HILBERTIAN MATRIX CROSS NORMED SPACES ARISING FROM NORMED IDEALS 7 0 0 2 TAKAHIROOHTA n a Abstract. GeneralizingPisier’sidea,weintroduceaHilbertianmatrix J crossnormedspaceassociated withapairofsymmetricnormedideals. 4 When the two ideals coincide, we show that our construction gives an 1 operator spaceifandonlyiftheidealistheSchatten class. Ingeneral, a pair of symmetric normed ideals that are not necessarily the Schat- ] ten class may give rise to an operator space. We study the space of A completelyboundedmappingsbetweenthematrixcrossnormedspaces O obtained in this way and show that the multiplicator norm naturally appears asthecompletelyboundednorm. . h t a m 1. Introduction [ An operator space is a subspace of the set of bounded operators on a 2 Hilbert space, which is abstractly characterized as a Banach space equipped v 5 with matrix norms satisfying certain properties. An operator space whose 9 base space is a Hilbert space is said to be a Hilbertian operator space. The 7 theoryofhomogeneousHilbertianoperatorspaceisoneofthecentraltopicsin 7 operator space theory and it plays an essential role in various situations. For 0 example, it is used to analyze the structures of the space of operator spaces 6 with the metric which is analogous to the Banach-Mazur distance (cf. [18]) 0 h/ and to obtain an embedding of operator spaces into noncommutative Lp- spaces (cf. [11] and [16]). t a The relationships between homogeneous Hilbertian operator spaces and m operatorideals arefirststudied by Mathes andPaulsen. Mathes andPaulsen : considered in [14] a larger category, called matricially normed spaces (m.c.n. v i spaces), than that of operator spaces. They showed that if H1 and H2 are X homogeneous Hilbertian m.c.n. spaces with the common base space H, then r thespaceofcompletelyboundedmappingsCB(H ,H )becomesasymmetric a 1 2 normedideal(s.n.ideal)[14,1.2.Proposition]andshowedthateverys.n.ideal on B(H) which is not equivalent to the ideal of compact operators or the idealoftrace class operatorsis isomorphicas a set to the space ofcompletely bounded mappings on some homogeneous Hilbertian m.c.n. spaces [14, 2.2. Theorem]. 1991 Mathematics Subject Classification. 47L25,47L20. 1 2 TAKAHIROOHTA G. Pisier showedthat the normofthe elements in the interpolating spaces between the row Hilbert space and the column Hilbert space is represented by the operator norm on the Schatten ideals [18, Theorem 8.4]. Inspired by thisanalysis,inourpaperweintroduceaHilbertianm.c.n.spaceH(Φ,Ψ)for a pair of symmetric norming functions (s.n. functions) Φ,Ψ with Φ Ψ and ≥ investigatethestructureofthe space. ThematrixnormofH(Φ,Ψ)isdefined by T xT∗ 1/2 T = supk i i kΨ , k kH(Φ,Ψ) x (cid:18) x Pk kΦ (cid:19) whereT = ξ T H M and(ξ )isanorthonormalbasisofaseparable i i n i ⊗ ∈ ⊗ HilbertspaceH. Wealsofocusonthespaceofcompletelyboundedmappings P betweentwospacesarisinginthisway. Them.c.n.spaceH(Φ,Ψ)isnotalways an operator space. In section 3 we show that if the m.c.n. space H(Φ,Ψ) is an operator space, then for all x,y,z S the following inequality Φ ∈ x y z y Ψ Φ k ⊗ k k ⊗ k x ≤ z Ψ Φ k k k k is satisfied, where S is the s.n. ideal arising from Φ. In particular, when Φ Φ = Ψ we show that the m.c.n. space H(Φ) = H(Φ,Φ) is an operator space if and only if Φ is the Schatten norm. However, the situation differs for Φ=Ψ. Indeed, when Φ is a Q∗-norm and Ψ is a Q-norm, H(Φ,Ψ) is always 6 an operator space. We also study the space of completely bounded mappings between m.c.n. spaceswe constructed. We determine the completely boundednormfromthe row Hilbert space R to H(Φ,Ψ) as 1/2 x2 y x = sup | | ⊗ Ψ . k kCB(R,H(Φ,Ψ)) y (cid:13) y Φ(cid:13) ! (cid:13) k k (cid:13) This implies that if H(Φ,Ψ) is anoperator space,then we have the isometric isomorphisms CB(R,H(Φ,Ψ)) = S and CB(C,H(Φ,Ψ)) = S for the Ψ˜ Φ˜∗ column Hilbert space C (see section 3 for the definition of Φ˜). The above result leads us to consider the condition: c>0, x y c x y , x,y S . Ψ Ψ Ψ Ψ ∃ k ⊗ k ≤ k k k k ∀ ∈ This condition implies that there exists a constant logn p= lim (P is any rank n projection) n n→∞log Pn Φ k k such that x c x , where x is the Schatten p-norm. This together p Φ p k k ≤ k k k k withadualversionimpliestheabovementionedfactthatH(Φ)isanoperator space only if Φ is the Schatten norm. HILBERTIAN MATRIX CROSS NORMED SPACES ARISING FROM NORMED IDEALS3 2. Preliminaries In this section we collect the basics of the theory of operator spaces and operator ideals, which are often used in the paper. We refer to [9] and [17] forthe theory ofoperatorspacesandto [10] for the theory ofoperatorideals. An operator space is abstractly characterized as follows. We consider a BanachspaceE suchthat for eachn N there is a norm onthe matrix n ∈ k·k space M (E) of n n matrices with entries in the elements of E and the n × family M (E), with equal to the original norm of E. Then we n n 1 { k·k } k·k can consider the two properties x 0 (M1) =Max x , y foranyx M (E), y M (E), 0 y {k km k kn} ∈ m ∈ n (cid:13)(cid:18) (cid:19)(cid:13)m+n a(cid:13)nd m,n(cid:13) N, and (cid:13) (cid:13)∈ (M2) (cid:13)axb (cid:13) a x b for any x M (E), a M , b M , n m m n×m m×n aknd mk ,n≤ kNk,kwkherkekM = M∈ (C) and a∈xb means th∈e matrix m×n m×n ∈ product. (M1) may be replaced with x 0 (M1)′ Max x , y ,foranyx M (E),y M (E), 0 y ≤ {k km k kn} ∈ m ∈ n (cid:13)(cid:18) (cid:19)(cid:13)m+n a(cid:13)nd m,n(cid:13) N. (cid:13) (cid:13)∈ For a H(cid:13)ilbert spa(cid:13)ce H an operator space E B(H) is a Banach space sat- ⊆ isfying the properties (M1) and (M2) under the identification of M (E) as a n subspaceofM (B(H))=B(Hn). Conversely,Ruan[15,Theorem3.1]showed n that a Banach space having the matrix norm structure with the properties (M1) and (M2) has an isometric embedding into the space B(H) for some HilbertspaceH suchthatthematrixnormscomefromM (B(H))=B(Hn). n The properties (M1) and (M2) are called Ruan’s axioms. In the operator space category, the morphisms are the completely bounded (c.b.) mappings. LetE, F be operatorspacesandu be alinear mapping fromE to F. We say that u is completely bounded if u =sup id u: M (E) M (F) < , cb n n n k k k ⊗ → k ∞ n where M (E) is identified with the algebraic tensor product M E. The n n ⊗ completely bounded norm of u is defined by u . An operator space E is cb k k said to be homogeneous if for any bounded linear mapping u on E we have u = u . We denote the Banach space of completely bounded mappings cb k k k k from E to F with norm by CB(E,F). cb k·k The categoryofmatrixcrossnormedspacesis largerthanthatofoperator spaces. Let H be a separable Hilbert space with a sequence of matrix norms ∞ onthefamily M (H) ∞ suchthat coincideswiththenorm {k·kn}n=1 { n }n=1 k·k1 of H. We call H a matrix cross normed space (m.c.n. space) if x A = x A k ⊗ kn k kk kMn 4 TAKAHIROOHTA for all x H, A M , and n N. n ∈ ∈ ∈ For a finite-dimensional or separable infinite-dimensional Hilbert space K with dimensionn, identifying B(K) with the matrix space M we denote the n matrix whose (i,j)-entry is 1 and the other entries are 0 by e . ij Next we introducethe basic theoryofthe operatorideals (cf. [10, Chapter III]). Let c , cˆ, and kˆ be the spaces of sequences of real numbers defined by 0 c = ξ = ξ :lim ξ =0 , 0 i i→∞ i { { } } cˆ= ξ = ξ c :only finitely many ξ ’s are nonzero , i 0 i { }∈ kˆ = ξ = ξ cˆ:ξ ξ ...ξ ... 0 , (cid:8) { i}∈ 1 ≥ 2 ≥ n ≥ ≥ (cid:9) respectively. A real valued function Φ on cˆ is called a symmetric norming (cid:8) (cid:9) (s.n.) function if it satisfies the followings: (1) Φ is a norm on cˆ; (2) Φ(1,0,0,...)=1; (3) Φ(ξ ,ξ ,...,ξ ,0,0,...) = Φ(ξ , ξ ,..., ξ ,0,0,...) for all ξ 1 2 n | j1| | j2| | jn| ∈ cˆ, where j ,j ,...,j is any permutation of 1,2,...,n . 1 2 n { } { } For an s.n. function Φ, we set c = ξ = ξ c :supΦ(ξ(n))< , Φ i 0 { }∈ ∞ n (cid:8) (cid:9) where ξ(n) =(ξ ,...,ξ ,0,0,...). We extend the domain of Φ by 1 n Φ(ξ)= lim Φ(ξ(n)), ξ c . Φ n→∞ ∈ For 1 p , we denote by Φ the ℓ -norm. p p ≤ ≤∞ Throughout the paper, H denotes a separable infinite-dimensional Hilbert space with an orthonormal basis ξ ∞ and S denotes the subspace of B(H) consisting of all compact o{peir}ait=o1rs on H∞. For x S we denote ∞ ∈ by s (x) ∞ the singular numbers (s-numbers) of x, i.e. the nonincreasing { j }j=1 rearrangementof eigenvalues of x. Let S be a two-sided ideal of|B|(H). A functional on S is said to be s k·k a symmetric norm if it satisfies the followings: (1) is a norm on S; s k·k (2) for any rank one operator x, x = x ; s (3) axb a x b ( a,b kB(kH), kxk S). s s k k ≤k kk k k k ∀ ∈ ∀ ∈ We call (S, ) a symmetrically normed ideal if is a symmetric norm s s on S and mkak·eks S a Banach space. k·k For ans.n. function Φ, we denote byS the set ofoperatorsx S with Φ ∞ ∈ s(x)= s (x) c , and put j Φ { }∈ x =Φ(s(x)). Φ k k Then S is an s.n. ideal with the norm . In this paper we often use the Φ Φ k·k property xx∗ S x∗x S and xx∗ = x∗x . Φ Φ Φ Φ ∈ ⇔ ∈ k k k k HILBERTIAN MATRIX CROSS NORMED SPACES ARISING FROM NORMED IDEALS5 Let Φ be an s.n. function. The function 1 Φ∗(η)=max η∗ξ . ξ∈kˆ (Φ(ξ) i i) i X makes sense for any η cˆand Φ∗ is an s.n. function. We call Φ∗ the adjoint ∈ of Φ. Note that for any s.n. function Φ, we have (Φ∗)∗ =Φ and the following duality x = sup Tr(yx). Φ k k | | kykΦ∗≤1 We introduce a few classesof normedidealsused inthis paper. We denote by S = S the Schatten ideal for 1 p . For 1 q p < , the p Φp ≤ ≤ ∞ ≤ ≤ ∞ Lorentz ideal S is an s.n. ideal whose norm is given by p,q 1/q ∞ s (x)q j x = . k kp,q  j1−q/p j=1 X   Let 1 = π π 0 be a sequence of nonincreasing positive numbers 1 2 suchthat lim≥ ≥π··=· ≥0 and ∞ π = . We say that sucha sequence is n→∞ n n=1 n ∞ binormalizing. The s.n. function Φ is defined by π P ∞ Φ (a)= π a∗, a=(a ), π n n n n=1 X where(a∗)isthenonincreasingrearrangementof(a ). Notethatifq =1,then n n the Lorentz ideal S is equal to the ideal S defined by the binormalizing p,1 Φπ sequence π =j1/p−1. j Finally we introduce an important class of operator spaces. If E ,E are 0 1 compatible Banach spaces, then we denote by (E ,E ) for 0 < θ < 1 the 0 1 θ complexinterpolationspaceofthem(see [5, Chapter4]). IfE ,E areopera- 0 1 tor spaces whose base spaces are compatible, we construct an operator space complexinterpolationbyidentifying M ((E ,E ) )with(M (E ),M (E )) n 0 1 θ n 0 n 1 θ foreachn N. WedenotebyRandC therowandcolumnoperatorspacere- ∈ spectively[9,Section3.4]. ThesespacesarehomogeneousHilbertianoperator spaces whose matrix norms are given by n n 1/2 n n 1/2 ξ T = T T∗ , ξ T = T∗T , (cid:13) i⊗ i(cid:13) (cid:13) i i (cid:13) (cid:13) i⊗ i(cid:13) (cid:13) i i(cid:13) (cid:13)Xi=1 (cid:13)R (cid:13)Xi=1 (cid:13) (cid:13)Xi=1 (cid:13)C (cid:13)Xi=1 (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) for a (cid:13)finite sequen(cid:13)ce of(cid:13)matrices(cid:13)T n(cid:13). Note tha(cid:13)t R∗ (cid:13)= C and (cid:13)C∗ = R in (cid:13) (cid:13) (cid:13) (cid:13){ i}i=(cid:13)1 (cid:13) (cid:13) (cid:13) the operator space category. We denote by R(θ) the operator space complex interpolation (R,C) for 0 < θ < 1, which is a homogeneous Hilbertian op- θ erator space. We set R(0) to be the row Hilbert space R and R(1) to be the column Hilbert space C. When θ = 1/2, we write OH = R(1/2). Pisier [18, 6 TAKAHIROOHTA Theorem1.1]introducedthesespacesandshowedthatforanyfinite sequence T it holds that i { } 1/2 ξ T = T T¯ , i i i i (cid:13) ⊗ (cid:13) (cid:13) ⊗ (cid:13) (cid:13)Xi (cid:13)OH (cid:13)Xi (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) where T¯ means the (cid:13)complex con(cid:13)jugate(cid:13)of T . Anoth(cid:13)er important property of i (cid:13) (cid:13) (cid:13) i (cid:13) OH is the self-duality. For an operatorspace E, the operator space E¯ means its complex conjugate. The matrix normsof the elements ofE¯ are defined by k(xij)kMn(E¯) =k(xij)kMn(E). Pisier showed in [18, Theorem 1.1] the completely isometric identification OH =OH∗. Another important example of a homogeneous Hilbertian operator space is the minimal operator space H . Let E be a Banach space. We can embed min E into a commutative C∗-algebra (for example the space of all continuous functionsontheunitballofE∗ equippedwiththeweaktopology). Wedenote bymin(E)theoperatorspacewhosematrixnormsariseformthisembedding. The minimal operator space norm is the minimal norm among all operator space norms. When E is a Hilbert space H, we denote the minimal operator space by H . The matrix norm on H satisfies min min m m ξ T =sup v T , i i i i (cid:13) ⊗ (cid:13) (cid:13) (cid:13) (cid:13)Xi=1 (cid:13)min (cid:13)Xi=1 (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) where the supremum(cid:13)is taken ove(cid:13)r all unit v(cid:13)ectors v(cid:13) of ℓm. (cid:13) (cid:13) (cid:13) { i(cid:13)} 2 3. Basic Properties of the m.c.n. space H(Φ,Ψ) Let K be a separable Hilbert space which is identified with a subspace of separable infinite-dimensional Hilbert space. For n N we denote by ∈ ∪{∞} I the identity operator on the Hilbert space of dimension n. Let T be a n finite sum T = ξ T in the algebraic tensor product H B(K) and i i ⊗ i ⊗ we set T∗ = ξ T∗. Pisier showed the identification of matrix norms of iPi⊗ i R(θ) (0 θ 1) in [18, Theorem 8.4] as follows: ≤ ≤P 1/2 ξ T =sup T xT∗ :x S , x 1 , (cid:13) i⊗ i(cid:13) (cid:13) i i (cid:13) ∈ p,+ k kp ≤  (cid:13)Xi (cid:13)R(θ)⊗minB(K) (cid:13)Xi (cid:13)p  (cid:13) (cid:13) (cid:13) (cid:13) w(cid:13)(cid:13)here p=θ−(cid:13)(cid:13)1. We define the opera(cid:13)(cid:13)tors ρT and(cid:13)(cid:13)ρT∗ on B(K) by  ρ (x)= T xT∗, x B(K), T i i ∈ X ρT∗(x)= Ti∗xTi, x∈B(K). X HILBERTIAN MATRIX CROSS NORMED SPACES ARISING FROM NORMED IDEALS7 sNideietdheirdeρaTl ninorBρ(TK∗)d,ewpeenhdasvoenρtTh(eSc)hoicSe oafntdheρbTa∗s(iSs){ξi}∞iS=1.. IFfoSr fiisxeadtws.on-. ⊆ ⊆ functions Φ and Ψ with Ψ Φ, we define a norm on the space of Φ,Ψ ≤ k·k finite sums T H B(K) by ∈ ⊗ T = ρ : S S 1/2. Φ,Ψ T Φ Ψ k k k → k Now we introduce an m.c.n. space H(Φ,Ψ) whose matrix norm structure is given by identifying M (H(Φ,Ψ)) with (H M , ). We write n n Φ,Ψ ⊗ k · k H(Φ) = H(Φ,Φ) for simplicity. Before proving that H(Φ,Ψ) is a homo- geneous m.c.n. space, we prove a useful formula. We denote by F(K) and U(K) the subsets of B(K) consisting of all finite-rank operators and all uni- tary operators, respectively. If S is a subset of B(K), we denote by S the + subset of S consisting of positive elements in B(K). Lemma 3.1. For any operator T we have the equality T 2 =sup Tr(aρ (b)) = T∗ 2 , k kΦ,Ψ { T } k kΨ∗,Φ∗ where the supremum is taken over all a,b F(K)+ with a Ψ∗ 1 and ∈ k k ≤ b 1. Φ k k ≤ Proof. Note first that for any b S it holds that Φ ∈ b = sup Tr(ab), Φ k k | | a∈F(K) kakΦ∗≤1 and if a is positive we can choose b to be also positive [10, proof of Theorem 12.2]. The trace duality implies ρ :S S = sup ρ (b) = sup Tr(aρ (b)). T Φ Ψ T Ψ T k → k k k | | kbkΦ≤1 kbkΦ≤1 kakΨ∗≤1 If we let a = ua and b = v b be the polar decompositions of a and b, | | | | respectively, by the Schwarz inequality we have 1/2 1/2 2 2 Tr(aρT(b)) Tr a 21Tiv b 21 Tr a 12u∗Ti b 12 | | ≤ | | | | ! | | | | ! Xi (cid:12) (cid:12) Xi (cid:12) (cid:12) = Tr(aρT(cid:12)(cid:12)(v bv∗))1/2(cid:12)(cid:12)Tr(uau∗ρT(b(cid:12)(cid:12)))1/2 (cid:12)(cid:12) | | | | | | | | sup Tr(xρ (y)). T ≤ x,y≥0 kxkΨ∗,kykΦ≤1 8 TAKAHIROOHTA Thus ρ :S S = sup Tr(xρ (y))= sup ρ (y) T Φ Ψ T T Ψ k → k k k x,y≥0 y≥0 kxkΨ∗,kykΦ≤1 kykΦ≤1 = sup Tr(xρT(y))= sup ρT∗(x) Φ∗ k k x∈F(K)+, y≥0 x∈F(K)+ kxkΨ∗,kykΦ≤1 kxkΨ∗≤1 = sup Tr(xρ (y)). T x,y∈F(K)+ kxkΨ∗,kykΦ≤1 2 (cid:3) Proposition 3.2. The space H(Φ,Ψ) is an m.c.n. space and satisfies the Ruan’s axiom (M2). Proof. Let T and S be finite sums defined by T = ξ T , S = ξ S , i i i i ⊗ ⊗ i i X X and let a,b F(K) . Then + ∈ Tr(aρ (b)) T+S = Tr(a(T +S )b(T∗+S∗)) i i i i i X = Tr(aρ (b))+Tr(aρ (b))+ (Tr(aT bS∗)+Tr(aS bT∗)) T S i i i i i X Tr(aρ (b))+Tr(aρ (b))+2 Tr(aT bT∗) Tr(aS bS∗) ≤ T S i i i i s i s i X X = Tr(aρ (b))+Tr(aρ (b))+2 Tr(aρ (b))Tr(aρ (b)) T S T S 2 = Tr(aρ (b))1/2+Tr(aρ (b))p1/2 . T S (cid:16) (cid:17) Thus T+S T + S . IfT =ξ Aisasimpletensorproduct Φ,Ψ Φ,Ψ Φ,Ψ k k ≤k k k k ⊗ with ξ =1, then k k ρ (x) = AxA∗ A x A A x A . T Ψ Ψ Ψ Φ k k k k ≤k kk k k k≤k kk k k k Conversely, T 2 sup ApA∗ =sup pA∗Ap = A 2, k kΦ,Ψ ≥ k kΨ k kΨ k k p p where p runs over all rank one projections. Thus ξ A = ξ A and Φ,Ψ k ⊗ k k kk k hence H(Φ,Ψ) is an m.c.n. space. Finally, if X and Y are scalar matrices, HILBERTIAN MATRIX CROSS NORMED SPACES ARISING FROM NORMED IDEALS9 then Tr( XT YaY∗T∗X∗b) XTY 2 = sup| i i i | k kΦ,Ψ a,b P kakΦkbkΨ∗ = sup|Tr( iXTiYaY∗Ti∗X∗b)|kYaY∗kΦkX∗bXkΨ∗ a,b kYPaY∗kΦkX∗bXkΨ∗ kakΦkbkΨ∗ T 2 X 2 Y 2. ≤ k kΦ,Ψk k k k This shows that H(Φ,Ψ) satisfies Ruan’s axiom (M2). (cid:3) Lemma 3.3. The space H(Φ,Ψ) is homogeneous. Proof. Let A B(H). It suffices to show that for any finite sequence ∈ m T = ξ T H M i i n ⊗ ∈ ⊗ i=1 X and x M , the norm inequality n,+ ∈ ρ (x) A 2 ρ (x) . (A⊗I)T Ψ T Ψ k k ≤k k k k holds. Let H be the finite-dimensional subspace of H spanned by Aξ m 0 { i}i=1 and η k be an orthonormal basis of H . Then k m and there is an { j}j=1 0 ≤ m k-matrix B = (b ) such that B A and Aξ = k b η . Note × ij k k ≤ k k i j=1 ij j that P (A I )T = Aξ T = η b T . n i i j ij i ⊗ ⊗ ⊗ ! i j i X X X Thus if we let S = b T for 1 j k, then j i ij i ≤ ≤ P ρ (x) (A⊗I)T Ψ (cid:13) (cid:13) = (cid:13) S xS∗ (cid:13) (cid:13) j j(cid:13) (cid:13) j (cid:13) (cid:13)X (cid:13)Ψ (cid:13)(cid:13)(cid:13) S1 ...(cid:13)(cid:13)(cid:13)Sk S.1∗ = (I x) . (cid:13)(cid:13)  k⊗  . (cid:13)(cid:13)(cid:13) (cid:13) S∗ (cid:13) (cid:13) (cid:13)  k (cid:13)Ψ (cid:13)   (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) 10 TAKAHIROOHTA S∗ 1 S1 ... Sk = (cid:13)(cid:13)(Ik⊗x12) ... (cid:13) (Ik⊗x12)(cid:13)(cid:13) (cid:13) S∗ (cid:13) (cid:13)  k  (cid:13) (cid:13)Ψ = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(Ik⊗x12)(B∗⊗In)T...1∗ (cid:13)T1... Tm(cid:13)(cid:13)(cid:13)(B⊗In)(Ik⊗x12)(cid:13)(cid:13) (cid:13) T∗ (cid:13) (cid:13)  m  (cid:13) (cid:13)Ψ ≤ (cid:13)(cid:13)(cid:13)kBk2(cid:13)(cid:13)(Im⊗x12)T...1∗ (cid:13)T1 ... Tm(Im⊗x12)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) (cid:13) T∗ (cid:13) (cid:13)  m  (cid:13) (cid:13)Ψ A 2(cid:13)(cid:13)ρT(x) Ψ.    (cid:13)(cid:13) ≤ k k k(cid:13) k (cid:13) (cid:3) Letusseesomeexamples. Thanksto[18,Theorem8.4],wehaveH(Φ )= ∞ R and H(Φ )=C. 1 LetH beahomogeneousHilbertianm.c.n.spaceandΦbeans.n.function. 1 MathesandPaulsen[14,p.1764]defineanewm.c.n.spaceH whosematrix 1,Φ norm is defined by T = sup (x I)T , T H B(K). k kH1,Φ x∈SΦ, kxkΦ≤1k ⊗ kH1 ∈ ⊗ It is easy to see that H is an m.c.n. space. For example, H = H and 1,Φ Φ∞ H = H (see [14, 1.3. Proposition]). If we are given an s.n. function Φ, Φ1 min let Φ˜ be the 2-convexificationof Φ defined by Φ˜(a ,...,a ,...)=Φ(a2,...,a2,...)1/2, a kˆ. 1 n 1 n ∈ Lemma 3.4. For any s.n. functions Φ and Ψ with Φ Ψ, we have the ≥ completely isometric identifications H(Φ ,Φ)=C , • H(Φ,1Φ )=RΦf∗, • ∞ Φ˜ H(Φ,Ψ) =H . • Φ2 min In particular, H(Φ ,Φ )=H . 1 ∞ min Proof. We first prove the second equation. Let T be a finite sum defined by T = ξ T H B(K). i i ⊗ ∈ ⊗ i X Then T 2 = sup Tr(bρ (a)). k kΦ,Φ∞ T a,b∈F(K)+ kakΦ,kbkΦ1≤1