J Mth Soc Jpn Vol 58, No, 006 Numericl rdius Hgerup norm nd squre fctoriztion through Hilbert spces By Tkshi Itoh nd Msru Ngis (Received Apr 7, 004) (Revised Mr, 005) Abstrct We study fctoriztion of bounded liner mps from n opertor spce A to its dul spce A It is shown tht T : A A fctors through pir of column Hilbert spce H c nd its dul spce if nd only if T is bounded liner form on A A by the cnonicl identifiction equipped with numericl rdius type Hgerup norm As consequence, we chrcterize bounded liner mp from Bnch spce to its dul spce, which fctors through pir of Hilbert spces 1 Introduction Fctoriztion through Hilbert spce of liner mp plys one of the centrl roles in the Bnch spce theory (cf [17]) Also in the C -lgebr nd the opertor spce theory, mny importnt fctoriztion theorems hve been proved relted to the Grothendieck type inequlity in severl situtions [8], [5], [18], [1] Let α be bounded liner mp from l 1 to l, {e i } the cnonicl bsis of l1, nd B(l ) the bounded opertors on l We regrd α s the infinite mtrix [α ij ] where α ij = e i, α(e j ) The Schur multiplier S α on B(l ) is defined by S α (x) = α x for x = [x ij ] B(l ) where α x is the Schur product [α ij x ij ] Let w( ) be the numericl rdius norm on B(l ) In [1], it ws shown tht w(α x) S α w = sup 1 x 0 w(x) if nd only if α hs the following fctoriztion with b 1: l 1 α l b where t is the trnsposed mp of Motivted by the bove result, we will show squre fctoriztion theorem of bounded liner mp through pir of column Hilbert spces H c between n opertor l l t 000 Mthemtics Subject Clssifiction Primry 46L07; Secondry 47L5, 46B8, 47L05 Key Words nd Phrses opertor spce, C -lgebr, Completely bounded mp, Numericl rdius Hgerup norm, Fctoriztion through Hilbert spces
364 T Itoh nd M Ngis spce nd its dul spce More precisely, let us suppose tht A is n opertor spce in B(H ) nd A A is the lgebric tensor product We define the numericl rdius Hgerup norm of n element u A A by { 1 u wh = inf [ ] x 1,, x n, y 1,, yn n u = x i y i } Let T : A A be bounded liner mp We show tht T : A A hs n extension T which fctors through pir of column Hilbert spces H c so tht C (A) H c T b C (A) H c with inf{ cb b cb T = b} 1 if nd only if T (A wh A) with T wh 1 by the nturl identifiction x, T (y) = T (x y) for x, y A We lso study vrint of the numericl rdius Hgerup norm in order to get the fctoriztion without using the structure As consequence, the bove result nd/or the vrint red squre fctoriztion of bounded liner mp through pir of Hilbert spces from Bnch spce X to its dul spce X The norm on X X corresponding to the numericl rdius Hgerup norm is s follows: { {( n )1 u wh = inf sup f(x i ) ( n f(y i ) )1 }}, where the supremum is tken over ll f X with f 1 nd the infimum is tken over ll representtions u = n x i y i X X The norm wh is equivlent to the norm H (see Remrk 44) introduced by Grothendieck in [7] However, wh will give us different view to fctoriztion problems of bounded liner opertors through Hilbert spces Let π () be the -summing norm (cf [17] or see section 4) of liner mp from X to H We show tht T : X X hs the fctoriztion X T X t H b H with inf{π () b T = t b} 1 if nd only if T (X wh X) with T wh 1 Moreover we chrcterize liner mp X X which hs squre fctoriztion by Lindenstruss nd Pelczynski type condition (cf [14] or see Remrk 44)
numericl rdius Hgerup norm 365 We refer to [6], [15], [0] for bckground on opertor spces, [17], [19] for fctoriztion through Hilbert spce, nd [16], [], [3], [4] for completely bounded mps relted to the numericl rdius norm Fctoriztion on opertor spces Let B(H ) be the spce of ll bounded opertors on Hilbert spce H Throughout this pper, let us suppose tht A nd B re opertor spces in B(H ) We denote by C (A) the C -lgebr in B(H ) generted by the opertor spce A We define the numericl rdius Hgerup norm of n element u A B by { 1 u wh = inf [ ] x 1,, x n, y 1,, yn n u = x i y i }, where [x 1,, x n, y 1,, y n] M 1,n (C (A+B)), nd denote by A wh B the completion of A B with the norm wh Recll tht the Hgerup norm on A B is { [ ] [ ] u h = inf x1,, x n t n y1,, y n u = x i y i }, where [x 1,, x n ] M 1,n (A) nd [y 1,, y n ] t M n,1 (B) By the identity λα + λ 1 β inf = αβ ( ) λ>0 for positive rel numbers α, β 0, the Hgerup norm cn be rewritten s { 1 ( [ ] u h = inf x1,, x n [ + y 1,, yn] ) n u = x i y i } Then it is esy to check tht 1 u h u wh u h nd u wh is norm We use the nottion xα y t for n m j=1 x iα ij y j, where x = [x 1,, x n ] M 1,n (A), α = [α ij ] M n,m (C) nd y t = [y 1,, y m ] t M m,1 (B) We note the identity xα y t = x αy t First we show tht the numericl rdius Hgerup norm hs the injectivity Proposition 1 Let A 1 A nd B 1 B be opertor spces in B(H ) Then the cnonicl inclusion Φ of A 1 wh B 1 into A wh B is isometric
366 T Itoh nd M Ngis Proof The inequlity Φ(u) wh u wh is trivil To get the reverse inequlity, let u = n x i y i A 1 B 1 We my ssume tht {y 1,, y k } B is linerly independent nd there exists n n k mtrix of sclrs L M nk (C) such tht [y 1,, y n ] t = L[y 1,, y k ] t We put z t = [y 1, y k ] t Then we hve u = x y t = x Lz t = xl(l L) 1/ (L L) 1/ z t nd [ xl(l L) 1/, ((L L) 1/ z t ) ] [ x, (y t ) ] So we cn get representtion u = [x 1,, x k ] [y 1,, y k ]t with [ x 1,, x k, y 1,, y k ] [ x, (y t ) ] nd {y 1,, y k } is linerly independent This implies tht x 1,, x k A 1 Applying the sme rgument for {x 1,, x k } insted of {y 1,, y n }, we cn get representtion u = [x 1,, x l] [y 1,, y l] t with [ x 1,, x l, y 1,, y l ] [ x, (y t ) ] nd x i A 1 nd y i B 1 It follows tht Φ(u) wh u wh We lso define norm of n element u C (A) C (A) by { [x1 u W h = inf,, x n ] t w(α) u = x i α ij x j }, where w(α) is the numericl rdius norm of α = [α ij ] in M n (C) A W h A is defined s the closure of A A in C (A) W h C (A) Theorem Let A be n opertor spce in B(H ) Then A wh A = A W h A Proof By Proposition 1 nd the definition of A W h A, it is sufficient to show tht C (A) wh C (A) = C (A) W h C (A) Given u = n x i y i C (A) C (A), we hve u = [ x 1,, x n, y1,, yn ] [ 0 n 1 n 0 n 0 n ] [ x 1,, x n, y 1,, y n ] t 1 n Since w ([ 0 n ]) 0 n 0 n = 1, u wh u W h To estblish the reverse inequlity, suppose tht u = n i,j=1 x i α ij x j C (A) C (A) with w(α) = 1 nd [x 1,, x n ] t = 1 It is enough to see tht there exist c i, d i
numericl rdius Hgerup norm 367 C (A)(i = 1,, m) such tht u = m c i d i with [c 1,, c m, d 1,, d m] By the ssumption w(α) = 1 nd Ando s Theorem [1], we cn find self-djoint mtrix β M n (C) for which P = [ 1+β α ] α 1 β is positive definite in Mn (C) Set [c 1,, c n ] = [x 1,, x n, 0,, 0]P 1 nd [d 1, d n ] t = P 1 [0,, 0, x 1,, x n ] t We note tht u = [c 1,, c n ] [d 1,, d n ] t Then we hve [ c 1,, c n, d 1,, d n] = [ x 1,, x n, 0,, 0 ] P [x 1,, x n, 0,, 0] t + [ 0,, 0, x 1,, x n] P [0,, 0, x1,, x n ] t = [ x 1,, x ] n (1 + β)[x1,, x n ] t + [ x 1,, x n] (1 β)[x1,, x n ] t = [x 1,, x n ] t = We recll the column (resp row) Hilbert spce H c (resp H r ) for Hilbert spce H For ξ = [ξ ij ] M n (H ), we define mp C n (ξ) by [ n ] C n (ξ) : C n [λ 1,, λ n ] λ j ξ ij H n nd denote the column mtrix norm by ξ c = C n (ξ) This opertor spce structure on H is clled the column Hilbert spce nd denoted by H c To consider the row Hilbert spce, let H be the conjugte Hilbert spce for H We define mp R n (ξ) by j=1 [ R n (ξ) : H n n ] [η 1,, η n ] (ξ ij η j ) C n nd the row mtrix norm by ξ r = R n (ξ) This opertor spce structure on H is clled the row Hilbert spce nd denoted by H r Let : C (A) H c be completely bounded mp We define mp d : C (A) H by d(x) = (x ) It is not hrd to check tht d : C (A) H r is completely bounded nd cb = d cb when we introduce the row Hilbert spce structure to H In this pper, we define the djoint mp of by the trnsposed mp of d, tht is, d t : ((H ) r ) = ((H ) r ) = (H ) c = H c C (A) (cf [5]) More precisely, we define j=1 (η), x = η, d(x) = (η (x )) for η H, x C (A) i i A liner mp T : A A cn be identified with the biliner form A A (x, y) x, T (y) C nd lso the liner form A A C We lso use T to denote both of the biliner form nd the liner form, nd use T β to denote the norm when A A is equipped with norm β
368 T Itoh nd M Ngis We re going to prove the min theorem The min result below cn be shown by modifying rguments for the originl Hgerup norm in [4] Theorem 3 Suppose tht A is n opertor spce in B(H ), nd tht T : A A C is biliner Then the following re equivlent: (1) T wh 1 () There exists stte p 0 on C (A) such T (x, y) p 0 (xx ) 1 p0 (y y) 1 for x, y A (3) There exist -representtion π : C (A) B(K ), unit vector ξ K nd contrction b B(K ) such tht T (x, y) = (π(x)bπ(y)ξ ξ) for x, y A (4) There exist n extension T : C (A) C (A) of T nd completely bounded mps : C (A) K c, b : K c K c such tht C (A) K c T b C (A) K c ie, T = b with cb b cb 1 Proof (1) () By Proposition 1, we cn extend T on C (A) wh C (A) nd lso denote it by T We my ssume T wh 1 By the identity ( ), it is sufficient to show the existence of stte p 0 S(C (A)) such tht T (x, y) 1 p 0(xx + y y) for x, y C (A) Moreover it is enough to find p 0 S(C (A)) such tht Re T (x, y) 1 p 0(xx + y y) for x, y C (A) Define rel vlued function T {x1,,x n,y 1,,y n}( ) on S(C (A)) by for x i, y i C (A) Set T {x1,,x n,y 1,,y n}(p) = n 1 p( x i x i + y i y i ) Re T (xi, y i ),
numericl rdius Hgerup norm 369 = { T {x1,,x n,y 1,,y n} x i, y i C (A), n N } It is esy to see tht is cone in the set of ll rel functions on S(C (A)) Let be the open cone of ll strictly negtive functions on S(C (A)) For ny x 1,, x n, y 1,, y n C (A), there exists p 1 S(C (A)) such tht p 1 ( x i x i + y i y i) = x i x i + y i y i Since T {x1,,x n,y 1,,y n}(p 1 ) = 1 p 1 ( xi x i + y i y i ) Re T (x i, y i ) = 1 x i x i + yi y i Re T (xi, y i ) 1 x i x i + yi y i T (xi, y i ) 0, we hve = By the Hhn-Bnch Theorem, there exists mesure µ on S(C (A)) such tht µ( ) 0 nd µ( ) < 0 So we my ssume tht µ is probbility mesure Now put p 0 = pdµ(p) Since T {x,y}, 1 p 0(xx + y y) Re T (x, y) = T {x,y} (p)dµ(p) 0 () (1) Since T (xi, y i ) p 0 ( xi x i 1 ) 1 ( ) 1 p 0 y i y i p0 ( xi x i + y i y i ) 1 [ x 1,, x n, y 1,, yn] for x, y A, we hve tht T (A wh A) with T wh 1 (1) (3) As in the proof of the impliction (1) (), we cn find stte p S(C (A)) such tht T (x, y) p(xx ) 1 p(y y) 1 for x, y C (A) By the GNS construction, we let π : C (A) B(K ) be the cyclic representtion with the cyclic vector ξ nd p(x) = (π(x)ξ ξ) for x C (A) Define sesquiliner form on K K by π(y)ξ, π(x)ξ = T (x, y) This is well-defined nd bounded since π(y)ξ, π(x)ξ p(x x) 1 p(y y) 1 = π(x)ξ π(y)ξ Thus there exists contrction b B(K ) such tht T (x, y) = (bπ(y)ξ π(x)ξ) (3) (4) Set (x) = π(x)ξ for x C (A) nd consider the column Hilbert structure for K Then it is esy to see tht : C (A) K c is complete contrction The composition T = b is n extension of T nd cb b cb 1
370 T Itoh nd M Ngis (4) (1) Since T (x, y) = (b(y) (x )) for x, y C (A), we hve n T ( x ) i α ij, x j = i,j=1 n (bα ij (x j ) (x i )) i,j=1 = b 0 ] [α ij (x 1) (x 1) 0 b (x n ) (x n ) w b 0 ] [α ij (x 1) 0 b (x n ) b cb w(α) x 1 cb x n for n i,,j=1 x i α ij x j C (A) C (A) At the lst inequlity, we use w(cd) c w(d) for double commuting opertors c, d s well s the fct tht B(K, K ) is completely isometric onto CB(K c, K c ) Hence we obtin tht T wh T wh 1 Remrk 4 (i) If we replce the liner mp T (x), y = T (x, y) with x, T (y) = T (x, y) in Theorem 3, then we hve fctoriztion of T through pir of the row Hilbert spces H r More precisely, the following condition (4) is equivlent to the conditions in Theorem 3 (4) There exist n extension T : C (A) C (A) of T nd completely bounded mps : C (A) K r, b : K r K r such tht C (A) K r T b C (A) K r ie, T = b with cb b cb 1 (ii) Let l n be n n-dimensionl Hilbert spce with the cnonicl bsis {e 1,, e n } Given α : l n l n with α(e j ) = i α ije i, we set the mp α : l n l n by α(ej ) = i α ijē i where {ē i } is the dul bsis For nottionl convenience, we shll lso denote α by α For n x i e i C (A) l n, we define norm by n x i e i = [x 1,, x n ] t Let T : C (A) C (A) be bounded liner mp Consider T α : C (A) l n C (A) l n with numericl rdius type norm w( ) given by { ( ) w(t α) = sup x } i e i, T α xi e i xi e i 1
numericl rdius Hgerup norm 371 Then we hve { w(t α) sup w(α) } α : l n l n, n N = T wh, since T ( x i α ij x j ) = x i e i, T α( x i e i ) (iii) Let u = x i y i C (A) C (A) It is strightforwrd from Theorem 3 tht ) u wh = sup w( ϕ(xi )bϕ(y i ) where the supremum is tken over ll - preserving complete contrctions ϕ nd contrctions b 3 A vrint of the numericl rdius Hgerup norm In this section, we study fctorizion of T : A A through column Hilbert spce K c nd its dul opertor spce K c Arguments required in this section re very similr to those in section, nd insted of repeting them we will just emphsize differences We define vrint of the numericl rdius Hgerup norm of n element u A B by { 1 u wh = inf [x1,, x n, y 1,, y n ] t n u = x i y i }, where [x 1,, x n, y 1,, y n ] t M n,1 (A + B), nd denote by A wh B the completion of A B with the norm wh We remrk tht wh nd wh re inequivlent, since h (resp h ) in [10] is equivlent to wh (resp wh ) while h nd h re inequivlent [10], [13] Proposition 31 Let A 1 A nd B 1 B be opertor spces in B(H ) Then the cnonicl inclusion Φ of A 1 wh B 1 into A wh B is isometric Proof The proof is lmost the sme s tht given in Proposition 1 In the next theorem, we use the trnsposed mp t : (K c ) C (A) of : C (A) K c insted of : K c C (A) We note tht (K c ) = (K ) r nd, t re relted by t ( η), x = η, (x) = ( η (x)) K for η K, x C (A) It seems tht the fourth condition in the next theorem is simpler thn the fourth one in Theorem 3, since we do not use -structure Theorem 3 Suppose tht A is n opertor spce in B(H ), nd tht T : A A C is biliner Then the following re quivlent:
37 T Itoh nd M Ngis (1) T wh 1 () There exists stte p 0 on C (A) such tht T (x, y) p 0 (x x) 1 p0 (y y) 1 for x, y A (3) There exist -representtion π : C (A) B(K ), unit vector ξ K nd contrction b : K K such tht T (x, y) = (bπ(y)ξ π(x)ξ) K for x, y A (4) There exist completely bounded mp : A K c nd bounded mp b : K c (K c ) such tht A K c T b A t (K c ) ie, T = t b with cb b 1 Proof (1) () (3) We cn prove these implictions by the similr wy s in the proof of Theorem 3 (3) (4) We note tht we use the norm for b insted of the completely bounded norm cb (4) (1) For x i, y i A, we hve n n T (x i, y i ) = (b(y i ) (x i )) K = 0 b 0 b 0 0 0 0 (x 1 ) (x 1 ) (x n ) (x n ) w (y 1 ) (y 1 ) (y n ) (y n ) 0 b 0 b 0 0 0 0 = 1 b cb [x1,, x n, y 1,, y n ] t 1 [x1,, x n, y 1,, y n ] t (x 1 ) (x n ) (y 1 ) (y n ) 4 Fctoriztion on Bnch spces Let X be Bnch spce Recll tht the miniml quntiztion Min(X) of X Let Ω X be the unit bll of X, tht is, Ω X = {f X f 1} For [x ij ] M n (X), [x ij ] min is defined by
numericl rdius Hgerup norm 373 [x ij ] min = sup{ [f(x ij )] f Ω X } Then Min(X) cn be regrded s subspce in the C -lgebr C(Ω X ) of ll continuous functions on the compct Husdorff spce Ω X Here we define norm of n element u X X by { {( n )1 u wh = inf sup f(x i ) ( n f(y i ) )1 }}, where the supremum is tken over ll f X with f 1 nd the infimum is tken over ll representtions u = n x i y i Proposition 41 Let X be Bnch spce Then Proof Min(X) wh Min(X) = Min(X) wh Min(X) = X wh X Let u = n x i y i Min(X) Then, using the identity ( ), we hve { 1 u wh = inf [ ] x 1,, x n, y 1,, yn n } u = x i y i { { 1 = inf sup [ f(x 1 ),, f(x n ), f(y 1 ),, f(y n )] } f Ω X u = { { ( 1 n ) } = inf sup f(x i ) + f(y i ) f Ω X u = { {( n )1 ( = inf sup f(x i ) n )1 } f(y i ) f ΩX u = = u wh n } x i y i n } x i y i n } x i y i The equlity u wh = u wh is obtined by the sme wy s bove Let T : X X be bounded liner mp As in Remrk 4(ii), we consider the mp T α : X l n X l n nd define norm for xi e i X l n by {( xi e i = sup f(xi ) ) 1 f Ω X } We note tht, given x X, x is regrded s x, f = f(x) for f X in the definition of w(t α), tht is, { ( ) w(t α) = sup x } i e i, T α xi e i xi e i 1
374 T Itoh nd M Ngis Let : X Y be liner mp between Bnch spces opertor if there is constnt C which stisfies the inequlity is clled -summing ( (xi ) ) 1 {( C sup f(xi ) ) 1 } f Ω X for ny finite subset {x i } X The smllest such constnt C is defined s π (), the -summing norm of The following might be well known Proposition 4 Let X be Bnch spce If is liner mp from X to H, then the following re equivlent: (1) : Min(X) H c cb 1 () : Min(X) H r cb 1 (3) π ( : X H ) 1 Proof (1) (3) For ny x 1,, x n X, we hve n (x i ) = [(x1 ),, (x n )] t cb [x1,, x n ] t Min n x i x i Min { n = sup f(x i ) f Ω X } (3) (1) For ny [x ij ] M n (Min(X)), we hve { [(x ij )] M = sup n(h c) λ j (x ij ) } λj = 1 i j { { ( ) } sup π () sup f } λ j x ij f Ω X λj = 1 j sup{ [f(x ij )] f Ω X } [x ij ] M n(min(x)) i () (3) It follows by the sme wy s bove Corollry 43 Suppose tht X is Bnch spce, nd tht T : X X is bounded liner mp Then the following re equivlent: (1) w(t α) w(α) for ll α : l n l n nd n N () T wh 1
numericl rdius Hgerup norm 375 (3) T fctors through Hilbert spce K nd its dul spce K by -summing opertor : X K nd bounded opertor b : K K s follows: X T X t K b K ie, T = t b with π () b 1 (4) T hs n extension T : C(Ω X ) C(Ω X ) which fctors through pir of Hilbert spces K by -summing opertor : C(Ω X ) K nd bounded opertor b : K K s follows: C(Ω X ) K T b C(Ω X ) K ie, T = b with π () b 1 Proof (1) () Suppose tht m ( m ) zi e i, T α z i e i 1 for ny m z i e i X l m with m z i e i 1 nd α M n (C) with w(α) 1 It is esy to see tht m i,j=1 z i, T (z j) α ij 1, equivlently m i,j=1 z i, T (z j ) α ij 1 Given n x i y i wh < 1, we my ssume tht 1 [x1,, x n, y 1,, y n ] t 1 Set 1 x i z i = 1 y i n i = 1,, n i = n + 1,, n [ ] 0n 1 n nd α = 0 n 0 n It turns out n z i e i 1 nd w(α) = 1 Then we hve T ( n x i y i ) = n i,j=1 z i, T (z j ) α ij 1 Hence T wh 1 () (1) Suppose tht T wh 1 Then T hs n extension T (C(Ω X ) wh C(Ω X )) with T wh 1 Given ε > 0 nd α M n (C), there exist x 1,, x n C(Ω X ) such tht n x i e i 1 (equivlently [x 1,, x n ] t 1) nd w(t α)
376 T Itoh nd M Ngis ε < n i,j=1 x i, T (x j ) α ij Hence we hve ( w(t α) w(t α) < T n i,j=1 x i α ij x j ) + ε [x 1,, x n ] t w(α) + ε w(α) + ε () (3) It is strightforwrd from Theorem 3 nd Propositions 41, 4 () (4) It is strightforwrd from Theorem 3 nd Propositions 41, 4 Remrk 44 Here we compre the bove corollry with the clssicl fctoriztion theorems through Hilbert spce Let X nd Y be Bnch spces Grothendieck introduced the norm H on X Y in [7] by { {( n )1 u H = inf sup f(x i ) ( n g(y i ) ) 1 }} where the supremum is tken over ll f X, g Y with f, g 1 nd the infimum is tken over ll representtions u = n x i y i X Y In [14], Lindenstruss nd Pelczynski chrcterized the fctoriztion by using T α : X l n Y l n for T : X Y, however the norm on X l is slightly different from the one in this pper Their theorems with modifiction re summrized for bounded liner mp T : X Y s follows: The following re equivlent: (1) T α α for ll α : l n l n nd n N () T H 1 (3) T fctors through Hilbert spce K by -summing opertor : X K nd b : K Y whose trnsposed b t is -summing s follows: X T K b Y ie, T = b with π ()π (b t ) 1 References [ 1 ] T Ando, On the structure of opertors with numericl rdius one, Act Sci Mth (Szeged), 34 (1973), 11 15 [ ] T Ando nd K Okubo, Induced norms of the Schur multiplier opertor, Liner Algebr Appl, 147 (1991), 181 199 [ 3 ] D Blecher nd V Pulsen, Tensor products of opertor spces, J Funct Anl, 99 (1991), 6 9 [ 4 ] E Effros nd A Kishimoto, Module mps nd Hochschild-Johnson cohomology, Indin Univ Mth J, 36 (1987), 57 76
numericl rdius Hgerup norm 377 [ 5 ] E Effros nd Z J Run, Self-dulity for the Hgerup tensor product nd Hilbert spce fctoriztion, J Funct Anl, 100 (1991), 57 84 [ 6 ] E Effros nd Z J Run, Opertor spces, London Mth Soc Monogr (NS), 3, Oxford Univ Press, 000 [ 7 ] A Grothendieck, Résumé de l théorie métrique des produits tensoriels topologiques, Bol Soc Mt São-Pulo, 8 (1956), 1 79 [ 8 ] U Hgerup, The Grothengieck inequlity for biliner forms on C -lgebrs, Adv Mth, 56 (1985), 93 116 [ 9 ] U Hgerup nd T Itoh, Grothendieck type norms for biliner forms on C -lgebrs, J Opertor Theory, 34 (1995), 63 83 [10] T Itoh, The Hgerup type cross norm on C -lgebrs, Proc Amer Mth Soc, 109 (1990), 689 695 [11] T Itoh nd M Ngis, Schur products nd module mps on B(H), Publ Res Inst Mth Sci, 36 (000), 53 68 [1] T Itoh nd M Ngis, Numericl Rdius Norm for Bounded Module Mps nd Schur Multipliers, Act Sci Mth (Szeged), 70 (004), 195 11 [13] A Kumr nd A M Sinclir, Equivlence of norms on opertor spce tensor products of C - lgebrs, Trns Amer Mth Soc, 350 (1998), 033 048 [14] J Lindenstruss nd A Pelczynski, Absolutely summing opertors in L p spces nd their pplictions, Studi Mth, 9 (1968), 75 36 [15] V Pulsen, Completely bounded mps nd opertor lgebrs, Cmbridge Stud Adv Mth, 78, Cmbridge Univ Press, 00 [16] V I Pulsen nd C Y Suen, Commutnt representtions of completely bounded mps, J Opertor Theory, 13 (1985), 87 101 [17] G Pisier, Fctoriztion of liner opertors nd the Geometry of Bnch spces, CBMS Reg Conf Ser Mth, 60, Amer Mth Soc, 1986 [18] G Pisier, The opertor Hilbert spce OH, complex interpoltion nd tensor norms, Mem Amer Mth Soc, 1, No 585, 1996 [19] G Pisier, Similrity problems nd completely bounded mps, nd expnded edit, Lecture Notes in Mth, 1618, Springer, 001 [0] G Pisier, Introduction to opertor spce theory, London Mth Soc Lecture Note Ser, 94, Cmbridge Univ Press, 003 [1] G Pisier nd D Shlykhtenko, Grothendieck s theorem for opertor spces, Invent Mth, 150 (00), 185 17 [] C-Y Suen, The numericl rdius of completely bounded mp, Act Mth Hungr, 59 (199), 83 89 [3] C-Y Suen, Induced completely bounded norms nd inflted Schur product, Act Sci Mth (Szeged), 66 (000), 73 86 [4] C-Y Suen, W ρ completely bounded mps, Act Sci Mth (Szeged), 67, (001), 747 760 Tkshi Itoh Deprtment of Mthemtics Gunm University Gunm 371-8510 Jpn E-mil: itoh@edugunm-ucjp Msru Ngis Deprtment of Mthemtics nd Informtics Chib University Chib 63-85 Jpn E-mil: ngis@mthschib-ucjp