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COMPLETELY CONTRACTIVE MAPS BETWEEN
C
∗-ALGEBRAS
W. T. SULAIMAN
Received 15 November 2001
We give a simple proof that any completely contractive map betweenC∗-algebras is the top right hand corner of a two completely positive unital matrix operator. Some well-known results are deduced.
2000 Mathematics Subject Classification: 47B65.
1. Introduction. LetAandBbeC∗-algebras,S⊂Abe a subspace, andφ:S→Bbe a linear map. We defineφn:Mn(S)→Mn(B)by
φnaij=φaij. (1.1)
We said that φ is n-positive if φn is positive and thatφ is completely positive if
φn is positive for all n. The mapφ is said to ben-bounded (resp.,n-contractive) ifφn ≤c (resp.,φn ≤1). The mapφis said to be completely bounded (resp., completely contractive) ifφcb=supnφn<∞(resp.,φcc=supnφn ≤1).n -positivity (resp.,n-boundedness orn-contractivity) implies(n−1)-positivity (resp., (n−1)-boundedness or(n−1)-contractivity). The converse is not true in general.
For anyC∗-algebraA,Mn(Mp(A))is identified withMp(Mn(A))because there is a canonical isomorphism betweenMn(Mp(A))andMp(Mn(A))by the rearrangement of ann×nmatrix ofp×pblocks as ap×pmatrix ofn×nblocks with the(i,j)th entry of the(k,)-block becoming the(k,)th entry of the(i,j)th block. This rearrangement corresponds to a pre- and post-multiplying of a given matrix by a unitary and its adjoint.
2. Main results
Lemma2.1(see [1]). Let A be aC∗-algebra,R,S, andT∈AwithT being positive and invertible. Then,
T S
S∗ R
≥0⇐⇒R≥S∗T−1S. (2.1)
Proof. The lemma follows from the identity
0 0
−S∗T−1 I
T S
S∗ R
0 0
−S∗T−1 I
∗
=
0 0
0 R−S∗T−1S
=
T S
S∗ R
−
T1/2 0
S∗T−1/2 0
T1/2 0
S∗T−1/2 0
∗
.
Letφ:A→Bbe a linear map. We denote byφ∗:A→Bthe linear map defined by
φ∗(a)=φa∗∗. (2.3)
LetS(A)be the linear subspace ofM2(A)given by
S(A)=
λ a b µ
:a,b∈A, λ,µ∈C
. (2.4)
LetΦ:S(A)→S(B)be defined by
Φ
λ a b µ
=
λ φ(a)
φ∗(b) µ
. (2.5)
Theorem2.2. The mapφisn-contractive which impliesΦisn-positive. Proof. Let
X=
λij aij
bij µij
∈MnS(A)+
. (2.6)
We may identifyXwith
Y=
λij aij
bij µij
. (2.7)
Therefore, Y is positive which implies [λij] and [µij]are positive in Mn(C). Since
Y ≥0 if and only ifY+(1/m)I >0 for everym∈I, we may assume that[λij]and
[µij]are invertible. We have
λij aij
a∗ji µij
≥0
⇐⇒ λij
aij
aij
∗
µij
≥0, via identification
⇐⇒µij
≥aij
∗
λij
−1
aij
, byLemma 2.1
⇐⇒I≥µij
aij
λij
λij
aij
µij
, whereµij
=µij
−1/2
, λij
=λij
−1/2
⇐⇒I≥λij
aij
µij
∗
λij
aij
µij
⇐⇒1≥λij
aij
µij
∗
λij
aij
µij
⇐⇒1≥0λij
aij
µij
2
⇐⇒1≥λijaijµ ij
⇐⇒1≥0 n
s,t=1
λusastµ tr u,r
⇐⇒1≥ φn
n
s,t=1
λusastµ tr u,r
⇐⇒1≥ n
s,t=1
λusφnastµ tr u,r ⇐⇒1≥λij
φaij
µij
⇐⇒1≥λij
φaij
µij
2
⇐⇒1≥λij
φaij
µij
∗
λij
φaij
µij
⇐⇒I≥λij
φaij
µij
∗
λij
φaij
µij
⇐⇒µij
≥φaij
∗
λij
−1
φaij
⇐⇒
λij φaij
φaij∗ µij
≥0
⇐⇒
λij φaij
φ∗a∗ji
µij
≥0, via identification
⇐⇒φn
λij aij
a∗ji µij
≥0.
(2.8)
This completes the proof of the theorem.
Theorem2.3. Letφ:E→Bbe a map from a selfadjoint subspaceEof aC∗-algebra
Ainto aC∗-algebraB. Define a mapΨ:S(E)→Bby
Ψ
λ a b µ
=(λ+µ)I+φ(a)+φ∗(b). (2.9)
(i) The mapφisn-contractive⇒Ψisn-positive. (ii) The mapΨis2n-positive⇒φisn-contractive.
Proof. (i) Define mapsΦ:S(E)→S(B),δ:M2(B)→Bby
Φ
λ a b µ
=
λ φ(a)
φ∗(b) µ
,
δ
a b c d
=a+b+c+d.
(2.10)
The mapΦisn-positive by Theorem 2.2. Asδisn-positive (in fact it is completely positive), thenΨ=δ◦Φisn-positive.
(ii) There are two methods to prove (ii).
Method1(see [4]). Via identification, we have
Ψn
λij
aij
aij
∗
µij
=λij+µijI+φnaij+φ∗ n
bij
,
λij,µij∈Mn(C); aij,bij∈Mn(A).
(2.11)
Let[aij] ≤1, we want to show thatφn[aij] ≤1. Now
aij≤1 ⇒ In
aij
aij
∗
In
≥0
⇒
In 0 0
aij
0 0 0 0
0 0 0 0
aij
∗ 0 0
In
≥0
⇒
Ψn
In 0
0 0
Ψn
0
aij
0 0
Ψn
0 0
aij
∗ 0
Ψn
0 0
0 In
≥0
⇒
In φn
aij
φ∗n
aij∗ In
≥0
⇒φn
aij≤1
⇒φn≤1.
Method2. SinceΨ2n=δ2n◦Φ2nandΨ2n,δ2nare both positive, thenΦ2nis also positive. AsΦ2nis unital, thenΦ2n ≤1. Hence
Ψ2n≤δ2nΦ2n=2·1=2. (2.13)
We identified
λij aij
bij µij
⊗Mn with
H A B K , (2.14) and write λij aij
bij µij
⊗Mn←→
H B A K , (2.15) where
H=λij, K=µij, A=aij, B=bij, H,K∈Mn, A,B∈Mn(A);
Ψ2n H A B K ⊗M2
=δ2n
Φ2n
H A
B K
⊗M2
=δ2n
H φ(A)
φ∗(B) K
⊗M2
←→δ2n
H⊗M2 φ(A)⊗M2
φ(B)⊗M2 K⊗M2
=
H φ(A)
φ∗(B) K
×4;
4φn(A)=
4
0 φn(A)
0 0 = Ψ2n 0 A 0 0 ⊗Mn
=Ψ2n
0 A 0 0 ⊗Mn =4A
⇒φn≤1.
(2.16) (iii) The proof of (iii) is obvious.
(iv) Via identification, we have
Ψ2n
H A
B K
=H+K+φn(A)+φ∗n(B)
φn(A)= Ψ2n 0 A 0 0 = δn◦Φn
0 A 0 0
=δnΦn
0 A 0 0 ≤2A
⇒φn≤1.
(2.17)
3. Applications
Proof. Define a mapΨ:S(E)→Bby
Ψ
λ a b µ
=(λ+µ)I+φ(a)+φ∗(b),
φis contractive⇒Ψis positive (Theorem 2.3)
⇒Ψis completely positive [1, Proposition 1.2.2] ⇒φis completely contractive (Theorem 2.3).
(3.1)
Theorem3.2(see [4, Theorem 1.1.13]). LetEbe a closed selfadjoint subspace of a C∗-algebra. LetBbe a commutativeC∗-algebra. LetΦ:E→Mn(B)be a linear map. If Φisn-positive then, it is completely positive.
The following theorem is a generalization ofTheorem 3.2.
Theorem3.3. LetEbe a selfadjoint subspace of aC∗-algebra. LetBbe a commu-tativeC∗-algebra. Letφ:E→Mn(B)be a linear map. Ifφisn-contractive then, it is completely contractive.
Proof. Define a mapΨ:S(E)→Mn(B)by
Ψ
λ a b µ
=(λ+µ)I+φ(a)+φ∗(b),
φisn-contractive⇒Ψisn-positive (Theorem 2.3)
⇒Ψis completely positive (Theorem 3.2) ⇒φis completely contractive (Theorem 2.3).
(3.2)
Theorem3.4(see [4, Theorem 1.1.12]). LetE be a closed selfadjoint subspace of a C∗-algebra, A containing the identity, and let φ:E →Mn =B(Cn) be n-positive map. Then,φpossesses a completely positive extensionΨ:A→Mnand therefore,φis completely positive.
In 1983 Smith [3] proved the following theorem.
Theorem3.5(see [3]). Letφ:A→Mnbe bounded. Thenφcb= φn. Here, we generalizeTheorem 3.5by giving the following theorem.
Theorem 3.6. LetE be a closed selfadjoint subspace of aC∗-algebra. If φ:E→
Mn(C)isn-contractive, thenφis completely contractive.
Proof. DefineΨ:S(E)→Mn(C)by
Ψ
λ a b µ
=(λ+µ)I+φ(a)+φ∗(b),
φisn-contractive⇒Ψisn-positive (Theorem 2.3)
⇒Ψis completely positive (Theorem 3.4) ⇒φis completely contractive (Theorem 2.3).
References
[1] M. D. Choi,Some assorted inequalities for positive linear maps onC∗-algebras, J. Operator Theory4(1980), no. 2, 271–285.
[2] R. I. Loebl,Contractive linear maps onC∗-algebras, Michigan Math. J.22(1975), no. 4, 361–366.
[3] R. R. Smith,Completely bounded maps betweenC∗-algebras, J. London Math. Soc. (2)27 (1983), no. 1, 157–166.
[4] W. T. Sulaiman,Some classes of linear maps betweenC∗-algebras, Ph.D. thesis, Heriot-Watt University, Edinburgh, 1985.
W. T. Sulaiman: College of Education, Ajman University, P.O. Box346, Ajman, United Arab Emirates
