%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%% MICIA %%%% 19.12.2010 15:20:48 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%% !!! % % \documentclass[12pt,oneside]{amsproc} \usepackage[T2A]{fontenc} %\usepackage[cp1251]{inputenc} %\usepackage[russian]{babel} \usepackage{amssymb} \usepackage{amsthm} \textheight=22cm \textwidth=15cm \oddsidemargin=5mm \topmargin=-5mm %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\def\ove#1{\overline{#1}} % %%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \pagestyle{plain} \title{Grothendieck's inequality and applications} \author{O.~I.~Reinov${ }^*$} \thanks{This note is a lecture that was given at the "International Conference on Mathematical Inequalities and Applications", March 07 -- March 12, 2010. Lahore, Pakistan.} \thanks{${ }^*$The research was supported by the Higher Education Commission of Pakistan.} \begin{document} \vphantom{} \maketitle %%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%% part 1 classic The famous Grothendieck inequality, which can be seen as a matrix inequality associated to certain bilinear operators and called by him "the fundamental theorem of the metric theory of tensor products of Banach spaces", is equivalent to the following assertion: {\it Let $\{a_{ij}\}_{i,j=1}^n$ be a finite matrix of real numbers such that $|\sum_{i,j=1}^n a_{ij}t_is_j|\le 1$ whenever $|t_i|, |s_j|\le1.$ Then for every set of unit vectors $\{x_i\}_{i=1}^n$ and $\{y_j\}_{j=1}^n$ in a Hilbert space $\|\sum_{i,j} a_{ij}(x_i,y_j)\|\le K,$ where $K$ is an absolute constant. }\ This theorem has a lot of generalizations and applications in very different directions. Some of them are investigations of multilinear extensions of the inequality as well as considerations of the cases of so-called operator spaces and of non-commutative $L_p$-spaces (such as the Schatten spaces $S_p).$ Let us mention just a few fields of applications: $\bullet$\ Theory of absolutely $p$-summing operators with application to the isomorphic classification of Banach spaces and to the geometry of normed spaces in general (an example: disk algebra $C_A(\Bbb T)$ is not isomorphic to a factor space of a $C(K)$-space); $\bullet$\ Investigations of uniform Banach algebras and, generally, of $Q$-algebras (commutative Banach algebras which are isomorphic as Banach algebras to the quotients of uniform Banach algebras). An example: the answers (for $1\le p\le \infty)$ to the old (essentially, due to Varopoulos) problem whether $S_p$-spaces (with their Schur products) should be $Q$-algebras; $\bullet$\ Problems of vector measures theory and related questions in geometric theory of Banach spaces (such as constructions of counterexamples to some long standing problems. For instance, to the question of whether a separable Banach space does not contain $l_1$ if and only if its dual space is separable). We shall be concerned with just some small (but hope, ones of the main) parts of the topic in connection with this beautiful Grothendieck inequality. %%%%%%%% %\newpage \vskip 0.35cm \noindent {\bf I.\, SOME CLASSICAL APPLICATIONS.} \vskip 0.2cm We begin with definitions of absolutely $p$-summing operators (for general notations, definitions and results, see, e.g., \cite{LiTz}, \cite{Woj}, \cite{LiPe}, \cite{PelA}). An operator $T$ from a Banach space $X$ into a Banach space $Y$ is said to be {\it $p$-absolutely summing}, notation $T\in \Pi_p(X,Y),$ where $00,$ $A$ and $B$ be subspaces of $L_p(\mu)$ and $L_p(\nu)$ respectively. Identify $A\otimes B$ with the set $$ \operatorname{span}_{L_p(\mu\times \nu)} \{h:\ h(s,t)=f(s)g(t),\, f\in A, g\in B\} $$ and put $A\otimes_p B= clos_{L_p(\mu\times\nu)} A\otimes B.$ A norm $\alpha$ on $X\otimes Y$ is said to be {\it $p$-regular}\ if for any $(S,\mu),$ $(T,\nu),$ $A$ and $B$ as above and for any operators $U:A\to X$ and $V: B\to Y$ the operator $U\otimes V$ can be extended to a continuous operator (still denoted by $U\otimes V)$ from $A\otimes_p B$ into $X\bar \otimes_\alpha Y$ (the completion with respect to $\alpha)$ and $||U\otimes V||\le ||U||\, ||V||.$ \vskip 0.1cm {\bf Theorem II.1.}\, {\it Let $C,D,X,Y$ be Banach spaces, $p>0,$ $S\in \Pi_p(C,X),$ $T\in \Pi_p(D,Y).$ If $\alpha$ is a $p$-regular norm on $X\otimes Y,$ then $S\otimes T\in \Pi_p(C\widehat{\widehat\otimes}D, X\bar \otimes_\alpha Y)$ and $\pi_p(S\otimes T)\le \pi_p(S)\, \pi_p(T).$ \rm \vskip 0.1cm {\it Proof}. Just apply Pietsch factorization theorem and the definition II.1. Thus, this theorem contains essentially only a modification of the definition; but it has some nice consequences which justify that the authors called it "the theorem". In follows immediately from the theorem, e. g. : \vskip 0.1cm {\bf Corollary II.1.}\, {\it If $T_i\in \operatorname{L}(l_1,l_2),$ $i=1,\dots, n,$ \, then $$T_1\otimes \dots \otimes T_n\in \Pi_1(l_1\widehat{\widehat\otimes}\dots \widehat{\widehat\otimes} l_1, l_2(\Bbb Z_{+}^n))\ \text{ and } \ \pi_1(T_1\otimes \dots \otimes T_n)\le K_G^n\, ||T_1||\dots ||T_n||.$$ \rm \vskip 0.1cm Therefore, the multi-dimensional generalization of Grothendieck's inequality is just a "right" definition (of a tensor norm) plus an application of the classical 1-dimensional inequality of A. Grothendieck. This "right" definition let us to get a lot of other applications. As an example, we obtain also another interesting consequence (with not very difficult proof). Bellow, it is denoted by $I_{\mu,p}$ the identity imbedding from $C(K)$ into $L_p(\mu)$ (where $\mu$ is a finite Radon measure on a compact $K).$ \vskip 0.1cm {\bf Corollary II.2.}\, {\it Let $X$ be a Banach spaces, $T$ is a linear continuous operator from $X$ to $C(K)$ and $p>0.$ Suppose that there is a sequence of finite dimensional projectors $\{P_n\}$ in $X$ with the following properties: $1)$\ $\sup_n ||P_n||<+\infty;$ $2)$\ $(id-P_n)X=(X_1^n\oplus\dots X_{k_n}^n)_p$ for some subspaces $X_1^n, \dots, X^n_{k_n}$ of the space $(id-P_n)X;$ $3)$\ for every $n$ there exists a family $I_1^n, \dots, I^n_{k_n}$ of pairwise disjoint Borel subsets of the compact $K$ such that all the functions from $T(X_j^n)$ vanish out of the set $I^n_{j}$ \, $(j=1, \dots, k_n).$ Let $\mu$ be a measure on $K$ with $\lim_n\, \sup_{1\le j\le k_n} \mu I^n_j=0.$ Then: \ $(a)$\, the operator $I_{\mu,p}T$ is compact for all $p, p>0;$ \ $(b)$\, if $1\le r<2$ and $10$ then every $p$-absolutely summing operator from $JT_r$ is compact.\ {\rm 2)\, } If $1\le r <2$ and $1 t \}= A \cup B, $ where each of the sets $ A $ and $ B $ has the lowest element and any two elements $ a $ and $ b, $ $ a \in A, b \in B $ are incomparable, and 3) no infinite chain in $ \mathcal T $ has an upper %border bound. Elements of $ \mathcal T $ are referred to as vertexes. The root of the tree is the %pinnacle vertex of the zero level, two vertexes (directly following it) are the vertexes of level 1, the next incomparable four vertexes are called the vertexes of level 2. In general, by induction, we can naturally define the vertexes of the $ n $-th level (there are exactly $ 2^n $ %pieces ones). %%%% next item If $ s \in \mathcal T $ then the set $ \{t: \, t \ge s \} $ is called a subtree growing from $s.$ The branch growing from a vertex $ s $ (of the $ n $-th level) is any totally ordered set in which $ s $ is the smallest element and that contains a vertex of the $ m $-th level for every $ m, m \ge n. $ By subtrees (branches) of the $ n $-th level we understand any subtrees (branches), growing from the vertexes of a $ n $-th level. Branches of the zero level are in natural bijective correspondence with the sequences of zeros and ones, that is, with the points of the dyadic Cantor set $ \mathcal C. $ In this correspondence (it is allowed some freedom of speech here), subtrees of the $n $-th level correspond to $ 2^n$ dyadic intervals of the $ n $-th rank (which form a partition of $ \mathcal C), $ which we denote by $ I_1^n, \dots, I^n_{2^n}. $ In what follows, if $ I $ is any dyadic interval then the corresponding subtree is denoted by $ \mathcal T_I; $ \, $ F_s^n $ is the branch of $ n $-th level, corresponding to $ s, s \in \mathcal C. $ Every branch $ F $ can be considered as a sequence (if numbering its elements in ascending order) and, therefore, the expression of the form $|| g|_F|| $ has a sense, where $ g $ is a (finite) function on $ \mathcal T $ and $ || \cdot || $ is a norm in some %!!! finite ? sequence space. %Recall The definition of the classical James's space $J$ can be found in \cite{Ja}. Recall it. The space $J$ is the completion of the set of all finite sequences with respect to the norm $||\cdot||_J:$ $$ ||x||_J:= \sup \bigg\{ \left( \sum_{j=1}^m |\sum_{k=n_j}^{n_{j+1}-1} x_k|^2\right)^{1/2}:\ 1\le n_1< n_2<\dots p\ge 1.$ It is {\it this case where the Grothendieck's inequality is used}. So, let us consider the part B of Theorem II.2 for $r\ge 2$ and $2>p\ge 1.$ % !!! So, ? Now Define an operator $ S $ from $ JT_r $ into $ C (\mathcal C) $ by $ (Sx) (s): = \lim_{a \in F^0_s} \sum_{b \le a} x (b). $ Let $ \mu $ be the Lebesgue measure on $ \mathcal C. $ We shall show that %if $ p $ and $ r $ satisfy the % conditions of B, then $ I_{\mu, p} S \notin \operatorname{N}_p^Q. $ Assume that the latter is not true. Let $ \varepsilon> 0, $ and find such a finite dimensional (say, $ m $-dimensional) operator $ U $ that $ \nu_p^Q (I_{\mu, p} S-U) <\varepsilon. $ Put $$ X_N: = \operatorname{span}_ {JT_r} \{e_1^{(N)}, \dots, e_{2^N}^{(N)} \}; $$ $ X_N $ is isometric to $ l_r^{(2^N)}$ (note that the vectors $ e_j^{(N)} $ correspond under this isometry to the standard basis of $ l_r^{(2^N)} $ and $ I_{\mu, p} S (e_j^{(N)}) $ are the characteristic functions of dyadic intervals of $ N $-th rank). Let $ P_N $ be the natural projection from $ L_p (\mu) $ onto $ \operatorname{span} \{I_{\mu, p} S (e_j^{(N)}) \}_{1 \le j \le 2^N}, $ $ || P_N || = 1.$ Operators $P_N I_{\mu, p} S|_ {X_N} $ and $P_N U|_{X_N} =: u_N $ can be considered as the operators acting from $ l_r^{(2^N)} $ into $ l_p^{(2^N)}, $ the first one being "identical" with $ 2^{-{N/p}} \, h_{r, p}^{(2^N)}, $ where $ h_{r, p}^{(2^N)} $ is the identity embedding of $ l_r^{(2^N)} $ into $ l_p^{(2^N)}, $ and $ \operatorname{rank} u_N \le m. $ Since $ \nu_p^Q (I_{\mu, p} S-U) <\varepsilon,$ we have $ 2^{-{N/p}} \, \pi_p (h_{r, p}^{(2^N)} -2^{N/p} \, u_N) \le \varepsilon. $ %!!! If $ \varepsilon $ is small, this contradicts Lemma 5. %!!! continue - OKnow! On the other hand, %consider ... if $r\ge 2$ and $2>p\ge 1$ then $\underline{\lim}_N\, 2^{-{N/p}} \, \pi_p (h_{r, p}^{(2^N)} -2^{N/p} \, u_N) \ge C>0, $ where $C$ is an absolute %positive constant. Indeed, to prove this we may and do assume that $r=2.$ Denote the operator $h_{2, p}^{(2^N)} -2^{N/p} \, u_N$ by $A,$ and let $M:=2^N$ and $b_1, \dots, b_{M}$ be the rows of the matrix $A.$ Since $p<2,$ $$ \pi_p(A)\ge \pi_2(A) \ge \gamma_\infty (A)=\gamma_1(A^*)\ge K_G^{-1}\, \pi_1(A^*) \ge K_G^{-1}\, \pi_p(A^*)% \ge K_G^{-1}\, \left(\sum_{1\le j\le M} ||b_j||_2^p\right)^{1/p} $$ (the second and third norms are the norms in the ideals of operators which can be factored through $C(K)$ and $L_1,$ respectively). Since $\operatorname{rank} 2^{N/p} \, u_N\le m$ for every $N,$ we can find (for all $N$ large enough) at least $M/2$ vectors $b_j$ with $l_2$-norms greater then, say, $1/4.$ Now, it is enough to recall that $M=2^N$ and to apply the definition of $\pi_p$-operators to get $\pi_p(A^*) \ge \left(\sum_{1\le j\le M} ||b_j||_2^p\right)^{1/p}\ge C_0 \, 2^{{N/p}}.$ Done. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Part III \vskip 0.35cm \noindent {\bf III.\, AN APPLICATION TO $S_p$-ALGEBRAS.} \vskip 0.2cm Let us recall that a uniform algebra is a closed subalgebra of $C(K)$ for some compact space $K.$ One says that a Banach algebra is a $Q$-algebra \cite{Werm} if there exists a uniform algebra $A$ and a closed ideal $I$ of $A$ so that $B$ is isomorphic as a Banach algebra to the quotient algebra $A/I.$ An operator algebra is a Banach algebra which can be identify up to norm equivalence with a closed subalgebra of $\operatorname{L}(H)$ for some Hilbert space $H.$ It is known that a quotient algebra of an operator algebra is also an operator algebra \cite{Werm}, \cite{Lum}, \cite{Var1}. This is a result of B. Cole who proved it for the quotient algebras of the uniform algebras; but the proof goes essentially even for the general case (see, e.g., \cite{Var1}). Thus, every $Q$-algebra is an operator algebra. It is also clear that a closed subalgebra of an operator algebra is an operator algebra. What is nice and what has been proved by N. Varopoulos \cite{Var1} is the fact that if a Banach $\mathcal L_\infty$ space admits a Banach algebra structure then it is necessarily the structure of an operator algebra. The theorem was stated for Banach $C(K)$-spaces $B,$ but as was mentioned in \cite{Var1}, the proof has a local Banach spaces technique character, so it suffice to suppose that $B$ is a $\mathcal L_\infty$-space in the sense of Lindenstrauss and Pelzcinski \cite{LiPe} %% !!! It is interesting that the first application of Grothendieck's theorem in the theory of operator algebras seems to be applied in \cite{Var1} by N. Varopoulos. More precisely, Varopoulos has proved a "multidimensional" analogue of the following theorem due to A.Grothendieck \cite{Grot} (see also \cite{LiPe}): {\it there is a $C>0,$ for which every complex bilinear form on $C(X)$ of norm 1, where $X$ is a compact, can be extended to a bilinear form on $L_2(X,\mu)\times L_2(X,\nu)$ of norm $\le C,$ for some probability measures $\mu, \nu$ on $X.$} Unfortunately, the formulation of this generalization of the Grothendieck theorem is a little bit complicated and there is no place to bring this nice result of Varopoulos to here (see Lemma 3.1 in \cite{Var1}.) One can say that the heart of the proof of Varopoulos Theorem \cite{Var1} is his criterium for a Banach algebra to be an operator algebra and a very clever application of the Grothendieck theorem. %%% In \cite{Var2} N. Varopoulos has proved a criterion for a Banach algebra to be a Q-algebra (the criterion was very close to the one of A.M. Davie \cite{Dav}, as Varopoulos noted, and the proof was also analogous). Varopoulos has introduced the new notion of so called {\it injective algebras}\ and used his criterion for proving (\cite{Var2}, Theorem 1) that any injective algebra is a Q-algebra. Recall the definition. A Banach algebra $R$ is said to be an injective algebra if the linear mapping induced by the algebra multiplication $$ m:\ R\otimes R \to R\quad (m(x\otimes y)= xy;\, x,y\in R) $$ is continuous for the injective norm of the tensor product $R\widehat{\widehat{\otimes}} R$ of A. Grothendieck. The next nice theorem in \cite{Var2} is an interpolation theorem, which was used not only by Varopoulos, but also by many other authors in considering of $S_p$-algebras. This result (Theorem 2 \cite{Var2}) asserts that for two Q-algebras that form an interpolation pair, the intermediate algebra is also a Q-algebra. As examples of applications, following Varopoulos, let us consider the algebras $l_p,$\, $1\le p\le \infty$ (with pointwise multiplications). A.M. Davie \cite{Dav} proved that the spaces $l_p$ are Q-algebras for $1\le p\le 2.$ It was communicated to Varopoulos by Sten Kaijser (see \cite{Var2}, p. 4) that $l_1$ is an injective algebra and thus, by Varopoulos, a Q-algebra . So, interpolating between $l_1$ and $l_\infty,$ Varopoulos got the fact that $l_p$ is Q-algebra for every $p\in[1,\infty].$ Returning to the paper \cite{Var1} by Varopoulos, let us mention, among the other results, the following one which gave a rise to a 35 years standing open question in the theory of operator algebras. To formulate this result we need some notations from Proposition 4.2 of \cite{Var1}. Let $H$ be a separable Hilbert space and fix $E:=\{e_n\}_{n=1}^\infty,$ an orthonormal basis in $H.$ Let $$ M:= \{m= (m_{ij};\ i,j= 1,2,\dots)\} $$ be the space of matrix representations of bounded operators on $H$ with respect to the basis $E$ (that is, $m_{ij}=\langle Te_i,e_j\rangle$ for $T\in L(H)$). {\it Then one can give on $M$ {commutative} Banach algebra structure by defining $$ m\cdot n = (m_{ij}\cdot n_{ij}:\ i,j=1,2,\dots) $$ for $m,n\in M,$ and that algebra is then an operator algebra}. For a proof that $M$ under the pointwise multiplication is {\it a normed algebra}\ see \cite{Var3}. Combined with a criterion of Varopoulos \cite{Var1} for a Banach algebra to be an operator algebra, the proof in \cite{Var3} gives more, namely, that $M$ is operator algebra. As mentioned in \cite{Var1}, $M$ appeared in the literature for the first time in \cite{Schur} in 1911. Thus, taking in account that the space $S_2(H)$ of all Hilbert--Schmidt operators with Schur ("pointwise") multiplication is evidently an operator algebra, Varopoulos in \cite{Var1} proved essentially that $S_p(H)$ (under the Schur multiplication) is an operator algebra for every $p\in[2,\infty].$ The fact that $S_p$ is an operator algebra for all $p\in[1,\infty]$ was proved later by D.P. Blecher and C. Le Merdy \cite{BlMe}. The main question, leaving open in \cite{Var1} was: {\it Is the above algebra $M$ a Q-algebra or is it not}? \vskip 0.12cm The mathematical community was solving, step by step, the problem of Varopoulos, or more generally, the problem of whether the commutative Schur $S_p$-algebras were Q-algebras for $1\le p\le \infty.$ The crucial thing was, surely, to solve the problem in the main cases where $p=1$ and $p=\infty$ (having in minds the beautiful interpolation result of Varopoulos). \vskip 0.12cm %%%% I would like to emphasize 3 main steps. 1) The case where $p=4$ was settled in 1998 by C. Le Merdy \cite{LeMe}. 2) The case where $p=1$ was settled in 2006 by David P{\'e}rez-Garc{\'i}a \cite{Pere}. 3) The case where $p=\infty$ was settled in 2009 by J. Bri{\"e}t, H. Buhrman, T. Lee and T. Vidick \cite{BriA}. Thus, by Varopoulos, there were solved, step by step, the cases 1)\, $2\le p\le 4,$\, 2)\, $1\le p\le 2$ and 3)\, $1\le p\le \infty.$ In any case, one of the main tool in proving the corresponding result was the Grothendieck's inequality in one or another formulation, or some of its generalizations. For instance, in \cite{LeMe}, among the other different and difficult facts, the author has used the little Grothendieck theorem (see the paper for details). We do not touch the technique from the last nice paper \cite{BriA} (see the short note by Jop Bri{\"e}t, "A problem of Varopoulos - Short survey on Schatten-Schur algebras"). Let us mention only that Grothendieck theorem was also one of the crucial point in the proof. \vskip 0.16cm When I was giving this small lecture, I was unaware of the result on the case where $p=\infty.$ So, a question, I recalled during the lecture, was "Is $S_p$ a Q-algebra for $4\le p\le\infty?$". Now, as we said, the problem is closed. For me (and, hope, not only for me), it is interesting to consider a main part of David P{\'e}rez-Garc{\'i}a's proof for the trace class. In fact, it is fairly to compare a multidimensional inequality proved in \cite{Pere} and a A. M. Davie's criterion which was used by David P{\'e}rez-Garc{\'i}a. Here they are. \vskip 0.16cm A multilinear Grothendieck's inequality (with a simple proof in that paper): {\bf Theorem III.1 \cite{Pere}\rm(Theorem 2.2.)}\ \it For every $m\in \Bbb N,$ $n\ge2,$ $(a_{i_1\dots i_n})\subset \Bbb C$ and $x^1_{i_1}, \dots, x^n_{i_n}\in B_{l_2^m}$ we have $$ \bigg| \sum_{i_j=1}^m a_{i_1\dots i_n}\, \sum_{k=1}^m x^1_{i_1}(k) \dots x^n_{i_n}(k) \bigg| \le K_G^{n-1}\, \sup_{|t_{i_j}|\le1}\, \bigg| \sum_{i_j=1}^m a_{i_1\dots i_n}\, t_{i_1} \dots t_{i_n} \bigg| $$ \rm (here $B_{l_2^m}$ is the closed unit ball of the $m$-dimensional Euclidean space $l_2^m).$ {\it Remark}: Compare with Corollary II.1. \vskip 0.14cm A. M. Davie's criterion \cite{Dav}: {\bf Theorem III.2.}\ \it A commutative Banach algebra $A$ is a Q-algebra if and only if there exists a positive constant $K$ such that $$ \bigg{\|} \sum_{i_j=1}^m a_{i_1\dots i_n}\, x_{i_1} \dots x_{i_n} \bigg{\|} \le K^n\, \sup_{|t_{i_j}|\le1}\, \bigg| \sum_{i_j=1}^m a_{i_1\dots i_n}\, t_{i_1} \dots t_{i_n} \bigg|, $$ for every sequence $x_1,\dots, x_m\in A$ with $||x_i||\le1$ and for every choice of $a_{i_1\dots i_n}\in \Bbb C.$ \rm \vskip 0.14cm I can guess that David P{\'e}rez-Garc{\'i}a was successfully looking at both of the theorems and then it was nothing for him to solve the problem for the case where $p=1$ (for the details, see the paper \cite{Pere} itself). \vskip 0.12cm {\bf Acknowledgments:}\, I would like to bring my sincere acknowledgments to Professor Josip Pe{\v{c}}ari{\'c} for inviting me to give a lecture during the "International Conference on Mathematical Inequalities and Applications" (March 07 -- March 12, 2010. Lahore, Pakistan). %%%%%%%%%% \vskip 0.25cm %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5 \frenchspacing \begin{thebibliography}{3} %% CUT REFERENCES BELOW AND PUT YOURS %% %Grothendieck, A.: R{\'e}sume de la th{\'e}eorie metrique des produits tensoriels topologiques. %- Bol. Soc. Mat. Sao Paulo 8, 1956, 1{79. \bibitem{Grot} Grothendieck, A., \emph{R{\'e}sum{\'e} de la th{\'e}eorie m{\'e}trique des produits tensoriels topologique}, Bol. Soc. Mat. Sao Paulo \textbf{8} (1956), 1-79. \bibitem{Pal33} Paley R. E. A. C., \emph{On the lacunary coefficients of power series}, Annals of Math. \textbf{34} (1933), 615-616. \bibitem{Pisi} Pisier G., \emph{Factorization of linear operators and geometry of Banach spaces}, CBMS Regional Conf. Series in Math. \textbf{60} (1986). \bibitem{LiTz} Lindenstrauss J., Tzafriri L., \emph{Classical Banach Spaces I}, Springer-Verla, \textbf{I} (1977). \bibitem{PelA} Pe{l}czy{\'n}ski A., \emph{Banach Spaces ofAnalytic Functions and Absolutely Summing Operators}, Amer. Math. Soc, Providence R.I., CBMS, \textbf{30} (1976). \bibitem{Woj} Wojtaszczyk P., \emph{Banach Spaces for Analysts}, Cambridge University Press, (1976). %Blei, R. C.: Multidimensional extensions of Grothendieck's inequality and applications. - %Ark. Mat. 17, 1979, 51{68. %% \bibitem{Blei} Blei, R. C., \emph{Multidimensional extensions of Grothendieck's inequality and applications}, Ark. Mat., \textbf{17} (1979). %Tonge, A.: The Von Neumann inequality for polynomials in several Hilbert-Schmidt %operators. - J. London Math. Soc. 18, 1978, 519{526. \bibitem{Tong} Tonge, A., \emph{The Von Neumann inequality for polynomials in several Hilbert-Schmidt operators}, J. London Math. Soc., \textbf{18} (1978). \bibitem{GKR} E.D. Gluskin, S.V. Kisljakov, O.I. Reinov, \emph{Tensor products of $p$-absolutely summing operators and right $(I\sb{p},\,N\sb{p})$-multipliers,} (Russian) Investigations on linear operators and the theory of functions, IX. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 92 (1979), 85--102, 319--320. \bibitem{LiSt} Lindenstãauss J., Stegall C., \emph{Examples of separable spaces which do not contain $l_1$ and whose duals are non-separable}, Studia Math. \textbf{54} (1975), 81--I05. \bibitem{Ja} James R. S., \emph{A non-reflexive Banach space isometric with its second conjugate}, Proc. Nat. Acad. Sci USA. \textbf{37} (1951), 174--177. \bibitem{Werm} Wermer J., \emph{Quotient algebras of uniform algebras,} in Symposium on function algebras and rational approximation, University of Michigan (1969). \bibitem{Lum} Lumer G., \emph{\'Etats, alg\`ebres quotients et sousespaces invariants}, C. R. Acad. Sci. Paris S\'er. A, \textbf{274} (1972), 1308--1311. \bibitem{Var1} Varopoulos N. T., \emph{A theorem on operator algebras}, Math. Scand., \textbf{37} (1975), 173-182. \bibitem{Var2} Varopoulos N. T., \emph{Some remarks on Q-algebras}, Ann. Inst. Fourier, \textbf{22} (1972), 1-11. %N.T. Varopoulos, Some remarks on Q-algebras, Ann. Inst. Fourier 22 (1972), 1-11. \bibitem{Var3} Varopoulos N. Th., \emph{On a inequality of von Neumann and an application of the metric theory of tensor products to operator theory}, J. Func. Analysis, \textbf{16} (1974), 83-100. \bibitem{Schur} Schur J., \emph{Bemerkungen zur Theorie der beschr{\"a}nkten Biliniarformen mit undendlich vielen Ver{\"a}nderlichen}, J. Reine Angew. Math., \textbf{140} (1911), 1-32. \bibitem{LiPe} Lindenstãauss J., Pelzcinski., \emph{Absolutely summing operators in $\mathcal L^p$-spaces and their applications}, Studia Math. \textbf{29} (1968), 275--326. %A.M. Davie, Quotient algebras of uniform algebras, London Math. Soc. 7 (1973), 31-40. \bibitem{Dav} Davie A. M., \emph{Quotient algebras of uniform algebras}, London Math. Soc., \textbf{7} (1973), 31-40. \bibitem{BlMe} Blecher D. P., Le Merdy C., \emph{On quotients of function algebras and operator algebra structures on $l_p$}, Operator Theory, \textbf{34} (1995), 315-346. %D.P. Blecher, C. Le Merdy, On quotients of function algebras and operator algebra structures %on `p, J. Operator Theory 34 (1995), 315-346. %Le Merdy, C.: The Schatten space S4 is a Q-algebra. - Proc. Amer. Math. Soc. 126, %1998, 715{719. \bibitem{LeMe} Le Merdy, C., \emph{The Schatten space S4 is a Q-algebra}, Proc. Amer. Math. Soc, \textbf{126} (1998), 715-719. %David P{\'e}rez-Garc{\'i}a, The trace class is a Q-algebra, Annales Academ. Scientiarum Fennic., %Mathematica %Volumen 31, 2006, 287{295 \bibitem{Pere} P{\'e}rez-Garc{\'i}a D., \emph{The trace class is a Q-algebra}, Annales Academ. Scientiarum Fennic. Mathematica,, \textbf{31} (2006), 287-295. %Bri{\"e}t J, Buhrman H, Lee T., Vidick T., %Multiplayer XOR games and quantum communication %complexity with clique-wise entanglement, \bibitem{BriA} Bri{\"e}t J, Buhrman H, Lee T., Vidick T., \emph{Multiplayer XOR games and quantum communication complexity with clique-wise entanglement,} ArXive: 0911.4007v1 [quant-ph] 20 Nov 2009, 1-25. %\textbf{31} (2006), 287-295. \end{thebibliography} \end{document} %% %%%%%%%%%%%% %%%%----