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\def\ove#1{\overline{#1}}    %
\def\ovs#1#2{\overset{#1}\to{#2}}
     \def\({\left(}       \def\al{\alpha}           \def\lee{\leqslant}
     \def\){\right)}      \def\e{\varepsilon}    \def\gee{\geqslant}
     \def\[{\left[}       \def\la{\lambda}
     \def\]{\right]}      \def\ffi{\varphi}
                          \def\be{\beta}
     %\def\{{\left\{}
                                      \def\ot{\otimes}
     \def\<{\langle}                 \def\wh{\widehat}
     \def\>{\rangle}                 \def\wt{\widetilde}
                 \def\sbs{\subset}
\def\tr{\operatorname{trace}\,}

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\def\Gr{\operatorname{Gr}}
\def\AP{\operatorname{AP}}
\def\BAP{\operatorname{BAP}}
\def\N{\operatorname{N}}
\def\I{\operatorname{I}}
\def\id{\operatorname{id}}
\def\L{\operatorname{L}}
\def\QN{\operatorname{QN}}
\def\J{\operatorname{J}}
\def\R{\operatorname{R}}
\def\reg{\operatorname{reg}}
\def\dual{\operatorname{dual}}
                     \def\sbs{\subset}

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    \def\QQ{$\quad\square$}      \def\small{\smallpagebreak}
    \def\Q{\quad\blacksquare}         \def\bigp{\bigpagebreak}

 \def\f{\vec}

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\documentclass[12pt,oneside]{amsproc}
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\textheight=22cm \textwidth=15cm \oddsidemargin=5mm
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\title{Approximation of $p$-summing operators by adjoints}
%{}
\author{Oleg Reinov}
\address{Department of Mathematics and Mechanics, St. Petersburg State University,
Saint Petersburg, RUSSIA.\newline
\phantom{Ao} Abdus Salam School of Mathematical Sciences, 68-B, New Muslim Town, Lahore 54600, PAKISTAN.
}
\email{orein51@mail.ru}



\thanks{%${ }^\maltese$
The research was supported by the Higher  Education Commission of Pakistan.}

\thanks{%${ }^\maltese$
AMS Subject Classification 2010: 47B10, 47A58.
%  47B10 Operators belonging to operator ideals (nuclear, p-summing, in the
% Schatten-von Neumann classes, etc.) [See also 47L20]
%  47A58 Operator approximation theory
}
\thanks{${ }$ Key words: absolutely $p$-summing operators, Banach tensor product, pointwise convergence. }




\begin{document}

                          $$ {} $$
\vphantom{} \maketitle


\begin{abstract}
We consider the following question for the ideals $\Pi_p$ of absolutely $p$-summing
operators:
Is it true that, for given Banach spaces $X$ and $Y,$
the unit ball of the space $\Pi_p(X,Y)$ is dense, for some natural topology, in the unit ball of the space
$\Pi_p(X,Y^{**})$ or in the unit ball of the corresponding space
$\Pi_p^{dual}(Y^*,X^*):= \{U: Y^*\to X^*\, |\ U^*|_X\in \Pi_p(X,Y^{**})\}?$
As "natural topologies", we consider strong and weak operator topologies, compact--open topology,
topology of $X\times Y^*$-convergence etc.
 \end{abstract}

\vskip 0.75cm
                 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

We discuss some questions of the following type.
Let $J$ be a normed operator ideal. Is it true that, for given Banach spaces $X$ and $Y,$
the unit ball of the space $J(X,Y)$ is dense, for some natural topology, in the unit ball of the space
$J(X,Y^{**})$ or in the unit ball of the corresponding space
$J^t(Y^*,X^*):= \{U: Y^*\to X^*\, |\ U^*|_X\in J(X,Y^{**})\}?$
As "natural topologies", we consider strong and weak operator topologies, compact--open topology,
topology of $X\times Y^*$-convergence etc.

In this paper, we consider the case where the ideals under investigations are the injective
ideals $\Pi_p$ of absolutely $p$-summing operators (here $1\le p \le\infty).$



  \vskip 0.33cm


\centerline{\bf \S1. Preliminaries}

 \vskip 0.23cm



All the spaces $X,Y,Z,W, \dots$ are Banach.   For a bounded subset $B$ of $X,$
we denote by $X_B$ the Banach space generated by $B,$ with the unit ball
$\ove{\Gamma}(B)$ (= the closed absolutely convex hull of $B);$
$\Phi_B: X_B\to X$ is the natural embedding (see, e.g., [1]).                             %!!!
$L(X,Y)$ is the space of all (bounded) linear operators from $X$ to $Y$
with its natural operator norm. For $Y=\mathbb K$ (scalar field), we write $X^*$
instead of $L(X,\mathbb K);$  we always consider the space $X$  as the  subspace
$\pi_X(X)$ of its second dual $X^{**}$ (denoting, if needed, by $\pi_X$ the canonical injection).
Some other notations (below $p,q\in[1,\infty] ).$

If $ X$ is a Banach space and
$ \mu$ is a measure then by
$ L_p(\mu; X)$ we understand
the $ L_p$-space of all (equivalent classes of) strongly
$ \mu$-measurable p-summable functions.
In the case where the measures are discrete,
we use also the notations of type
$ l_p(X)$, $ l_p(\Gamma; X)$, $ c_0(X)$, etc.
For the quasinorm of  a sequence  $ (x_k)_k$ from
the space $ l_p(X)$, we use the notation
$ \alpha_p(x_k):= \left(\sum\limits_k \|x_k\|^p\right)^{1/p}$
(if $p=\infty,$ the corresponding changes are needed).
Recall that
$ l_p(\Gamma;X)^*=l_{p'}(\Gamma;X^*)$ and $ c_0(\Gamma;X)^*=l_1(\Gamma;X)$
for all finite $ p\geqslant 1$.
A family $(x_k)_{k=1}^\infty\subset X$,
for which the value $ \alpha_p(x_k)$ is finite,
is said to be {\it absolutely $ p$-summable}.
A family $ (x_k)_{k=1}^\infty$ is called
 {\it weakly $ p$-summable,}\
if  $ (\langle x_k, x' \rangle)\in l_p$ for all $ x'\in X^*$.
We set
$$
\varepsilon_p(x_j):=
\sup_{\|x'\|\leqslant 1} \biggl(\sum\limits_j
|\langle x_j,
x'\rangle |^p\biggr)^{1/p}.
$$
This is a norm  in the space
$ l_p \{ X\}$ of all weakly $ p$-summable sequences
in $ X$.

We use the following notations for the classical operator ideals
(all the information on the theory of operator ideals can be found in
[3]; in our work, we follow, however, the other notation and terminology;
see [4, 5]):

$[\operatorname{\Pi}_p,\pi_p]$ ---
the ideal of the absolutely $ p$-summing operators;

$[\operatorname{QN}_p,\pi_p]$ ---
the ideal of quasi-$p$-nuclear operators;

$[\operatorname{N}_p, \nu_p]$ ---
the ideal of $p$-nuclear operators;

$[\operatorname{ I}_p,i_p]$ ---
the ideal of (strictly) $ p$-integral operators;


Recall some important definitions (to be selfcontained; see [3, 4, 7]).

Let $ T\in \operatorname{L}(X,Y)$.
%For $ 0< p\leqslant \infty,$
The operator  $ T:X\to Y$ is called
{\it absolutely $p$-summing},  if there is a constant $C>0$
such that, for any finite family
$\{x_n\}^M_{n=1} \subset X,$ the following inequality holds
$ \alpha_p(Tx_n) \leqslant C
\varepsilon_p(x_n);$ corresponding norm
($ \inf C$) is denoted by $ \pi_p(T)$.
Note that $ [\Pi_{\infty}, \pi_{\infty}]$
is exactly the operator ideal $ [\operatorname{L},\|\cdot\|]$.
%%%============
%For $ 0< p\leqslant \infty,$
The operator  $ T:X\to Y$ is called
 {\it quasi-$p$-nuclear},
$ T\in \operatorname{ QN}_p(X,Y)$, if for some isometric embedding
 $ i:Y\to L_\infty(\mu)$ the composition
$iT$ is in the closure of the space of all finite rank operators
in
$ \operatorname{N}_p(X, L_\infty(\nu))$.
The  norm  in $\operatorname{ QN}{_p}(X,Y)$
is induced from the space
$ \operatorname{\Pi}_p(X, L_\infty(\nu))$.

  % ....

%Let $ p\in[1,\infty]$.
We say that an operator
$ T\in \operatorname{L}(X,Y)$ {\it belongs to the ideal}
$ \operatorname{N}_p$ $(p$-nuclear), if it can be represented in the form
$$
T:= \sum\limits_{n=1}^{\infty} x'_n\otimes y_n,
$$
where the sequences $ (x'_n)$ and $ (y_n)$ are such that
$ \varepsilon_{p'}(y_n)<\infty$, $ \alpha_p(x'_n)<\infty$.
With the norm, given by $ \nu^p(T):= \inf \varepsilon_{p'}(y_n)\alpha_p(x'_n),$
the class $ \operatorname{N}_p$ is a normed operator ideal.
Let us give other characteristics of the operators $T$ from
$ \operatorname{N}_p(X,Y)$.
An operator $ T$ is an  $ \operatorname{N}_p$-operator iff
it factors in the following way:
$$
%\CD
X \overset{A}{\rightarrow} l_{\infty} \overset{\Delta}{\rightarrow} l_p \overset{B}{\rightarrow} Y,
%\endCD
$$

\noindent
where $ \Delta$ is a  diagonal operator with a diagonal  from $ l_{p}$,
$ A$ and $B$ are the operators of norm 1; moreover,
$ \nu_p(T)$ is just the infimum (over all possible factorizations) of the norms of the diagonals in $ l_p$.   %!!!

Let us say that an
operator $ T\in \operatorname{L}(X,Y)$
{\it belongs to the ideal}
$ \operatorname{I}_p$ (strictly $p$-integral), if it admits a factorization of the kind
$$
X \overset{A}{\rightarrow} L_{\infty}(\mu) \overset{j}{\hookrightarrow} L_p(\mu)
\overset{B}{\rightarrow} Y,
$$
where $ \mu$ is a probability measure, $ j$ is the identity injection,
$ A$ and $ B$ are continuous operators.
We put
$ i_p(T)= \inf \|A\|
\|B\|$,
where the $ \inf$ is taken over all the factorizations of  $ T$
of the mentioned kind.
%%%===================
       \vskip 0.02cm


Now, finally, some important words on the notions
of the Banach tensor products.
%           $$ * * *$$
We consider, mainly, the tensor norms on the tensor products of the kind
$ X^*\otimes Y.$
In this case, the tensor product $ X^*\otimes Y$
can be identified naturally with the linear space of all
finite dimensional
operators from $ X$ to $ Y$.
In the general case, the tensor product $ X\otimes Y$
can be considered as the linear space of all  $weak^*$-$to$-$weak$ continuous
finite rank linear mappings from $X^*$ to $Y$ (or from $Y^*$ to $X$).

On the class $ {\mathfrak T}$ of all such tensor products there is a
maximal (the strongest)
and a minimal (the weakest) tensor norms
[3]. The strongest tensor norm $ \nu_1^0$ on $ X^*\otimes Y$
generates (after completion with respect to this norm) the
{\it projective tensor product of Grothendieck} $ X^*\widehat\otimes Y$ [3],
and the weakest one --- $ \|\cdot\|$ --- {\it injective tensor product}
$ X^*\widehat{\widehat\otimes} Y$, which can be considered as the completion
of the linear space of all finite dimensional operators from $ X$ to $ Y$,
equipped with the usual operator norm $ \|\cdot\|$
   (for the case of $ X\otimes Y,$ we can say the analogous words).
Thus, $ X^*\widehat{\widehat\otimes} Y$ can be identified with
the closed linear  subspace of
$ \operatorname{L}(X,Y)$.
Therefore, for any tensor norm $ \alpha$  between
$\nu^0_1$ and $||\cdot||,$
the natural mapping
$ X^*\otimes_{\alpha} Y\to \operatorname{L}(X,Y)$
can be extended to the
 {\it canonical map} from
$ X^*\widehat\otimes_{\alpha} Y$ to $\operatorname{L}(X,Y)$.
    Analogously, in the general case of products of the kind $ X\otimes Y$,
    the natural mapping    $ X\otimes_{\alpha} Y\to \operatorname{L}(X^*,Y)$
    can be extended to the   {\it canonical map} from
    $ X\widehat\otimes_{\alpha} Y$ to $\operatorname{L}(X^*,Y),$
    and the image of this map belongs to the subspace (of $\operatorname{L}(X^*,Y)=\operatorname{L}(Y^*,X)$)\,  of all
  $ weak^*$-$to$-$weak$ continuous operators from $X^*$ to $Y$ (or from $Y^*$ to $X$).

%%%%%%========================

Let us give the main examples of the tensor products we will working with.


{\it The finite $p$-nuclear tensor norm}
$\|\cdot\|_p$  for
$p\in [1,+\infty]$ is defined on the product $X\otimes Y$ by
the following way:
if $z\in X\otimes Y$, then
$$
\|z\|_p:= \inf
\biggl(\;\sum\limits_{k=1}^N
\|x_k\|^p\biggr)^{1/p}\!
\sup_{\|y'\|\leqslant 1}
\biggl\{ \left(\sum\limits_{k=1}^N
|\langle y_k,y'
\rangle |^{p'}\right)^{1/p'}\biggr\},
%\tag{**}
$$
where $\ 1/p+1/p'=1\ $ and
the infimum is taken over all representations of the tensor element
 $z$ in the space  $X\otimes Y$ in the form
$
z= \sum\limits_{k=1}^Nx_k\otimes y_k$ (formally,
(**) has sense only for finite exponents
$p>1$, and for the case  $p=1$ and $p=+\infty,$
the definition have to be modified).
The completion of the tensor product $ X\otimes Y$
with respect to the norm $ \|\cdot\|_p$,
$ 1\le p\leqslant \infty$, is denoted by $ X\widehat\otimes_p Y$.

%%%%%%%%%%%%            TO LATER PLACE?:
If $ p\in[1,\infty]$,
then the conjugate space to the tensor product $ X\widehat\otimes_p Y$
is equal to $ \Pi_{p'}(Y,X^{**})$
(with the natural duality defined by trace).

%%%%%%%%%%%

For a tensor $z\in X^*\wh\ot X,$ the trace of $z$ is well defined:
if $z=\sum_{k=1}^\infty x'_k\otimes x_k$ is a representation of $z$ in the space $X^*\wh\ot X$
(see [1])                                                                                                                              %!!!
  then
  $$
  \tr z= \sum_{k=1}^\infty \<x'_k, x_k\>.
  $$

For $T\in \Pi_p(X,Y^{**})$ and $z\in Y^*\wh\ot_{p'} X,$
the trace of $T\circ z$ is well defined (since $T\circ z$ belongs to the
projective tensor product $Y^*\wh\ot Y^{**}).$
This trace gives us a possibility to consider the space $\Pi_p(X,Y^{**})$
as the Banach dual to $Y^*\wh\ot_{p'} X.$
Moreover, the dual to any $p$-projective tensor product $Z\wh\ot_{p'} W$
is $\Pi_p(W,Z^*)$    (again, with duality defined by trace).
Analogously,  let $Y^*\wh{\wh\ot}_p X$ be the closure of finite rank operators
in $\Pi_p(Y,X);$  then the dual to the space $Y^*\wh{\wh\ot}_p X$ is $\I_{p'}(X,Y^{**})$
(and the Banach dual to any product $Z\wh{\wh\ot}_p W$ of such a kind
is $\I_{p'}(W,Z^*)$).
Note that in the case $p=1$ we can write $X\wh\ot_1 Y=Y\wh\ot_1 X$
(and only in this case, generally). For $p=\infty,$ \,
$(\Pi_\infty, \pi_\infty)=(L, ||\cdot||).$



%%%%%%%%%%%%%% Paragr 1.

  \vskip 0.33cm


\centerline{\bf \S2. Results}

 \vskip 0.23cm




We need the following notation. For an operator $T\in \L(X,Y),$ we write
$T\in \Pi_p^d(X,Y)$ iff $T^*\in \Pi_p(Y^*,X^*),$ and we define a norm
on the linear space $\Pi_p^d(X,Y)$ by setting $\pi_p^d(T):=\pi_p(T)$
(so, $\Pi_p^d$ is the ideal which is dual to the ideal $\Pi_p$ in the sense
of [3]).

       \vskip 0.1cm

{\bf Lemma 1.}\
Let $T\in \L(X,Y).$ We have: $T\in \Pi_p(X,Y)$ iff $T^*\in \Pi_p^d(Y^*,X^*).$

\vskip 0.03cm

{\it Proof}.\ If $T^*\in \Pi_p^d(Y^*,X^*)$ then $T^{**}\in \Pi_p(X^{**},Y^{**});$
so, by injectivity of $\Pi_p,$ one has $T\in \Pi_p(X,Y).$
If $T\in \Pi_p(X,Y)$ then $T^{**}\in \Pi_p(X^{**},Y^{**}).$ It is clear, but let us
explain this: it is enough, e.g., to consider an isometric imbedding of $Y$
into an $L_\infty$-space and to use the fact that the second adjoint to a $p$-integral
(with values in the space with the metric approximation property) is
$p$-integral itself.
          \vskip 0.1cm

{\bf Lemma 2.}\
Let $C>0,$ $\{A_\beta\}_{\beta\in \mathcal B}$ be a net in $\Pi_p(Z,W),$\, $A\in\Pi_p(Z,W).$
The following are equivalent:

1)\
for each $\beta$\ $\pi_p(A_\beta)\le C$
and
for every $x\in Z$ we have $A_\beta x\underset\beta\to Ax$ in $W;$

2)\
for each $\beta$\ $\pi_p(A_\beta)\le C$
and for every $\e>0$ and any compact subset $K\subset Z$
there is a $\beta_\e$ so that for each $\beta>\beta_\e$ and
for every $k\in K$\, $||A_\beta k-Ak||\le\e.$

\vskip 0.03cm

{\it Proof}.\ We have to prove only that the second part of the assertion 1)
is equivalent to the second part of the assertion 2), if we suppose only that
the usual norms of all operators from the net $\{A_\beta\}_{\beta\in \mathcal B}$
are bounded by $C.$ But in this case, the lemma is proved, e.g., in [6].

          \vskip 0.1cm



Let us say that a net $\{B_\alpha\}_{\alpha\in \mathcal A}$ of operators from $X$
to $Y\,(\subset Y^{**})$ is $X\times Y^*$-pointwise convergent to an operator $B\in \L(X,Y^{**})$
(or, to an operator $D\in \L(Y^*,X^*)$), if
for all $x\in X$ and $y'\in Y^*$ one has $\<B_\alpha x,y'\>\underset\alpha\to \<Bx,y'\>$
(or, $\<B_\alpha x,y'\>\underset\alpha\to \<x,Dy'\>).$

Recall also the definition of the topology $\tau_p$ of $\pi_p$-compact convergence.
For Banach spaces $X, Y,$ the {\it topology $ \tau_p$\ of
$ \pi_p$\!-compact convergence}\
in the space $ \Pi_p(Y,X)$ is the topology, a local base (in zero) of which
is defined by sets of type
$$ \omega_{K,\e}= \left\{ U\in \Pi_p(Y,X):\ \pi_p(U\Phi_K)<\e\right\},
$$
where $ \e>0,$\, $ K=\ove{\Gamma}(K)$ is a compact subset of $ Y.$



  \vskip 0.1cm

{\bf Proposition 3.}\
Let $C>0,$ $B\in \Pi_p(X,Y^{**})$ and $\{B_\alpha\}_{\alpha\in \mathcal A}$ be a net in $\Pi_p(X,Y^{**}).$
The following are equivalent:

(i)\ for every $\alpha$\ $\pi_p(B_\alpha)\le C$
and the net $\{B_\alpha\}_{\alpha\in \mathcal A}$
is $X\times Y^*$-pointwise convergent to the operator $B;$

(ii)\ for every $\alpha$\ $\pi_p(B_\alpha)\le C$
and the net $\{B_\alpha\}_{\alpha\in \mathcal A}$
is $\sigma(\Pi_p(X,Y^{**}),Y^*\wh\otimes_{p'} X)$ convergent to the operator $B.$

In both cases, we have $\pi_p(B)\le C.$

If, in addition, $\{B_\alpha\}_{\alpha\in \mathcal A}\subset \Pi_p(X,Y)$ and
$B\in \Pi_p(X,Y),$ then every of these two assertions  implies
the following ones (and (iii)$\iff$(iv)$\iff$(v)\,):

(iii)\  the operator $B$ is in the closure of the ball of radius $C$ of the space  $\Pi_p(X,Y)$
in the strong operator topology, i.e.  there exists a net  $\{A_\beta\}_{\beta\in \mathcal B}$
such that
for every $x\in X$ we have $A_\beta x\underset\beta\to Bx$ in $Y;$

(iv)\  the operator $B$ is in the closure of the ball of radius $C$ of the space  $\Pi_p(X,Y)$
in the topology of compact convergence;

(v)\  the operator $B$ is in the closure of the ball of radius $C$ of the space  $\Pi_p(X,Y)$
in the topology of $\tau_p$-convergent to the operator $B.$

In all the cases, we have $\pi_p(B)\le C.$

\vskip 0.03cm

{\it Proof}.\ The natural mapping $j_p: Y^*\wh\ot X \to Y^*\wh\ot_{p'} X$
has a dense image, so its adjoint gives us a homeomorphism from
the unit ball of $\Pi_p(X,Y^{**})$ with its weak${}^*$-topology
onto a weak${}^*$-compact subset of $\L(X,Y^{**}).$ To prove $(i)\iff (ii),$
it remains to note
that $X\times Y^*$-pointwise convergence of our net to $B$ is just
its convergence to the operator $B$ in the topology
$\sigma(\L(X,Y^{**}),Y^*\wh\otimes X).$

In the cases (iii)--(v), the topology
$\sigma(\L(X,Y^{**}),Y^*\wh\otimes X),$ considered on the linear subspace
$\Pi_p(X,Y),$ gives the same closure of the ball of radius $C$ of the space  $\Pi_p(X,Y)$
as the topologies of compact convergence (follows from [1]) and $\tau_p$ [5].
Therefore, (i)--(ii) imply (iv) and (v), and (iv)$\iff$(v). The fact, that these last two assertions are
equivalent to the assertion (iii), follows from Lemma 2.

          \vskip 0.1cm

By Lemma 1, we can identify the Banach space $\Pi_p(X,Y^{**})$ with the Banach space
$\Pi_p^d(Y^*,X^*).$ Also, these two Banach spaces give us realizations of the dual space
to the tensor product $Y^*\wh\ot_{p'} X.$ Considering in the first part of the proof of
Proposition 3 the space $\Pi_p^d(Y^*,X^*)$ instead of $\Pi_p(X,Y^{**}),$ we get

  \vskip 0.1cm

{\bf Proposition 4.}\
Let $C>0,$ $B\in \Pi_p^d(Y^*,X^{*})$ and $\{B_\alpha\}_{\alpha\in \mathcal A}$ be a net in $\Pi_p^d(Y^*,X^{*}).$
The following are equivalent:

(i)\ for every $\alpha$\ $\pi_p^d(B_\alpha)\le C$
and the net $\{B_\alpha\}_{\alpha\in \mathcal A}$
is $Y^*\times X$-pointwise convergent to the operator $B;$

(ii)\ for every $\alpha$\ $\pi_p^d(B_\alpha)\le C$
and the net $\{B_\alpha\}_{\alpha\in \mathcal A}$
is $\sigma(\Pi_p^d(Y^*,X^{*}),Y^*\wh\otimes_{p'} X)$ convergent to the operator $B.$

In both cases, we have $\pi_p^d(B)\le C.$
 \vskip 0.1cm

{\bf Corollary 5.}\
1)\ With notations of Proposition 3, if a $\pi_p$-bounded net
$\{B_\alpha\}_{\alpha\in \mathcal A}$ converges in the space $\Pi_p(X,Y^{**})$ to $B$
in the topology of compact convergence (or strongly), then this net
is $\sigma(\Pi_p(X,Y^{**}),Y^*\wh\otimes_{p'} X)$ convergent to the operator $B.$

2)\
With notations of Proposition 4, if a $\pi_p^d$-bounded net
$\{B_\alpha\}_{\alpha\in \mathcal A}$ converges in the space $\Pi_p^d(Y^*,X^{*})$ to $B$
in the topology of compact convergence (or strongly), then this net
is $\sigma(\Pi_p^d(Y^*,X^{*}),Y^*\wh\otimes_{p'} X)$ convergent to the operator $B.$

\vskip 0.1cm

{\bf Corollary 6.}\
With notations of Proposition 3, if a $\pi_p$-bounded net
$\{B_\alpha\}_{\alpha\in \mathcal A}$ converges $X\times Y^*$-pointwise to $B,$
then $B_\alpha^*|_{Y^*}\underset\alpha\to B^*|_{Y^*}$ \, $Y^*\times X$-pointwise and vice versa.
\vskip 0.1cm

We are going now to reformulate the definitions of the  approximation properties of
order $q,\, q\in[1,\infty],$ which were considered in [7] and, e.g., in [4], in terms of some
convergences (by analogue with the approximation properties of Grothendieck; see [1]).
Recall the "tensor language" definitions. We say that the Banach space $X$ has the $\AP_q,$
if for every Banach space $Y$ the natural mapping $Y^*\wh\ot_q X\to N(Y,X)$ is one-to-one
(and thus, an isometric isomorphism between these spaces). In [4], there were introduced
the notions of bounded approximation properties of order $q.$ To recall them, we need some new
notations.

Consider, in the space $I_q(Y,X^{**}),$ the subspace $Y^*\wt\ot_q X,$ the closure of
the space of all finite rank operators from $Y$ to $X$ in that space $I_q(Y,X^{**}).$
Let $C\in[1,\infty).$ We say that the space $X$ has the property $C$-$MAP_q,$ if
for each Banach space $Y$ the natural map $i_q: Y^*\wh\ot_q X\to Y^*\wt\ot_q X$
is  $C$-isometric, i.e., $i_q$ is one-to-one and $||i_q^{-1}||\le C.$
This is the same as to tell that for each Banach $Y$ the adjoint map $i_q^*$
takes the unit ball of the space $G_{q'}(X,Y^{**}),$ dual to $Y^*\wt\ot_q X$
(it is, evidently, some Banach space of operators), into the weak${}^*$-compact set, containing
the ball of radius $1/C$
of the space $\Pi_{q'}(X,Y^{**})$ (which is a representation of the dual space to the
space $Y^*\wh\ot_q X).$ If the space $X$ has $C$-$MAP_q$ for some constant $C,$ we say
that it has the property $BAP_q.$
Clearly, $BAP_q$ implies $AP_q$ (the inverse is not true [4]).

Now, consider the subspace $X^*\ot_{q'} Y$ of the space $X^*\wh{\wh\ot}_{q'} Y,$
which is, in turn, the closure of the space of all finite rank operators $X^*\ot Y$
in $\Pi_{q'}(X, Y^{**}).$ The dual to the space $X^*\ot_{q'} Y$ is just
$I_q(Y,X^{**}),$ so, by the bipolar theorem, the unit ball  of the space
$X^*\ot_{q'} Y$ is weak${^*}$-dense in the unit ball of the space $G_{q'}(X,Y^{**}).$
Thus, the space $X$ has the $C$-metric approximation property of order $q,$\,
$C$-$MAP_q,$ if and only if {\it for any Banach space $Y$}
the  ball of radius $C$ of the space $X^*\ot_{q'} Y$
is weak${^*}$-dense in the unit ball
of the space $\Pi_{q'}(X,Y^{**})$ (or, if one wishes, of the space $\Pi_{q'}^d(Y^*,X^{*})$).
 Also, we can see that the assertion {\it
"for each reflexive Banach space $Y,$ the natural map $i_q: Y^*\wh\ot_q X\to Y^*\wt\ot_q X$
is  $C$-isometric"
 } is equivalent to the assertion {\it
"for any reflexive Banach space $Y,$
the  ball of radius $\le C$ of the space $X^*\ot_{q'} Y$
is weak${^*}$-dense in the unit ball
of the space $\Pi_{q'}(X,Y)$".
 }
By Proposition 3, the last is equivalent to the assertion {\it
"for any reflexive Banach space $Y,$
the  ball of radius $C$ of the space $X^*\ot_{q'} Y$
is dense in the unit ball
of the space $\Pi_{q'}(X,Y)$ in the topology of compact convergence (or, in $\tau_{q'}$)".
}
Consider the case where $q\in (1,\infty).$ Since the subspaces of $L_{q'}$ in Grothendieck-Pietsch
factorizations for absolutely $q'$-summing operators are reflexive (see also [3], 17.3.11),
we get easily (taking in account
the second part of Proposition 3):


  \vskip 0.1cm

{\bf Proposition 7.}\
Let $q\in (1,\infty).$ A Banach space $X$ has the $C$-metric approximation property of order $q,$\,
$C$-$MAP_q,$ if and only if,
for any  Banach space $Y,$
the  ball of radius $\le C$ of the space $X^*\ot_{q'} Y$
is dense in the unit ball
of the space $\Pi_{q'}(X,Y)$" in the topology of compact convergence (or, in the
topology $\tau_{q'}).$ Also, this is equivalent to the fact that
for each reflexive Banach space $Y,$ the natural map $i_q: Y^*\wh\ot_q X\to Y^*\wt\ot_q X$
is  $C$-isometric.

 \vskip 0.1cm

Of course, the proposition is valid in the case $q=1$ (A. Grothendieck [1]). What about the case
$q=\infty,$ we have no place to discuss this at the moment.

We are able to get now, as a consequence of our considerations, the first result,
concerning the question in the very beginning of our paper.

\vskip 0.1cm

{\bf Theorem 8.}\ Let $p\in [1,\infty].$
If the Banach space $X$ has the metric approximation property of order $p$
(i.e., the $1$-$MAP_p),$ then, for every Banach space $Y,$
the unit ball of $\Pi_p(X,Y)$ is dense in the unit ball of the space $\Pi_p(X,Y^{**})$
in the topology of compact convergence.
 \vskip 0.1cm

By Corollaries 5 and 6, we get also

\vskip 0.1cm

{\bf Theorem 9.}\ Let $p\in [1,\infty].$
If the Banach space $X$ has the metric approximation property of order $p$
then, for every Banach space $Y,$
the unit ball of $\Pi_p(X,Y)$ is dense in the unit ball of the space $\Pi_p(Y^*,X^{*})$
in the topology of $Y^*\times X$-pointwise convergence.
 \vskip 0.1cm

The proof of the following result use the main theorem from J. Lindenstrauss' paper [2].

\vskip 0.1cm

{\bf Theorem 10.}\ Let $p\in [1,\infty].$
Let $Z$ be such a space that for every Banach space $W,$  each operator
$U\in \Pi_p^d(W^*,Z^*)$ with $\pi_p^d(U)=1$ belongs to the closure
in the topology of $W^*\times Z$-convergence of the unit ball
of the space $\Pi_p(Z,W).$
Then, for every Banach space $E,$ each operator $S\in\Pi_p(Z,E)$ with $\pi_p(S)=1$
belongs to the closure in the topology of compact convergence (or,
in the $\tau_p$-topology) of finite rank operators with $\pi_p$-norms $\le1.$


\vskip 0.03cm

{\it Proof}.\ Fix $E$ and $S.$ We can assume that the space $E$ is separable.
By [2], there is a separable Banach space $Y$ such that $Y^*$ has the metric approximation property
of Grothendieck and  $Y^{**}=E^*\oplus Y,$ with the natural projector $P$ from $Y^{**}$ onto $E$
having the norm one, and with the natural inclusion $j:E^*\hookrightarrow Y^{**}$
being weak${^*}$-to-weak${}^*$ continuous.
Consider the operator $S^*P: Y^{**}\to E^*\to Z^*.$ It can be approximated, $Y^{**}\times Z$-pointwisely,
by $T$'s$: Z\to Y^*$ with $\pi_p(T)\le1.$
Apply Proposition 4 with $C=1$ to the operator $S^*P\in\Pi_p^d(Y^{**},Z^*)$ to get that
there is a net $\{T_\alpha\}\subset \Pi_p(Z,Y^*)$ such that $\pi_p(T_\alpha)\le1$ for all $\alpha$
and $T_\alpha \underset\alpha\to S^*P$ is in the weak${}^*$-topology
$\sigma(\Pi_p^d(Y^{**},Z^*), Y^{**}\wh\ot_{p'}Z).$
The family  $\{T_\alpha\}$ belongs to the closure in $\sigma(\Pi_p^d(Y^{**},Z^*), Y^{**}\wh\ot_{p'}Z)$
of finite rank operators from the unit ball of the space $\Pi_p(Z,Y^*)$
(since the space $Y^*$ has the $MAP).$
The operator $S^*P$ belongs to the closure in the same topology of the set $\{T_\alpha\}.$
Thus, $S^*P$ belongs to the closure in $\sigma(\Pi_p^d(Y^{**},Z^*), Y^{**}\wh\ot_{p'}Z)$
of finite rank operators from the unit ball of the space $\Pi_p(Z,Y^*).$
Since $S^*=S^*Pj: E^*\to Y^{**}\to E^*\to Z^*,$
we have that $S$ lies in the closure in the weak${}^*$-topology
$\sigma(\Pi_p(Z,E^{**}), E^{*}\wh\ot_{p'}Z)$
of finite rank operators from the unit ball of the space $\Pi_p(Z,E),$
hence in the closure of finite rank operators from the unit ball of the space $\Pi_p(Z,E)$
in $\tau_p$topology (or, if one wish, in the topology of compact convergence).

 \vskip 0.1cm

Taking in account the last three theorems and Proposition 7 before them,
we obtain, as by-product (at least, for $C=1),$ that the proposition 7 is true
also in the cases where $q=1$ or $\infty.$ It is not hard to see that the theorems
can be formulated and proved for general cases "for any constant $C\ge1$", ---
e.g., for $C$-$MAP_p,$ for "closures of $C$-balls" etc.
But this is the theme of the next papers (as well as the considerations of some
"(counter)examples").





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\begin{thebibliography}{0}

\bigskip
\medskip

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 \textit{Produits tensoriels topologiques et \'espaces nucl\'eaires},
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\bibitem{2} J. Lindenstrauss:
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   \bibitem{3} A. Pietsch:
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North Holland (1980).


%\bibitem{4} O.I. Reinov:
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%Theory of operators and theory of functions (LGU) 1(1983), 145-165.

\bibitem{4} O.I. Reinov:
\textit{Approximation properties of order p and the existence of
 non-p-nuclear operators with p-nuclear second adjoints},
Math. Nachr.  109(1982), 125-134.



\bibitem{5} O.I. Reinov:
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Application of functional analysis in the approximation theory (KGU, Kalinin)
  (1985), 128-142.


%\bibitem{5} O.I. Reinov:
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%Journal of Mathematical Sciences \textbf{115}, No. 3 (2003), 2243-2250.

\bibitem{6} O.I. Reinov:
\textit{Approximation of Operators in Dual Spaces by Adjoint Operators},
Journal of Mathematical Sciences \textbf{173}, No. 5 (2011), 632-641.






%\bibitem{1} Sinha D.P., Karn A.K.:
%%\textit{Compact operators which factor through subspaces of $l_p$},
%Math. Nachr.  281(2008), 412-423.


%\bibitem{4} O.I. Reinov:
%\textit{Disappearance of tensor elements in the scale of p-nuclear operators},
%Theory of operators and theory of functions (LGU) 1(1983), 145-165.


%\bibitem{5} O. Reinov: \textit{Approximation properties of order p and the existence of
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%\ref \no133   \by Reinov O.I.  \pages 597-599
%\paper Un contre-exemple a une conjecture de A.Grothendieck
%\yr 1983 \vol 296 \issue
%\jour C. R. Acad. Sc. Paris, Serie I
%\finalinfo
%\endref



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\ref \no134 \by Reinov O.I.  \pages 905-907
\paper Sur les operateurs p-nucl\'eaires entre espaces de Banach avec bases
\yr 1993\vol  316
\jour  C. R. Acad. Sc. Paris. ---  Serie I
\endref

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 %\comment

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


{\bf Proposition 2} [3]. \it
If the canonical mapping
$ j: X^*\wh\ot_{p'} Y\to \N_{p'}(X,Y)$
is one-to-one then
$ \Pi_p(Y,X)= \ove{Y^*\ot X}^{\,\tau_p}.$
\vskip 0.1cm


{\bf Corollary 2.}\it \, For a reflexive space
$ X$
the canonical mapping
$ j: X^*\wh\ot_{p'} Y\to \N_{p'}(X,Y)$
is one-to-one iff the set of finite rank operators is dense in the space
$ \Pi_p(Y,X)$
in the topology
$ \tau_p$\ of $ \pi_p$-compact convergence. \rm

\vskip 0.1cm


