COMPLETELY CONTRACTIVE MAPS BETWEEN C -ALGEBRAS

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1 IJMMS 32:8 2002) PII. S Hindwi Publishing Corp. COMPLETELY CONTRACTIVE MAPS BETWEEN C -ALGEBRAS W. T. SULAIMAN Received 15 November 2001 We give simple proof tht ny completely contrctive mp between C -lgebrs is the top right hnd corner of two completely positive unitl mtrix opertor. Some well-known results re deduced Mthemtics Subject Clssifiction: 47B Introduction. Let A nd B be C -lgebrs, S A be subspce, nd φ : S B be liner mp. We define φ n : M n S) M n B) by φ n = φ ). 1.1) We sid tht φ is n-positive if φ n is positive nd tht φ is completely positive if φ n is positive for ll n. Thempφ is sid to be n-bounded resp., n-contrctive) if φ n c resp., φ n 1). The mp φ is sid to be completely bounded resp., completely contrctive) if φ cb = sup n φ n < resp., φ cc = sup n φ n 1). n- positivity resp., n-boundedness or n-contrctivity) implies n 1)-positivity resp., n 1)-boundedness or n 1)-contrctivity). The converse is not true in generl. For ny C -lgebr A, M n M p A)) is identified with M p M n A)) becuse there is cnonicl isomorphism between M n M p A)) nd M p M n A)) by the rerrngement of n n n mtrix of p p blockssp p mtrix of n n blocks with the i,j)th entry of the k, l)-block becoming the k, l)th entry of the i, j)th block. This rerrngement corresponds to pre- nd post-multiplying of given mtrix by unitry nd its djoint. 2. Min results Lemm 2.1 see 1). nd invertible. Then, Let A be C -lgebr, R,S, nd T A with T being positive ) T S 0 R S T 1 S. 2.1) R S The lemm follows from the identity ) ) ) T S S T 1 I S R S T 1 I ) ) ) ) T S T 1/2 0 T 1/2 0 = 0 R S T 1 = S R S T 1/2 0 S T 1/2. 0 S 2.2)

2 508 W. T. SULAIMAN Let φ : A B be liner mp. We denote by φ : A B the liner mp defined by φ ) = φ )). 2.3) Let SA) be the liner subspce of M 2 A) given by ) λ SA) = :,b A, λ,µ C. 2.4) Let Φ : SA) SB) be defined by ) ) λ λ φ) Φ = φ. 2.5) b) µ Theorem 2.2. The mp φ is n-contrctive which implies Φ is n-positive. Let ) X = λ ) +. Mn SA) 2.6) b We my identify X with Y = ) λ. 2.7) Therefore, Y is positive which implies λ nd re positive in M n C). Since Y 0 if nd only if Y + 1/m)I > 0 for every m I, we my ssume tht λ nd re invertible. We hve ) λ 0 ji λ ) 0, vi identifiction µ 1 λ, by Lemm 2.1 I µ λ λ µ I λ µ ) λ, where µ ) µ 1/2, 1/2 = µ λ = λ

3 COMPLETELY CONTRACTIVE MAPS BETWEEN C -ALGEBRAS 509 λ ) λ 1 µ µ ) λ ) µ λ 1 µ ) n 1 0 λ us st µ tr s,t=1 u,r n 1 φ n λ us st µ tr s,t=1 u,r n 1 λ ) usφ n st µ tr s,t=1 u,r 1 λ ) φ µ 1 λ ) 2 φ µ λ ) ) λ ) 1 φ µ φ µ ) λ ) ) λ ) ) I φ µ φ µ ) 1 ) φ λ φ )) λ φ ) 0 φ µ λ φ )) φ ) 0, vi identifiction ji ) φ n λ 0. ji 2.8) This completes the proof of the theorem. Theorem 2.3. Let φ : E B be mp from selfdjoint subspce E of C -lgebr A into C -lgebr B. Define mp : SE) B by ) λ = λ+µ)i +φ)+φ b). 2.9) i) The mp φ is n-contrctive is n-positive. ii) The mp is 2n-positive φ is n-contrctive. iii) The mp φ is completely contrctive is completely positive. iv) The mp is n-positive φ n 2.

4 510 W. T. SULAIMAN i) Define mps Φ : SE) SB), δ : M 2 B) B by ) ) λ λ φ) Φ = φ, b) µ ) b δ = +b +c +d. c d 2.10) The mp Φ is n-positive by Theorem 2.2. Asδ is n-positive in fct it is completely positive), then = δ Φ is n-positive. ii) There re two methods to prove ii). Method 1 see 4). Vi identifiction, we hve λ ) n = λ + µ I +φn +φ n b, µ 2.11) λ, µ Mn C);, b Mn A). Let 1, we wnt to show tht φ n 1. Now I n 1 0 I n I n 0 I n n I n 0 n 0 I n φ n φn 0 I n n 0 n 0 0 I n 2.12) φn 1 φn 1.

5 COMPLETELY CONTRACTIVE MAPS BETWEEN C -ALGEBRAS 511 Method 2. Since 2n = δ 2n Φ 2n nd 2n, δ 2n re both positive, then Φ 2n is lso positive. As Φ 2n is unitl, then Φ 2n 1. Hence 2n δ2n Φ2n = 2 1 = ) We identified ) λ M n b ) H A with, 2.14) nd write ) ) λ H B M n, 2.15) A K b where H = λ, K = µ, A=, B = b, H,K Mn,A,B M n A); 2n H A ) M 2 = δ2n H A Φ2n ) M 2 = δ2n H φa) ) M φ 2 B) K ) ) H M2 φa) M 2 H φa) δ 2n = φb) M 2 K M 2 φ 4; B) K 4φn A) ) 0 φn A) = 4 = 0 A 2n ) M n = ) 2n 0 A M n = 4 A φn ) iii) The proof of iii) is obvious. iv) Vi identifiction, we hve ) H A 2n = H +K +φ n A)+φ nb) φn A) ) ) 0 A = 2n = 0 A δ n Φ n 2.17) = ) δn Φn 0 A 2 A φn Applictions Theorem 3.1 see 2). Let E be liner subspce of C -lgebr nd let B be commuttive C -lgebr. Let φ : E B be liner mp. If φ is contrctive, then it is completely contrctive.

6 512 W. T. SULAIMAN Define mp : SE) B by λ ) = λ+µ)i +φ)+φ b), φ is contrctive is positive Theorem 2.3) is completely positive 1, Proposition ) φ is completely contrctive Theorem 2.3). Theorem 3.2 see 4, Theorem ). Let E be closed selfdjoint subspce of C -lgebr. Let B be commuttive C -lgebr. Let Φ : E M n B) be liner mp. If Φ is n-positive then, it is completely positive. The following theorem is generliztion of Theorem 3.2. Theorem 3.3. Let E be selfdjoint subspce of C -lgebr. Let B be commuttive C -lgebr. Let φ : E M n B) be liner mp. If φ is n-contrctive then, it is completely contrctive. Define mp : SE) M n B) by λ ) = λ+µ)i +φ)+φ b), φ is n-contrctive is n-positive Theorem 2.3) is completely positive Theorem 3.2) 3.2) φ is completely contrctive Theorem 2.3). Theorem 3.4 see 4, Theorem ). Let E be closed selfdjoint subspce of C -lgebr, A contining the identity, nd let φ : E M n = BC n ) be n-positive mp. Then, φ possesses completely positive extension : A M n nd therefore, φ is completely positive. In 1983 Smith 3 proved the following theorem. Theorem 3.5 see 3). Let φ : A M n be bounded. Then φ cb = φ n. Here, we generlize Theorem 3.5 by giving the following theorem. Theorem 3.6. Let E be closed selfdjoint subspce of C -lgebr. If φ : E M n C) is n-contrctive, then φ is completely contrctive. Define : SE) M n C) by λ ) = λ+µ)i +φ)+φ b), φ is n-contrctive is n-positive Theorem 2.3) is completely positive Theorem 3.4) 3.3) φ is completely contrctive Theorem 2.3).

7 COMPLETELY CONTRACTIVE MAPS BETWEEN C -ALGEBRAS 513 References 1 M.D.Choi,Some ssorted inequlities for positive liner mps on C -lgebrs, J.Opertor Theory ), no. 2, R. I. Loebl, Contrctive liner mps on C -lgebrs, Michign Mth. J ), no. 4, R. R. Smith, Completely bounded mps between C -lgebrs, J. London Mth. Soc. 2) ), no. 1, W. T. Sulimn, Some clsses of liner mps between C -lgebrs, Ph.D. thesis, Heriot-Wtt University, Edinburgh, W. T. Sulimn: College of Eduction, Ajmn University, P.O. Box 346, Ajmn, United Arb Emirtes E-mil ddress: wd@jmn.c.e