to N ( a opu) in the weak-operator topology on L(Dn, D:). Combining this with (2.4) and (2.5), we obtain

This inequality remains true if p is replaced by a finite sum of projections with positive coefficients.

Let y E %(En), y 1 0. By Lemma 2.4 there exists an increasing sequence (gk) of !B(3-1,,) with aupremum y, such that each yk is a finite sum of projections of B(3-1,) with positive coefficients. We have already shown Tn(a o yka) 1 Nn(a o g k a ) for all k. Again, a o g k a is an increasing net in L(Dn, D:) with supremum a o va and it follows immediately

Tn(a 0 W ) 1 &(a 0 w). l’he proof is complete, because any x E L(D,,, D;) is of the form x = a o ya by Schmuedgen, Proposition 3.2.31. 0

Now we prove the uniqueness assertion. Assume there are decompositions T = Vl + Sl = N2 + Sa. The mapping N1 - N2 = 5’2 - 5’1 is normal and singular, ,herefore, it vanishes on L(D, D+). The proof of Theorem 2.1 is complete.

T(a opal 2 N(. ow).

1. Normal completely positive mappings

Suppose that N is a normal completely positive linear mapping on B(’H). By the ltinespring-theorem [Stinespring 551 there exist a *-representation ?r of B(N) into 3 ( K ) and a bounded linear mapping V : ‘H 4 K such that Nx = V*lr(x)V for all 2 in 3(3-1). Because each normal *-representation R of B(31) is a direct sum of identical epresentations of B(’H) [Neumark, 22.33, we can write N as

3.1)

,here each vi is a bounded mapping on 3-1 and the sum converges with respect to ie -topology (see [Kraus]). (On B(7f) this topology is usually called ultraweak ,pology. )

234 Math. Nachr. 185 (1997)

Now we prove a similar representation for mappings on &(D, D+).

Theorem S.l. Assume that D[t] is an ( F ) -domain. Let N be a normal completely positive mapping on &(D, V+). Then there are an indez set K , continuous linear mappings y : D[t] + D[t] for all i E K such that

N X = C C ~ + X Q , f o r all x E L(D, D+), i € K

where the sum converges with respect to the crW -topology. Momover, we have the following representation for the pndual mapping

N , t = C yicr, for all t E !&(D), i€ K

where the sum converges with respect to the T, -topology.

Remark 3.2. 1. The expression c t x y is defined by

(3.2) ( (c txci)cp,I jr) := ( x c i c p , c i + ) , for all P , + E D .

Generally, Q is an element of t ( D , D+). Provided that y is in C+(D), then C ~ Z C , coincides with the partial product c t o xy. 2. Each element t of B@) possesses an extension i : D+ -+ D, so the expression

Q ~ C : is a composition of mappings. Writing t as in (l.l), we have

(3.3) nn 1

3. The theorem is also true for mappings N : L(D1, Dr) -t t ( D 2 , D:), where Vl;

4. In the case of a separable Hilbert space the index set K is countable. 5. The dual mapping of c: . y leaves the set of all vector functionals wV,*(. ) :=

( '9, +), cp, rl, E D, invariant. Such mappings are described in [Paul 194, Paul]. 6. If N is defined on &+(D), there is a rp-continuous extension (see Lemma 2.2)

and the theorem applies. Unfortunately, the summands c: . y are not mappings 011 t + ( D ) in general.

Proof. Although, there exists a Stinespring-representation for the mapping N (see [Paul]), the proof is different from the bounded case, because there is not iL

similar characterisation of the representations of L(D, D+). We will only sketch the proof. For a comprehensive version see [Paul] or [Paul 1941,

which contains a detailed proof for unital mappings ( N I = I). Fix a E &+(D), a 2 I with N I E t,. As in the proof of Lemma 2.2, we haw

NLI C C,. According to the identification of t, and 23(EH4), N induces a mapping

Dz are (F) -domains.

fi : B(3.1) --+ B(3.1,) by

((Nz) cp,rl,) = ((fiz) cp, +),, for all cp, 1c, E D , and for all z E B(3.1).


We will apply the Kraus-representation (3.1) to this mapping. Step 1: We have to check the assumptions for fi. By (2.1)) we have

L(Nz) I (81:(NO)b(z) , for all hermitian z E f 1 . Then, the continuity of i? is an easy consequence.

Considering the dexxsity of D in there is no problem to carry the complete positivity from N to N.

It remains to prove the continuity of fi with respect to the ultraweak topologies of B(X) and !Y3(Xa), this is equivalent to the existence of a predual mapping E, : !&(Xa) + ?&('Id), for which "r,(fi,t)z = T r ~ , , t ( % i ~ ) for all t E !B1(7&) and z E %(X). For this, we define a mapping S : !&('Ha) n S ( D ) -+ !&(X). Fix t = C,"==, +,, 8 Vn E !&(D), &, qn E D. We denote by t". the finite dimensional operator of Bl(Xa) which is defined by the same vectors +,, (P, E 2, C ' Ida, i.e.,

N N

n= 1 n= 1

The predual mapping N, exists by the u,,, -continuity of N. So we can define Stxa :=

Next we show that S is continuous with respect to the norm- topologies of !&(?la) Not E bl(Z)) c !Bl(X).

and b1(1.i). By definition of S it follows that

Becauee Nz is in La, we can use the trace of !B(X,,), therefore

Finally, the properties of the trace yield

This inequality proves the desired continuity of S.

following equation S extends continuously to ! B I ( ~ & ) ~ because S is defined on a dense subspace. The

h

Tk(StHo) z = trN,tz = t d N z = "rxatRa%z, for all z E CI ,

shows that S is precisely the predual mapping of fi. Step 2: By the Kraus-representation (3.1) there are bounded operators v, from

X,, into 1.i such that

(3.4)

236 Math. Nachr. 186 (1997)

where the sum converges for each x E B(7.1) with respect to the ultraweak topology of B(%).

Step 3: We define the mappings ci : 2, + 'H by q := v i (p . In case x = I it follows

and, hence, q E C(D, D+). Step 4: With the help of an idea of the proof of Proposition 5.3.1 of [Schmuedgen]

we can show ciD C 2). Each ci is closable on D[t] by (3.5). The continuity results from the closed graph theorem.

It follows immediately that c: .ci is a rp -continuous mapping on B(D) into C(D, D+) and the extension to the whole of C(D, D + ) is precisely determined by (3.2). Moreover, for any finite subset KO c K and x E t ( D , D+) the partial sum

is an dement of C(D, D+).

and, therefore, the partial sum Once more the continuity of cj and [Schmuedgen, Proposition 5.1.121 imply (3.3)

(3.7) iEKo

is an element of B@) for each t E 'Is@).

x E L(D, D+), x 1 0 and t E B@), t 10. We will show that Step 5: It remains to prove that the partial sums (3.6) and (3.7) converge. Fix

For this, x is approximated by positive bounded operators xa E CI (see [Schmuedgen, Theorem 3.4.11). Then we write (3.8) in the form

iE K

This sum converges to ' I k ~ ~ t " ~ f i z , by the Kraus-representation (3.1). From (3.8) it follows easily that the sum CiEK v~(Q&$) is finite for any T~ -seminorm

Vb. Furthermore, the sum CiEK ciic' is T= -convergent, because !&(Z))[T,] is afidchet space.

For the seminorm UI we have

The uniqueness of the limit yields CiEK ciic; = N,t. This remains true for any t of Bl(D), because t is a linear combination of four positive elements of Bl(3.1).


0 The T~ -convergence of this sum and (1.2) imply the cr,-convergence of (3.6).

Acknowledgements

The author would like to h a n k Pmf. W. TIMMERMANN for many helpful diScws8ona.

References (Davies] [Fried - Schm 89)

[Kad. Ring.]

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[Neumark]

[Paul]

[Paul 941

[Schaefer: W S ] (Schmuedgen 781

(Schmuedgen]

[Schole 911

[Stinespring 551

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Ztutiiui fir Analysb Fachn'chiung Maihcmatik Fokuliii Maihcmatik und Naiurwisscnschaften Technirche UniversiiP Dresden 0-01063 Dnrdcn Germany

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Decomposition of Completely Positive Maps