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\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
$0<p\le \infty,$
if there is a constant $C\in (0,\infty)$ such that for all finite families $(x_i)_{i=1}^n\subset X$
$$
 \sum_{i=1}^n ||Tx_i||^p \le C^p\, \sup \{\sum_{i=1}^n |\langle x',x_i\rangle|^p:\ x'\in X,\, ||x'||\le1\}.
$$
The $p$-summing norm $\pi_p(T)$ is defined as $\inf C.$

The Grothendieck's inequality can be easily reformulated in terms of
1-absolutely summing operators: every linear continuous operator
from a $L_1(\mu)$-space into a Hilbert space is 1-absolutely summing.
So, the first application of the inequality is its reformulation in terms of absolutely summing operators
(the proofs can be found in \cite{PelA}, \cite{Woj}, \cite{LiPe} etc. ).
\vskip 0.1cm

{\bf Theorem I.1.}\ \it
$\operatorname{L}(l_1,l_2)=\Pi_1(l_1,l_2).$
 \rm
\vskip 0.1cm

{\it Proof}.\,
Put $P(f):=(\widehat{f}(2^n))_{n=1}^\infty$ for $f\in H^1(\Bbb T).$
By Paley's inequality \cite{Pal33}, the associated operator with operator $P$ is a projection
("Paley projection") in $H^1(\Bbb T),$ which can be considered as
an operator from $H^1(\Bbb T)$ onto $l_2.$
 Let $T$ be any operator from $l_1$ to $l_2,$
 $J:C_A(\Bbb T)\to H^1(\Bbb T)$
is the identity embedding. Then the operator $PJ: C_A(\Bbb T)\to l_2$
is "onto" and therefore, there is a factor map
$Q: l_1\to C_A(\Bbb T)$ such that $T=PJQ.$ Since $J$ is 1-absolutely summing,
we are done.
\vskip 0.1cm

The next application is the so-called "little Grothendieck Theorem"
(see, for example, \cite{Pisi},Theorem 5.4):
\vskip 0.1cm

{\bf Theorem I.2.}\  \it
Any operator from a $C(K)$-space to a Hilbert space is
2-absolutely summing. \rm
\vskip 0.1cm

{\it Proof}.\,
It is not difficult to see that an operator $T\in \operatorname{L}(X,Y)$
is 2-absolutely summing iff for any $U\in \operatorname{L}(l_2,X)$
the operator $TU$ is of type $\Pi_2.$ In the case where
$T$ maps $X=C(K)$ into $l_2,$
this is the same as $TU$ (or $U^*T^*$) is a Hilbert-Schmidt
operator. By Grothendieck, $U^*: C^*(K)\to l_2$ is of type
$\Pi_1;$ thus, 2-absolutely summing. Done.
\vskip 0.1cm

One more nice application:
\vskip 0.1cm

{\bf Theorem I.3.}\  \it
Hardy space $H^1(\Bbb T)$ is not isomorphic to any complemented
subspace of $L_1(\mu)$-space.  \rm
\vskip 0.1cm

{\it Proof}.\,
There is an operator from $H^1(\Bbb T)$ to $l_2$ which is not
1-absolutely summing (for one of the possible proof, see \cite{LiPe}, pp. 300-301,
where a Hardy inequality is used; another proof consists of
considering the Paley projection: in above notation, if $P$
is 1-absolutely summing then $PJ$ is a nuclear (hence, compact)
from $C_A(\Bbb T)$ {\it onto}\, $l_2!$).
\vskip 0.1cm

And also, we have a nice characterization of Hilbert spaces (see \cite{LiPe}):
\vskip 0.1cm

{\bf Theorem I.4.}\ \it
A Banach space is isomorphic to a Hilbert space iff
it is isomorphic to a subspace of an $L_1(\nu)$-space
and to a quotient of an $L_\infty(\mu)$-space.  \rm
\vskip 0.1cm

{\it Proof}.\
Every operator from $L_\infty(\mu)$ to $L_1(\nu)$
is 2-absolutely summing (see, e.g., \cite{LiPe}, Theorem 5.1),
hence, can be factored through a Hilbert space.
\vskip 0.1cm

The following interesting application will be given without any proof
(a proof can be found, e.g., in \cite{LiTz} or in \cite{LiPe}).
\vskip 0.1cm

{\bf Theorem I.5.}\ \it
All normalized unconditional bases in $l_1(\Gamma)$
are equivalent to the unit vector basis in $l_1(\Gamma).$
The same is true for the space $c_0(\Gamma).$  \rm
\vskip 0.1cm

We conclude this section by a result on the disc algebra (see \cite{PelA} or
\cite{Woj}).
\vskip 0.1cm

{\bf Theorem I.6.}\ \it
 The disk algebra $C_A(\Bbb T)$ is not isomorphic to a factor space of $C(K).$
\rm
\vskip 0.1cm

{\it Proof.}\
 Consider
$S: C(K) \to C_A(\Bbb T)$
and
$J: C_A(\Bbb T)\to H^1(\Bbb T),$
where $J$ is the natural inclusion.
If $S$ is "onto" then $JS$ is 1-integral, so
it is nuclear (values in a separable dual).
Thus, $J$ is compact.



%%%%%%%%%%

 %%%%%%%%%%%%%%%%%%%%%%%%% Part: GKR's
%\newpage

		      \vskip 0.35cm

\noindent
{\bf II.\, AN APPLICATION TO $(\operatorname{I}_p,\operatorname{N}_p)$-MULTIPLICATORS.}
		    \vskip 0.2cm

 One of the simplest way to get  a multi-dimensional
 analogue of Grothendieck's inequality (see \cite{Blei}, \cite{Tong},)by using Grothendieck inequality itself,
 can be found in \cite{GKR}.
The authors define there (p. 95) a notion of the so-called $p$-regular norm on the tensor product
$X\otimes Y$ of two Banach spaces in such a way that this definition gives them, almost
immediately, the multi-dimensional inequality of A. Grothendieck (%assuming
%1-dimensional inequality is given;
see \cite{GKR}, Theorem 3 and its Corollary 1).
Namely:
         \vskip 0.1cm

{\bf Definition II.1.}\
 Let $(S,\mu)$ and $(T,\nu)$ be finite measure spaces, $p>0,$ $A$ 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 $1<p<r'$ then $I_{\mu,p}T\in \operatorname{N}_p(X,L_p(\mu));$ \
$(c)$\,  if $1\le r<2$ and $0<p<r'$ then $I_{\mu,p}T\in \operatorname{QN}_p(X,L_p(\mu)).$
      \rm
  \vskip 0.1cm


Now, we will consider  one more of the applications.
Namely, we will show (following \cite{GKR}) how to obtain
James-tree-like spaces $JT_r$ with some unusual properties (applying
Corollary II.2 for checking these properties).                         %!!!
All the difficulties in the construction of such a space is
knowing the corollary II.2 and applying just some beliefs in its existence
and some mathematical thinking.

 We will sketch a construction of the spaces $JT_r$ for $r\in [1,\infty)$
with the properties, mentioned in the following theorem.
	    \vskip 0.1cm

{\bf Theorem II.2}\ \it
{\rm A.\, 1)\, } If $1\le r<\infty,$ and $ p>0$ then
every $p$-absolutely summing operator from $JT_r$ is compact.\
{\rm 2)\, } If $1\le r <2$ and $1<p<r'$ then every $p$-integral
operator from $JT_r$ is $p$-nuclear.\
{\rm 3)\, } If $1\le r <2$ and $1\le p<r'$ then every $p$-absolutely summing
operator from $JT_r$ is quasi-$p$-nuclear.\

{\rm B.\,} If $1<r<2$ and $p\ge r'$ or $r\ge 2$ and $p\ge 1$
then there exists an operator
from $JT_r$ which is $p$-integral but not quasi-$p$-nuclear.
 \rm
	    \vskip 0.1cm

As a simple consequence of this theorem we get some more examples [cf. \cite{LiSt}] of
separable Banach spaces having non-separable duals and not containing $l_1.$

Note %again
that the proof of the part A of the theorem (when the space is constructed) is %an easy
a simple application of the corollary II.2 (and thus, of the theorem II.1).

Let us describe shortly the James-tree-like spaces from \cite{GKR}.
The separable Banach space $JT_r$  consists of functions
on a dyadic tree. The norm in $JT_r$ is defined in such a way that
every trace of $JT_r$ on each branch of the tree gives us the classical
James's space $J$ (of codimension 1 in its second dual);
for every level of the tree, say $n$-th, consisting of $2^n$ vertexes,
the corresponding restriction of $JT_r$ onto $2^n$ natural subtrees of the tree
(growing from those vertexes)
gives  the direct $l_r^{2^n}$-sum of $2^n$ 1-complemented subspaces of $JT_r$
(which are isometric to $JT_r$ itself).

More precisely,
dyadic tree  is a partially ordered
set $ \mathcal T $ which is uniquely (up to isomorphism) determined
by the following requirements: I)  there is the smallest
element in $ \mathcal T $ ("the root of the tree"), 2) if $ t \in \mathcal T $
then $ \{s \in \mathcal T: \ s> 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  $||\cdot||_J:$
$$
 ||x||_J:= \sup \bigg\{
  \left(
   \sum_{j=1}^m |\sum_{k=n_j}^{n_{j+1}-1} x_k|^2\right)^{1/2}:\ 1\le n_1< n_2<\dots <n_m,\, m=1,2, \dots
  \bigg\}.
$$


Finally, we define the space $ JT_r, $ \, $ 1 \le r <\infty. $
$ JT_r $ is the space of functions on $ \mathcal T $, obtained by %enlarging
completion of the  set of finite functions %under the norm
with respect to the norm $ ||| \cdot |||_r,$
$$
 |||x|||_r:= \sup_n\, \sup\bigg\{
   \left(
   \sum_{j=1}^{2^n} ||x|_{F_{s_j}^n}||_J^r
   \right)^{1/r}: \    s_j\in I_j^n,\, 1\le j \le 2^n
 \bigg\}.
$$

We consider only  the proof of Part B of Theorem II.2, ---                                      %!!! N2
moreover, only the case where $r\ge 2$ and $2>p\ge 1.$
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
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