CONTRACTIVITY AND COMPLETE CONTRACTIVITY FOR FINITE DIMENSIONAL BANACH SPACES. 1. Introduction

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1 CONTRCTIVITY ND COMPLETE CONTRCTIVITY FOR FINITE DIMENSIONL BNCH SPCES GDDHR MISR, VIJIT PL ND CHERIN VRUGHESE bstract. Choose an arbitrary but fixed set of n n matrices 1,..., m and let Ω C m be the unit ball with respect to the norm, where z 1,..., z m = z z m m op. It is known that if m 3 and B is any ball in C m with respect to some norm, say B, then there exists a contractive linear map L : C m, B M k which is not completely contractive. The characterization of those balls in C for which contractive linear maps are always completely contractive thus remains open. We answer this question for balls of the form Ω in C and the balls in their norm dual. 1. Introduction In 1936 von Neumann see [17, Corollary 1.] proved that if T is a bounded linear operator on a separable complex Hilbert space H, then, for all complex polynomials p, pt p,d := { pz : z < 1} if and only if T 1. Or, equivalently, the homomorphism ρ T induced by T on the polynomial ring P [z] by the rule ρ T p = pt is contractive if and only if T is contractive. The original proof of this inequality is intricate. couple of decades later, Sz.-Nazy see [17, Theorem 4.3] proved that a bounded linear operator T admits a unitary power dilation if and only if there exists a unitary operator U on a Hilbert space K H such that P H pu H = pt, for all polynomials p. The existence of such a dilation may be established by actually constructing a unitary operator U dilating T. This construction is due to Schaffer cf. [14]. Clearly, the von Neumann inequality follows from the existence of a power dilation via the spectral theorem for unitary operators. Let P = p ij be a k k matrix valued polynomial in m variables. Let P,Ω = { p ij z op : z Ω}, where Ω C m is a bounded open and connected set. Define P T to be the operator p ij T, 1 i, j k. The homomorphism ρ T is said to be completely contractive if P T P,Ω, k = 1,,.... deep theorem due to rveson cf. [1] says that T has a normal boundary dilation if and only if ρ T is completely contractive. Clearly, if ρ T is completely contractive, then it is contractive. The dilation theorems due to Sz.-Nazy and ndo cf. [17] give the non-trivial converse in the case of the disc and the bi-disc algebras. However, Parrott cf. [15] showed that there are three commuting contractions for which it is impossible to find commuting unitaries dilating them. In view of rveson s theorem this naturally leads to the question of finding other algebras OΩ for which all contractive homomorphisms are 010 Mathematics Subject Classification. 46L07, 47L5. The work of G.Misra was ported, in part, through the J C Bose National Fellowship and UGC-CS. The work of. Pal was ported, in part, through the UGC-NET and the IFCM Research Fellowship. The results of this paper are taken from his PhD thesis submitted to the Indian Institute of Science in 014, with some of the proofs significantly simplified. 1

2 GDDHR MISR, VIJIT PL ND CHERIN VRUGHESE necessarily completely contractive. t the moment, this is known to be true of the disc, bi-disc cf. [17], symmetrized bi-disc cf. [3] and the annulus algebras cf. []. Counter examples are known for domains of connectivity cf. [8] and any ball in C m, m 3, as we will explain below. Neither ndo s proof of the existence of a unitary dilation for a pair of commuting contractions, nor the counter example to such an existence theorem due to Parrott involved the notion of complete contractivity directly. In the papers [10, 11, 1], it was shown that the examples of Parrott are not even contractive. In these papers, for any bounded, connected and open set Ω C m, the homomorphism ρ V : OΩ M p+q, induced by an m-tuple of p q matrices V = V 1,..., V m, modeled after the examples of Parrott, was introduced. This was further studied, in depth, by V. Paulsen [18], where he showed that the question of contractive vs completely contractive for Parrott like homomorphisms ρ V is equivalent to the question of contractive vs completely contractive for the linear maps L V from some finite dimensional Banach space X to M n C. The existence of linear maps of the form L V which are contractive but not completely contractive for m 5 were found by him. refinement see remark at the bottom of p. 76 in [16] includes the case m = 3, 4, leaving the question of what happens when m = open. This is Problem 1 on page 79 of [16] in the list of Open Problems. For the normed linear space C,, we show, except when the pair 1, is simultaneously diagonalizable, that there is a contractive linear map on C, taking values in p q matrices, which is not completely contractive. We point out that the results of Paulsen used deep ideas from geometry of finite dimensional Banach spaces. In contrast, our results are elementary in nature, although the computations, at times, are somewhat involved.. Preliminaries The norm z = z z m m op, z C m, is obtained from the embedding of the linear space C m into the C algebra of n n matrices via the map P z := z z m m. Let Ω C m be the unit ball with respect to the norm. Let OΩ denote the algebra of functions each of which is holomorphic on some open set containing the closed unit ball Ω. Given p q matrices V 1,..., V m and a function f OΩ, define m fwip i=1.1 ρ V f := ifw V i 0 fwi q for a fixed w Ω. Clearly, ρ V : OΩ, M p+q C, op defines an algebra homomorphism. t the outset we point out the interesting and useful fact that ρ V is contractive on OΩ if and only if it is contractive on the subset of functions which vanish at w. This is the content of the following lemma. The proof is reproduced from [18, Lemma 5.1], a direct proof appears in [10, Lemma 3.3]. Lemma.1. f =1{ ρ V f op : f OΩ } 1 if and only if g =1{ ρ V g op : g OΩ, gw = 0} 1. Proof. The implication in one direction is obvious. To prove the converse, assume that ρ V g 1 for every g such that gw = 0 and g = 1. For f OΩ with f = 1 let φ fw be the Möbius map of the disc which maps fw to 0. We let g = φ fw f. Then gw = 0, g = 1 and, from our assumption, ρ V g 1. So ρ V f = ρ V φ 1 fw g = φ 1 fw ρv g since ρ V is a homomorphism 1.

3 CONTRCTIVITY ND COMPLETE CONTRCTIVITY FOR FINITE DIMENSIONL BNCH SPCES 3 In the last step we use the von Neumann inequality since φ 1 disc to itself. fw is a rational function from the Note: For the rest of this work, we restrict to the case where w = 0 in the definition.1 of ρ V above. The following lemma provides a characterization of the unit ball Ω with respect to the dual norm in Cm, that is Ω = Cm, 1. Lemma.. The dual unit ball Ω = { 1 f0, f0,, m f0 : f HolΩ, D, f0 = 0 }. Proof. Given z C m such that z = 1 and f HolΩ, D, f0 = 0, we define g z : D Ω by g z λ = λz, λ D. Then f g z : D D with f g z 0 = 0. pplying the Schwarz Lemma to the function f g z we get 1 f g z 0 = f g z 0 g z0 = f 0 g z0 = f 0 z. In the above, f 0 z = m i=1 if0z i, etc. Hence 1 f0, f0,, m f0 Ω. Conversely, given w Ω, we define f wz = w z so that i f w 0 = w i..1. The Maps L k V : From Lemma.1 above it follows that m i f0 V i op 1.. ρ V 1 if and only if f =1,f0=0 Considering Lemma. and the equivalence. above it is natural to consider the induced linear map L V : C m, M p,qc given by It follows from. above that i=1 L V w = w 1 V w m V m. ρ V 1 if and only if L V 1. We will show now that the complete contractivity of ρ V and L V are also related similarly. For a holomorphic function F : Ω M k with F = z Ω F z, we define.3 ρ k V F := ρ VF ij m i,j=1 =. F 0 I m i=1 if 0 V i 0 F 0 I Using a method similar to that used for ρ V it can be shown that m ρ k V 1 if and only if { i F 0 V i : F HolΩ, M k 1, F 0 = 0} 1, F i=1 that is, by repeating the argument used for ρ V we have where is the map ρ k V 1 if and only if Lk V 1, L k V : Cm M k,,k M k M p,q, op L k V Θ 1, Θ,, Θ m = Θ 1 V 1 + Θ V + + Θ m V m for Θ 1, Θ,, Θ m M k.

4 4 GDDHR MISR, VIJIT PL ND CHERIN VRUGHESE.. The polynomial P. very useful construct for our analysis is the matrix valued polynomial P with P Ω M n, op 1 defined by P z 1, z,, z m = z z + + z m m, with the norm P = z1,,z m Ω P z 1,, z m op. Note that P = 1 by definition. The typical procedure used to show the existence of a homomorphism which is contractive but not completely contractive is to construct a contractive homomorphism ρ V by a suitable choice of V and to then show that its evaluation on P, that is, ρ n V P, has norm greater than Homomorphisms induced by m-vectors. We now consider the special situation when the matrices V 1,, V m are vectors in C m realized as row m-vectors. For w = w 1,..., w m in some bounded domain Ω C m, the commuting m-tuple of m + 1 m + 1 matrices of the form wi V i 0 w i I m, 1 i m, induce the homomorphism ρ V via the usual functional calculus, that is, ρ V f := f w 1 V 1 0 w 1 I m,..., wm V m 0 w mi m, f OΩ, see.1. The localization of a commuting m - tuple T of operators in the class B 1 Ω, introduced in [5, 6], is also a commuting m - tuple of m + 1 m + 1 matrices, which is exactly of the form described above. The vectors V 1,..., V m appearing in such localizations are given explicitly in terms of the curvature of the holomorphic Hermitian vector bundle corresponding to T as shown in [6]. The contractivity of the homomorphism ρ V then results in curvature inequalities see [9, 11, 1, 13]. Let V i = v i1 v i v im, i = 1,, m. The propositions below are useful to study contractivity and complete contractivity in this special case, where, as before, we assume that Ω = Ω and w = 0. Proposition.3. The following are equivalent: i ρ V is contractive, ii m j=1 z j 1 m j=1 z jb j op 1, where B j = m i=1 v ij i. Proof. We have shown that the homomorphisms ρ V OΩ M m+1 C is contractive if and only if the linear map L V C m, Cm, is contractive equivalently if L V C m, C m, is contractive. v11... v 1m The matrix representation of L V is v m1 v m1... v mm Hence the contractivity of L V is given by the condition that v11... v 1m z1. m j=1 z j v mm z m From the definition of it follows that L V C m, C m, 1 if and only if m j=1 z j 1 m z j B j op 1 where B j = m i=1 v ij i. In particular, if V 1 = u 0 and V = 0 v, the condition ii above becomes z 1 u 1 + z v 1, j=1 z j 1 which is equivalent to the following two conditions: j=1

5 CONTRCTIVITY ND COMPLETE CONTRCTIVITY FOR FINITE DIMENSIONL BNCH SPCES 5 i u 1 or v 1 1 ii inf β C, β =1 { 1 u 1β v β + uv 1β β 1 β, β } 0. Proposition.4. The following are equivalent: i ρ n V P 1, ii T he n mn matrix B 1 B B m is contractive, where Bj = m i=1 v ij i. Proof. Since P 0 = 0, it follows from the definition.3 that ρ n V P 1 if and only if For V i = v i1 v im, we have 1 V m V m 1. 1 V m V m = B 1 B B m Thus ρ n V P 1 if and only if B 1 B B m 1. In particular if V 1 = u 0 and V = 0 v the condition ii above becomes { 1 u 1β v β } 0. inf β C, β =1 Note: For most of this paper we will restrict to the two dimensional case. That is, we consider C with the norm defined by a matrix pair 1,. In fact, for the most part, we even restrict to the situation where 1, are matrices. This is adequate for our primary purpose of constructing homomorphisms of OΩ which are contractive but not completely contractive. Many of the results can be adapted to higher dimensional situation. 3. Defining Function and Test Functions Recall the matrix valued polynomial P : Ω M, op 1 defined earlier by P z 1, z = z z, where M, op 1 is the matrix unit ball with respect to the operator norm. For z 1, z in Ω, the norm P := P z 1, z op = 1 z 1,z Ω by definition of the polynomial P. Let B be the unit ball in C. For α, β B B, define p α,β : Ω D to be the linear map p α,β z 1, z = P z 1, z α, β = z 1 1 α, β + z α, β. The norm p α,β, for any pair of vectors α, β in B B, is at most 1 by definition. Let P denote the collection of linear functions {p α,β : α, β B B }. The map P which we call the Defining Function of the domain and the collection of functions P which we call a family of Test Functions encode a significant amount of information relevant to our purpose about the homomorphism ρ V. For instance ρ V is contractive if its restriction to P is contractive. lso the lack of complete contractivity can often be shown by evaluating ρ V on P. Some of the details are outlined in the lemma below.

6 6 GDDHR MISR, VIJIT PL ND CHERIN VRUGHESE Lemma 3.1. In the notation fixed in the preceding discussion, we have i ρ V p α,β ρ V P, ii ρ V is contractive if and only if ρ V p α,β 1. Proof of i. Since by definition, it follows that Hence 3.1 ρ V p α,β = 0 1 p α,β 0 V 1 + p α,β 0 V 0 0 ρ V p α,β = 1p α,β 0 V 1 + p α,β 0 V op = 1 α, β V 1 + α, β V op = u = v =1 1 α, β V 1 u, v + α, β V u, v. ρ V p α,β = = = ρ V P. u = v =1 u = v =1 u = v =1 1 α, β V 1 u, v + α, β V u, v 1 V 1 + V α u, β v ρ V P α u, β v Proof of ii. s indicated earlier the contractivity of ρ V is equivalent to the contractivity of L V : C, M p,q, op given by the formula L V ω 1, ω = ω 1 V 1 + ω V. So we identify the conditions for the contractivity of L V : L V = ω 1,ω 1 ω 1 V 1 + ω V op = ω 1,ω 1 Hence, since ω 1, ω lies in the dual of Ω, u = v =1 ω 1 V 1 u, v + ω V u, v. L V 1 V 1 u, v, V u, v Ω u, v such that u = v = 1 u = v =1 V 1 u, v 1 + V u, v op 1 u = v =1 1 α, β V 1 u, v + α, β V u, v 1 ρ V p α,β 1 from 3.1 above. s mentioned earlier, by choosing a pair V 1, V such that the inequality in i above is strict, we can often construct a contractive homomorphism which is not completely contractive. We illustrate below choices of V 1, V for the Euclidean ball for which the inequality is strict.

7 CONTRCTIVITY ND COMPLETE CONTRCTIVITY FOR FINITE DIMENSIONL BNCH SPCES 7 Example 3.. Euclidean Ball Choosing = , , we see that Ω defines the Euclidean ball B in C. Choose V 1 = v 11 v 1, V = v 1 v. We will prove that ρ V p α,β < ρ V P op, if V 1 and V are linearly independent. In fact we can choose V 1, V such that ρ V p α,β 1 and ρ V P op > 1. This example of a contractive homomorphism of the ball algebra which is not completely contractive was found in [10, 11]. Theorem 3.3. For Ω = B, let V 1 = v 11 v 1, V = v 1 v. Then i ρ V p α,β = 11 1 v 1 v op ii ρ V P op = 11 1 v 1 v HS HS represents the Hilbert Schmidt norm Consequently, ρ V p α,β < ρ V P op if V 1 and V are linearly independent. Proof. By the definition of ρ V we have ρ V p α,β = 1 α, β V 1 u, v + α, β V u, v α = β = u = v =1 On the other hand, we have ρ = α 1 v 11 u 1 + v 1 u + α v 1 u 1 + v u β 1 α = β = u =1 = α 1 v 11 u 1 + v 1 u + α v 1 u 1 + v u α = u =1 = v 11 u 1 + v 1 u + v 1 u 1 + v u u =1 = v 11 v 1 v 1 v op. V P op = V 1 + V = 11 1 v 1 v HS. If V 1 and V are linearly independent 11 1 v 1 v op < 11 1 v 1 v HS and we have ρ V p α,β < ρ V P op. Now choose V 1 = 1 0 and V = 0 1. From Lemma 3.1 and Theorem 3.3 it follows that ρ V is contractive but ρ V P =. 4. Unitary Equivalence and Linear Equivalence If U and W are unitary matrices and à = U 1W, U W, then z 1, z = z z op = z 1 U 1 W + z U W op = z 1, z Ã. There are, therefore, various choices of the matrix pair 1, related as above which give rise to the same norm. We use this freedom to ensure that 1 is diagonal. Consider the invertible linear transformation z 1, z z 1, z on C defined as follows:

8 8 GDDHR MISR, VIJIT PL ND CHERIN VRUGHESE For z = z 1, z in C, let where p, q, r, s C. Then z 1 = p z 1 + q z z = r z 1 + s z, z 1, z = z 1, z Ã, where à is related to as follows: à 1 = p 1 + r à = q 1 + s. More concisely, if T is the linear transformation above on C, then In particular T maps Ωà onto Ω. T z = z T I. Lemma 4.1. For k = 1,,..., the contractivity of the linear maps L k V C M k, determine the contractivity of the linear maps Lk Ã,k and conversely, where à = T I and Ṽ = T IV. Proof. For k = 1,,..., we have to show that defined on Ṽ defined on C M k,,k L k V C M k, Ã,k M k M p,q, op 1 L k Ṽ C M k,,k M k M p,q, op 1. We prove this result for the case k = 1, that is, for the map L V. The proof for the general case is similar. Consider the bijection between the spaces {f HolΩ, D, f0 = 0} and { f HolΩÃ, D, f0 = 0} defined as follows: Using this bijection f f = f T, f f = f T 1 L V C, Mp,q, op 1 { D f0 V op : f HolΩÃ, D, f0 = 0} 1 à f { Df T 0 V op : f HolΩ, D, f0 = 0} 1 f { Df0 T V op : f HolΩ, D, f0 = 0} 1 f { Df0 T IV op : f HolΩ, D, f0 = 0} 1 f L T IV C, Mp,q, op 1 In the above, Df is a row vector, T is a matrix and by an expression of the form X Y we mean i=1 X iy i. It follows that, in our study of the existence of contractive homomorphisms which are not completely contractive, two sets of matrices = 1, and à = Ã1, à which are related through linear combinations as above yield the same result. We can, therefore, restrict our attention to a subcollection of matrices. Since 1 has already been chosen to be diagonal, we consider transformations as above with r = 0 to preserve the diagonal structure of 1. By choosing the parameters p, q, s suitably we can ensure that one diagonal entry of 1 is 1 and the diagonal entries of are 1 and 0. By further

9 CONTRCTIVITY ND COMPLETE CONTRCTIVITY FOR FINITE DIMENSIONL BNCH SPCES 9 conjugating with a diagonal unitary and a permutation matrix it follows that we need to consider only the following three families of matrices: Table 1. Cases modulo unitary and linear equivalence d 1 d C 1 b c C, b R + d d C 1 b c C, b R d d C 0 b c C, b R + In the above, R + represents the set of non-negative real numbers Simultaneously Diagonalizable Case. For the study of contractivity and complete contractivity in this situation we consider two possibilities. The first when 1 and are simultaneously diagonalizable and the second when they are not. The simultaneously diagonalizable case reduces to the case of the bi-disc where we know that any contractive homomorphism is completely contractive. In all the other cases when 1 and are not simultaneously diagonalizable we show that there exists a contractive homomorphism which is not completely contractive. Consider first the case when 1 and are simultaneously diagonalizable. Based on the discussion of linear equivalence above we need to study only the following possibilities: Table. Simultaneously diagonalizable cases d 1 d C d d C pplying linear transformations as before, both cases can be reduced to = , 0 0 which represents the bi-disc. s mentioned earlier, it is known that any contractive homomorphism is completely contractive in this case. We now study the situation when 1 and are not simultaneously diagonalizable. 0 1

10 10 GDDHR MISR, VIJIT PL ND CHERIN VRUGHESE 5. Contractivity, Complete Contractivity and Operator Space Structures We recall some notions about operator spaces which are relevant to our purpose. Definition 5.1. cf. [17, Chapter 13, 14] n abstract operator space is a linear space X together with a family of norms k defined on M k X, k = 1,, 3,..., where 1 is simply a norm on the linear space X. These norms are required to satisfy the following compatibility conditions: 1 T S p+q = max{ T p, S q } and SB p op S q B op for all S M q X, T M p X and M p,q C, B M q,p C. Two such operator spaces X, k and Y, k are said to be completely isometric if there is a linear bijection T : X Y such that T I k : M k X, k M k Y, k is an isometry for every k N. Here we have identified M k X with X M k in the usual manner. We note that a normed linear space X, admits an operator space structure if and only if there is an isometric embedding of it into the algebra of operators BH on some Hilbert space H. This is the well-known theorem of Ruan cf. [16]. We recall here the notions of MIN and MX operator spaces and a measure of their distance, αx, following [17, Chapter 14]. Definition 5.. The MIN operator structure M INX on a finite dimensional normed linear space X is obtained by isometrically embedding X in the C algebra C X 1, of continuous functions on the unit ball X 1 of the dual space. Thus for v ij in M k X, we set v ij MIN = v ij = { fv ij : f X 1 }, where the norm of a scalar matrix fv ij in M k is the operator norm. For an arbitrary k k matrix over X, we simply write v ij MINX to denote its norm in M k X. This is the minimal way in which we represent the normed space as an operator space. There is also a maximal representation which is denoted M XX. Definition 5.3. The operator space M XX is defined by setting v ij MX = { T v ij : T : X BH}, and the remum is taken over all isometries T and all Hilbert spaces H. Every operator space structure on a normed linear space X lies between M INX and MXX. The extent to which the two operator space structures MINX and MXX differ is characterized by the constant αx introduced by Paulsen cf.[17, Chapter 14], which we recall below. Definition 5.4. The constant αx is defined as αx = { v ij MX : v ij MIN 1, v ij M k X, k N}. Thus αx = 1 if and only if the identity map is a complete isometry from MINX to M XX. Equivalently, we conclude that there exists a unique operator space structure on X whenever αx is 1. Therefore, those normed linear spaces for which αx = 1 are rather special. Unfortunately, there aren t too many of them! The familiar examples are C,, and consequently C with the l 1 norm. It is pointed out in [16, pp. 76] that αx > 1 for dimx 3, refining an earlier result of Paulsen that αx > 1 whenever dimx 5. This leaves the question open for normed linear spaces whose dimension is. Returning to the space C, with z 1, z = z z op, we show below that αω > 1 in a large number of cases. From [18, Theorem 4.], it therefore follows that, in all these

11 CONTRCTIVITY ND COMPLETE CONTRCTIVITY FOR FINITE DIMENSIONL BNCH SPCES 11 cases, there must exist a contractive homomorphism of OΩ into the algebra BH which is not completely contractive. In the remaining cases, the existence of a contractive homomorphism which is not completely contractive is established by a careful study of certain extremal problems. The norm z 1, z = z z op defines a natural isometric embedding into M C given by z 1, z z z. However, note that z 1, z = z z op = z 1 t 1 + z t op = z 1, z t. We, therefore, get another isometric embedding into M C given by z 1, z z 1 t 1 + z t. In a variety of cases the operator spaces determined by these two embeddings are distinct and the parameter α > 1 in these cases. Therefore, the existence of contractive homomorphisms which are not completely contractive is established in these cases. We present the details below. Recall the map P defined earlier by P z 1, z = z z. Let P = P I. For the three families of matrices = 1, characterized in Table 1 we show that and t define distinct operator space structures unless d = 1 or b = c. Theorem 5.5. Let Z 1 = and Z = P Z t 1, Z op. Proof. We illustrate the proof for the case 1 = d similarly. For this case 5.1 P Z 1, Z op = = Z1 +Z bz. If d 1 and b c then P Z 1, Z op, = 1 b. The other cases can be proved op Z1 +Z cz cz dz 1 bz dz1 Z1 +Z Z 1 +Z +b Z Z cz 1 +Z Z +b dz Z1 cz Z 1 +Z +bdz 1 Z c Z Z + d Z 1 Z1 op. Similarly we have P Z t 1, Z Z1 +Z op = Z 1 +Z + c Z Z bz 1 +Z Z +c dz 5. Z1 op bz Z 1 +Z + cdz 1 Z b Z Z + d Z 1 Z. 1 ssume P Z 1, Z op = P Z t 1, Z op. Using the form of Z 1, Z this is equivalent to +b c op = c c + d + c b b b + d op i.e. b c 1 d = 0 note that the matrices on the left and right have the same trace, from which the result follows. Since αω = 1 if and only if the two operator spaces MINΩ and MXΩ are completely isometric, it follows from the Theorem we have just proved that if d = 1 and b c, then αx > 1. Consequently, there exists a contractive homomorphism of OΩ into BH, which is not completely contractive. Example 5.6. Euclidean Ball The Euclidean ball B is characterized by 1 = , = So, in Theorem 5.5, we have d 1 and b c. Hence and t give rise to distinct operator space structures and, consequently, there exists a contractive homomorphism which is not completely contractive. 6. Cases not menable to the Operator Space Method Theorem 5.5 shows that there is a contractive homomorphism which is not completely contractive for all the choices of 1, listed in Table 1 except when d = 1 or b = c. We are, therefore, left with the following families of 1, to be considered:

12 1 GDDHR MISR, VIJIT PL ND CHERIN VRUGHESE Table 3. Cases not covered by the operator space approach i e iθ 1 θ R 1 b c C, b R + ii e iθ θ R 0 b c C, b R + iii e iθ θ R 1 b c C, b R + iv d d C 1 c c C v d d C 0 c c C vi d d C 1 c c C These six families are not disjoint and have been classified as such on the basis of the method of proof used Dual norm method. We first consider a special case of type ii in Table 3 with 1 = , = lthough this case is covered by the more general method to be outlined later we present an alternate, interesting procedure for this example since it is possible to explicitly calculate the dual norm in this case. Equipped with the information about the dual norm we can directly construct a pair V = V 1, V such that L V 1 and L V P > 1. Note that in this case z 1, z = z + z + 4 z 1 and the unit ball Ω = {z 1, z : z 1 + z < 1}. Lemma 6.1. Let 1 = , = If ω 1, ω C, then the dual norm { ω1 +4 ω ω 1, ω = 4 ω if ω ω 1 ; ω 1 if ω ω 1. Proof. Let f ω1,ω be the linear functional on C, defined by f ω1,ω z 1, z = ω 1 z 1 + ω z.

13 Then CONTRCTIVITY ND COMPLETE CONTRCTIVITY FOR FINITE DIMENSIONL BNCH SPCES 13 ω 1, ω = f ω1,ω z 1, z z 1,z Ω = z 1 z 1 ω 1 z 1 + ω z = ω 1 z 1 + ω z z 1 z 1 = ω1 z 1 + ω 1 z 1. z 1 1 If ω ω 1 the expression on the right attains its maximum at z 1 = ω 1 ω 1 and the maximum value is ω 1 +4 ω 4 ω. If ω ω 1 the expression on the right is monotonic in z 1 and the maximum is attained at z 1 = 1. The maximum value in this case is ω 1. Theorem 6.. Let 1 = , = and V 1 = 1 0, V = 0 1. Then i L V C, C, = 1 ii L 3 V P =. Consequently ρ V, for this choice of V = V 1, V, is contractive on OΩ but not completely contractive. Proof of i. L V C, C, = ω 1,ω =1 ω 1 V 1 + ω V = ω 1,ω =1 ω1 + ω. We now consider two cases: Case a: ω ω 1 and 1 = ω 1, ω = ω 1 +4 ω 4 ω from Lemma 6.1. These two constraints together can be seen to be equivalent to the constraints 1 ω 1 and ω 1 = 4 ω 1 ω. Hence the remum above for this range of ω 1, ω is given by 1 ω 1 ω ω = 1. Case b: ω ω 1 and 1 = ω 1, ω = ω 1 from Lemma 6.1. The remum for this range of ω 1, ω is given by ω ω = 3 4. Taking the larger of the remums in Case a and Case b we get that L V = 1.

14 14 GDDHR MISR, VIJIT PL ND CHERIN VRUGHESE Proof of ii. L V P = 1 V 1 + V = = = 0 1 using the form of 0 1, = General cases not amenable to the operator space method. The various families of 1, listed in Table 3 require a case by case analysis to show that there is a contractive homomorpism which is not completely contractive. We first present a general outline of the method used. We choose the pair V = V 1, V to be of the form V 1 = u 0, V = 0 v, u, v R +. L V : C, C, then becomes the linear map z 1, z z 1 u, z v. We show, in each case, that by a suitable choice of u and v we can ensure that L V is contractive while L V P > 1 although P = 1 by definition. We list the contractivity conditions see Propositions.3 and.4 for details. a L V is contractive if and only if the following two conditions are satisfied: i u 1 1 or v 1 and ii 6.1 inf β C, β =1 { 1 u 1β v β + u v 1β β 1 β, β } 0. b L V P 1 if and only if { 6. inf 1 u 1β v β } 0. β C, β =1 Note that the term in parenthesis in 6.1 is non-negative by the Schwarz inequality and that the expression 6. is the same as the first three terms in 6.1. We show that, in each case, we can choose u, v such that the infimum in 6.1 is exactly 0. lso that this infimum is attained at β = β 0 such that the term in parenthesis in 6.1 is positive that is, the Schwarz inequality referred to above is a strict inequality at β 0. It then follows that the expression in braces in 6. is negative when β = β 0 and, consequently, the infimum in 6. is negative. Taken together it follows that L V and consequently ρ V is contractive but L V P > 1 and, as a result, ρ V is not contractive. Let η i, i = 1,, be the vectors such that 1 ηi and ηi are linearly dependent. That is, the term in parenthesis in 6.1 vanishes when β = η i. We now provide the details of the argument which proceeds in two steps. Step 1: Show that there are certain ranges of the parameters u, v such that the infimum in 6.1 is not attained at η 1 or η for those values of u, v.

15 CONTRCTIVITY ND COMPLETE CONTRCTIVITY FOR FINITE DIMENSIONL BNCH SPCES 15 Let g u,v β = 1 u 1β v β + u v 1β β 1 β, β. We need to show that there exists β such that g u,v β < g u,v η i, i = 1,, when u, v take values in a range of interest. That is, we need to find β such that 6.3 g u,v η i g u,v β = a i βu + b i βv cβu v > 0. Here 6.4 Consider the functions a i β = 1β 1η i b i β = β η i cβ = 1β β 1 β, β 0. f i u, v, β = a i βu + b i βv cβu v with cβ 0, i = 1,. The following result is evident from the nature of the functions f i u, v, β. Lemma 6.3. i ssume a i β > 0 for some fixed β and i = 1,. Then, given any u 0 > 0, there exists v 0 > 0 depending on u 0 such that f i u, v, β > 0 in the region u < u 0, v < v 0 u 0 u, that is, inside the triangle with vertices 0, 0, u 0, 0 and u 0, v 0. ii ssume b i β > 0 for some fixed β and i = 1,. Then, given any v 0 > 0, there exists u 0 > 0 depending on v 0 such that f i u, v, β > 0 in the region v < v 0, u < u 0 v 0 v, that is, inside the triangle with vertices 0, 0, 0, v 0 and u 0, v 0. iii If f i u 0, v 0, β > 0 then f i tu 0, tv 0, β > 0 for 0 < t < 1. We will show below that, in each of the six cases in Table 3, it is possible to ensure the positivity of a i β, i = 1, or b i β, i = 1, for some choice of β. Consequently, it will follow that the inequality 6. will be true for that vector β with u, v in the region characterized in Lemma 6.3 above. Hence, for u, v in this range, the infimum is not attained at η i, i = 1,. Consider first the cases i, ii and iii. i e iθ 1 θ R 1 b c C, b R + ii e iθ θ R 0 b c C, b R + iii e iθ θ R 1 b c C, b R + We use the unitary equivalence described in Section 4. In cases i and ii multiply 1 and on the left by the unitary matrix e iθ so that 1 becomes the identity matrix. In case iii

16 16 GDDHR MISR, VIJIT PL ND CHERIN VRUGHESE multiply 1 and on the left by the unitary matrix e iθ so that 1 becomes the identity matrix. Now conjugate 1 and by the unitary which makes upper triangular so that cases i,ii and iii reduce to the situation 1 = and = µ σ 0 ν with µ ν, σ 0 In this case a i β = 0 for all β but it is possible to choose β such that b i β > 0. η i satisfies the equation λ i 1 ηi = 0. So in this case η i is a unit eigenvector of with eigenvalue λ i. Since the eigenvalues of are µ and ν it follows that ηi = µ or ν. Hence we can take β = 1 0 so that b iβ σ > 0. Now consider cases iv and v. iv d 1 d C, d 1 1 c c C v d d C, d 1 0 c c C In cases iv and v we have, in Equation 6.4, a i β = β 1 + d β η i 1 d η i = β 1 η i 1 + d β η i = 1 d η i β. If η i = 0 or 1 then c = 0 and it reduces to the simultaneously diagonalizable case. If ηi 0, 1 we can choose β such that β < η i resp. β > η i if d < 1 resp. d > 1 to ensure that a i β > 0 for i = 1,. The methods used in cases iv and v can be adapted to the last case vi: vi d d C, d 1 1 c c C Step : Show that, in each case, there is a choice of u, v in the region characterized in Lemma 6.3 for which the infimum in 6.1 is, in fact, zero. We choose ˆβ to ensure that a i ˆβ or b i ˆβ is positive as described in Step 1. Note that g u,v ˆβ 1 vanishes at the two points u, v = 1 ˆβ, 0, u, v = 0, 1 and also ˆβ along a curve joining these two points. We now consider two cases: Case i: a i ˆβ > 0 Choose u 0, v 0 such that 0 < v 0 < 1, f iu 0, v 0, ˆβ > 0 and g u0,v 0 ˆβ = 0. This is possible using Lemma 6.3 and the above note about the vanishing of g u,v ˆβ.

17 CONTRCTIVITY ND COMPLETE CONTRCTIVITY FOR FINITE DIMENSIONL BNCH SPCES 17 Let x 0 = inf{u : inf β g u,λ 0 uβ 0} where λ 0 = v 0 u 0. Note that x λ 0. lso, from Lemma 6.3, it is clear that f i x 0, λ 0 x 0, ˆβ > 0. We now show that inf β g x0,λ 0 x 0 β = 0. To prove this we first show that g x0, λ 0 x 0 β 0 for all β with β = 1 as follows. ssume there exists β = µ such that g x0,λ 0 x 0 µ < 0. Then there exists a neighborhood U of x 0 such that g u,λ0 uµ < 0 for all u U. For any u U, inf β g u,λ0 uβ < 0, since g u,λ0 uµ < 0 for all u U. Since U is a neighborhood of x 0 there exists a u U such that u < x 0. By the previous assertion, inf β g u,λ0 uβ 0 for this smaller value of u, which is a contradiction. Since inf β g x0,λ 0 x 0 β 0 by the definition of x 0 it follows that inf β g x0,λ 0 x 0 β = 0. Case ii: b i ˆβ > 0 The arguments in this case are similar to Case i. This time choose u 0, v 0 such that 0 < u 0 < 1, f iu 0, v 0, ˆβ > 0 and g u0,v 0 ˆβ = 0. 1 Let y 0 = inf{v : inf g λ 0 v,vβ 0} where λ 0 = u 0. β v 0 s in Case i we can see that y 0 1 λ and from Lemma 6.3 that f i λ 0 y 0, y 0, ˆβ > 0. Using a procedure similar to that used in Case i it follows that inf β g λ0 y 0,y 0 β = 0. We have therefore shown that for all the cases in Table 3 which were not covered by the operator space approach it is possible to choose u, v such that the infimum in 6.1 is zero and this infimum is attained at a vector β not equal to η 1 or η, so that the last term in parenthesis in 6.1 is positive at β. It follows that, in each of these cases, there exists a contractive homomorphism which is not completely contractive. From [10, Theorem 4.1], it follows that, except in the case where is simultaneously diagonalizable, there must exist a contractive linear map on the dual space C, which is not completely contractive. Thus we have proved the following theorem. Theorem 6.4. Suppose that = 1, and the matrices 1, are not simultaneously diagonalizable. Then there exists a contractive linear map on C, which is not completely contractive. lso, there exists a contractive linear map on the dual space C, which is not completely contractive. 7. n Interesting Operator Space Computation In Section 5 the existence of contractive homomorphisms which are not completely contractive was shown in many cases by studying different isometric embeddings of the space C, into M, op which led to distinct operator space structures. The two embeddings considered there were z 1, z z z and z 1, z z 1 t 1 + z t. In this section we show that we can, for some choices of 1,, construct large collections of isometric embeddings of the space C, into various matrix spaces. lthough the embeddings are into very distinct matrix spaces, we show that the operator space structures thus obtained are equivalent. result which is very useful in this context is the following proposition due to Douglas, Muhly and Pearcy cf. [7, Prop..]. Proposition 7.1. For i = 1,, let T i be a contraction on a Hilbert space H i and let X be an operator mapping H into H 1. necessary and sufficient condition that the operator on H 1 H

18 18 GDDHR MISR, VIJIT PL ND CHERIN VRUGHESE defined by the matrix into H 1 such that T1 X 0 T be a contraction is that there exist a contraction C mapping H X = 1 H1 T 1 T 1 C 1 H T T. The operator norm of the block matrix αi m 0 αi n, where B is an m n matrix and α C, is not hard to compute cf. [10, Lemma.1]. The result can be easily extended to a matrix of the form α1 I m B 0 α I n, for arbitrary α 1, α C. Lemma 7.. If B is an m n matrix and α 1, α C then. α1 I m B α1 B = 0 α I n 0 α Proof. Consider the following two sets and S 1 = { α 1, α ; B : α 1 I m B 0 α I n 1 } S = { α 1, α ; B : α 1 B 0 α 1 }. To prove the lemma, it is sufficient to show that these unit balls are the same. From Proposition 7.1 the condition for the contractivity of the elements of S 1 and S is the same, that is, B 1 α 1 1 α The important observation from the lemma above is that, for fixed α 1, α, the norm of the matrix α1 I m B 0 α I n depends only on B. Now consider the pair = 1, with 1 = α α, = 0 β 0 0. Given any m n matrix B with B = β we have the following isometric embedding of C, into M m+n, op z1 α z 1, z 1 I m z B. 0 z 1 α I n For various choices of the dimensions m, n and the matrix B, this represents a large collection of isometric embeddings. For fixed α 1, α, we let X B represent the above embedding of C, into M m+n, op. We now show that the operator space structures determined by these embeddings depend only on B. If V is the space C,, then X B I k gives the embedding of M k V into M k M m+n C. n element of M k V is defined by a pair of k k matrices Z 1, Z, and the corresponding embedding into M k M m+n C has the form α1 Z 1 I m Z B. 0 α Z 1 I n It now remains to show that the operator norm of this matrix depends only on B. Using Proposition 7.1 it can be shown that α1 Z 1 I m Z B α1 Z 1 if and only if 1 Z B 1. 0 α Z 1 I n 0 α Z 1 Hence it follows that these two norms are in fact equal. We have therefore proved the following theorem. B

19 CONTRCTIVITY ND COMPLETE CONTRCTIVITY FOR FINITE DIMENSIONL BNCH SPCES 19 Theorem 7.3. For all m n matrices B with the same operator norm, the operator space structures on C m+n, determined by the different embeddings α1 I z 1, z z m 0 0 B 1 + z 0 α I, α n 0 0 1, α C, are completely isometric irrespective of the particular choice of B. Moreover all of them are completely isometric to the MIN space. cknowledgment. The authors gratefully acknowledge the help they have received from Sayan Bagchi, Michael Dritschel and Dmitry Yakubovich. References [1] W. rveson, Subalgebras of C -algebras II, cta Math., , [] J. gler, Rational dilation on an annulus, nn. of Math., , [3] J. gler and N. J. Young, Operators having the symmetrized bidisc as a spectral set, Proc. Edinburgh Math. Soc., , [4] B. Bagchi and G. Misra, Contractive homomorphisms and tensor product norms, Integral Equations and Operator Theory, , [5] M. J. Cowen and R. G. Douglas, Complex geometry and operator theory, cta Math., , [6] M. J. Cowen and R. G. Douglas, Operators possessing an open set of eigenvalues, Functions, series, operators, Vol. I, II Budapest, 1980, , Colloq. Math. Soc. Janos Bolyai, 35, North-Holland, msterdam, [7] R. G. Douglas, P. S. Muhly and Carl Pearcy, Lifting Commuting Operators, Michigan Math. Journal, , [8] M. Dritschel and S. McCullough, The failure of rational dilation on a triply connected domain, J. mer. Math. Soc., , [9] G. Misra, Curvature inequalities and extremal properties of bundle shifts, J. Operator Th., , [10] G. Misra, Completely contractive Hilbert modules and Parrott s example, cta Math. Hungar., , [11] G. Misra and N. S. N. Sastry, Contractive modules, extremal problems and curvature inequalities, J. Funct. nal., , [1] G. Misra and N. S. N. Sastry, Completely contractive modules and associated extremal problems, J. Funct. nal., , [13] G. Misra and. Pal, Contractivity, complete contractivity and curvature inequalities, to appear, J. d nalyse Mathematique. [14] B. Sz.-Nagy and C. Foias, Harmonic analysis of Hilbert space operators, NorthHolland, [15] S. Parrott, Unitary dilations for commuting contractions, Pac. J. Math., , [16] G. Pisier, n Introduction to the Theory of Operator Spaces, Cambridge University Press, 003. [17] V. Paulsen, Completely Bounded Maps and Operator lgebras, Cambridge University Press, 00. [18] V. Paulsen, Representations of function algebras, abstract operator Spaces and Banach space geometry, J. Funct. nal., , G. Misra Department of mathematics, Indian Institute of Science, Bangalore , India address, G. Misra: gm@math.iisc.ernet.in. Pal Department of Mathematics and Statistics, Indian Institute of Science Education nd Research Kolkata, Mohanpur address,. Pal: avijitmath@gmail.com C. Varughese Renaissance Communications, Bangalore address, C. Varughese: cherian@rcpl.com