%%%%%%%%%% % \input trtrtrtr.tex %%%%%%%%----------------------------------- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%% after Kalinin 1985 Primenen. FA v t. pribl. %%%% суббота, Февраль 27, 2010 12:32 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%% poisk voprosov po !!! \def\nor#1{||{#1}||} % \def\md#1{|{#1}|} % \def\sp#1#2{\(#1,#2\)} % \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}\,} %%%%%%%%%%%%% after 30.01.00 02:44:40 Sat: \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} %%%%%%%%%%%%%%%%%%%%%%%%% % \def\{\quad\blacksquare} \def\med{\medpagebreak} \def\QQ{$\quad\blacksquare$} \def\small{\smallpagebreak} \def\Q{\quad\blacksquare} \def\bigp{\bigpagebreak} \def\f{\vec} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \documentclass[12pt,oneside]{amsproc} \usepackage[T2A]{fontenc} %\usepackage[cp1251]{inputenc} \usepackage[english]{babel} %\usepackage[russian]{babel} \usepackage{amssymb} \usepackage{amsthm} \textheight=22cm \textwidth=15cm \oddsidemargin=5mm \topmargin=-5mm \title{Banach spaces without approximation properties of type $p$} %{Неизоморфные банаховы пространства с одним линейным носителем} \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. } %\address{Abdus Salam School of Mathematical Sciences, 68-B, New Muslim Town, Lahore 54600, PAKISTAN.} \email{orein51@mail.ru} \author{Qaisar Latif} \address{Abdus Salam School of Mathematical Sciences, 68-B, New Muslim Town, Lahore 54600, PAKISTAN.} %\thanks{Работа выполнена при поддержке Российского фонда фундаментальных исследований, грант %№~06-01-00457 .} \thanks{${ }^\maltese$ AMS Subject Classification 2000: 46B28. Spaces of operators; tensor products; approximation properties } \thanks{${ }$ Key words: $p$-nuclear operators, approximation properties, compact approximation, nuclear tensor norms, tensor products. } \begin{document} \vphantom{} \maketitle \begin{abstract} The main purpose of this note is to show that the question posed in the paper of Sinha D.P. and Karn A.K. ("Compact operators which factor through subspaces of $l_p$ Math. Nachr. 281, 2008, 412-423; see the very end of that paper) has a negative answer, and that the answer was known, essentially, in 1985 after the papers [2], [3] by Reinov O.I. have been appeared in 1982 and in 1985 respectively. \end{abstract} \vskip 0.3cm The main purpose of this note is to show that the question posed in [1] %!!! (see the very end of that paper) has a negative answer, and that the answer was known, essentially, in 1985 after the papers [2], [3] have been appeared %!!! in 1982 and in 1985 respectively. For the sake of completeness (and since the article [3] is now quite %!!! difficult to get) %obtain, find?? %!!! ? we will reconstruct some of the facts from [3], %!!! maybe, in a somewhat changed form (a translation of [3] with some remarks can be found in arXiv:1002.3902v1 [math.FA]). % !!! %"ArxiveM" and "SPbPr" %for FA in RUS!!! utochnit'! All the notation and terminology, we use, are from [2], [4], or [5] and more or less standard. %!!! For the general theory of absolutely $p$-summing, $p$-nuclear and other operator ideals, we refer to [6]. %!!! Pietsch Now, we recall the main definition from [3] %!!! of the topology $\tau_p$ of the $\pi_p$-compact convergence, which is just the same as the $\lambda_p$-topology in [1]\footnote{ %!!! Definition 4.3 from [1]:\, %!!! "Given a compact subset $K\sbs X$ we define a seminorm $||\cdot||$ on $\Pi_p(X,Y)$ given by $||T||_K=\inf\{\kappa_p^d(Ti_Z):\, i_Z:Z\to X \text{ as above}\}.$ The family of seminorms $\{||\cdot||:\, K\sbs X \text{ is compact}\}$\, determines a locally convex topology $\lambda_p$ on $\Pi(X,Y).$"; --- for details, and for the definition of maps $i_Z,$ see [1], Lemma 4.2. }, where it was proved, essentially, that that $T\in K_p^d(X,Y)$ if and only if $T\in QN_p(X,Y).$ 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)}$ --- a compact subset of $ Y.$ \vskip 0.2cm {\bf Proposition 1} [3].\ {\it Let $ \operatorname{R}$ be a linear subspace in $ \Pi_p(Y,X),$ containing $ Y^*\ot X.$ Then $ (\operatorname{R},\tau_p)'$ is isomorphic to a factor space of the space $ X^*\wh\ot_{p'} Y.$ More precisely, if $ \ffi\in (\operatorname{R}, \tau_p)',$ then there exists an element $ z=\sum_1^\infty x'_n\ot y_n\in X^*\wh\ot_{p'} Y$ such that $$ \ffi(U)= \tr\, U\circ z,\ \, U\in \operatorname{R}. \eqno(*) $$ On the other hand, for every $ z\in X^*\wh\ot_{p'} Y$ the relation $ (*)$ defines a linear continuous functional on $ (\R,\tau_p).$ } \vskip 0.1cm %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% OK till here !!! %The\, {\it Proof.}\ %of this proposition will be given in detail. %However, we will omit details in analogues cases hereinafter. %So, Let $ \ffi$ be a linear continuous functional on $ (\R,\tau_p).$ Then one can find a neighborhood of zero $ \omega_{K}=\omega_{K,\e},$ such that $ \ffi$ is bounded on it: $ \forall\, U\in\omega_K, \ |\ffi(U)|\lee 1.$ We may assume that $ \e=1.$ Consider the operator $ U\Phi_K:\,Y_K\ovs{\Phi_K}\longrightarrow Y \ovs{U}\longrightarrow X.$ Since the mapping $ \Phi_K$ is compact, $ U\Phi_K\in \QN_p(Y_K,X).$ Put $ \ffi_k(U\Phi_K)=\ffi(U)$ for $ U\in \R.$ On the linear subspace $ \R_K= \left\{ V\in \QN_p(Y_K,X):\ V=U\Phi_K \right\}$ of the space $ \QN_p(Y_K,X),$ the linear functional $ \ffi_K$ is bounded: if $ V=U\Phi_K\in \R_K$ and $ \pi_p(V)\lee 1,$ then $ |\ffi_K(V)|=\ffi(U)|\lee 1.$ Therefore, $ \ffi_K$ can be extended to a linear continuous functional $ \wt\ffi$ on the whole $ \QN_p(Y_K,X);$ moreover, because of the injectivity of the ideal $ \QN_p,$ considering $ X$ as a subspace of some space $ C(K),$ we may assume that $ \wt\ffi\in\QN_p(Y_K, C(K))^*.$ Let us mention that $$ \wt\ffi(jU\Phi_K)=\ffi_K(U\Phi_K)=\ffi(U) \eqno(1) $$ (here $ j$ is an isometric embedding of $ X$ into $ C(K)$). Furthermore, since $ \QN_p(Y_K, C(K))^*= \I_{p'}(C(K), (Y_K)^{**}),$ we can find an operator $ \Psi: C(K)\to (Y_K)^{**},$ for which $$ \wt\ffi(A)= \tr \Psi A,\ \, A\in (Y_K)^*\ot C(K). $$ Let $ A_n\in (Y_K)^*\ot C(K),\, $ $ \pi_p(A_n-jU\Phi_K)\to 0.$ Then $$ \wt\ffi(jU\Phi_K)= \lim\, \tr \Psi A_n. \eqno(2) $$ Consider the operator $ \Phi_K^{**}\Psi: C(K)\ovs{\Psi}\longrightarrow (Y_K)^{**} \ovs{\Phi_K^{**}}\longrightarrow Y.$ Since $ \Psi\in \I_{p'},$ and $ \Phi_K$ is compact, we have $ \Phi_K^{**}\Psi \in \N_{p'}(C(K), Y)= C(K)^*\wh\ot_{p'} Y.$ % Ho!: go via 2 compact operators and refl. space by Johnson!!! Let $ \sum_1^\infty \mu_n\ot y_n\in C(K)^*\wh\ot_{p'} Y$ be a representation of the operator $ \Phi^{**}_K\Psi.$ Put $ z=\sum j^*(\mu_n)\ot y_n.$ The element $ z$ generates an operator $ \Phi_K^{**}\Psi j$ from $ X$ to $ Y.$ We will show now that $ \tr U\circ z=\wt\ffi(jU\Phi_K)$ (note that $ U\circ z$ is an element of the space $ X^*\wh\ot X,$ so the trace is well defined). We have: where $ (jU\Phi_K)^{**}\Psi: C(K)\ovs{\Psi}\longrightarrow (Y_K)^{**} \ovs{\Phi_K^{**}}\longrightarrow Y \ovs{U}\longrightarrow X\ovs{j}\longrightarrow C(K).$ % identify X with its image in X**; come through with a compact act!!!! Since $ \pi_p(A_n- jU\Phi_K)\to 0,$ then $ \pi_p\(A^{**}_n - (jU\Phi_K)^{**}\)\to 0.$ Moreover, if $A:= A_n=\sum_1^N w_m\ot f_m\in (Y_K)^*\ot C(K),$ then $$ \tr A_n^{**}\Psi= \sum_m \< \Psi^*w_m, f_m\> = \sum_m \< w_m, \Psi f_m\>= \tr \Psi A. $$ \begin{multline} \tr U\circ z= \tr \( \sum j^*(\mu_n)\ot Uy_n\)= \sum \< j^*(\mu_n), Uy_n\> =\\ = \sum \< \mu_n, jUy_n\>= \tr jU\Phi_K^{**}\Psi=\tr (jU\Phi_K)^{**}\Psi, \label(3) \end{multline} Hence, $ \tr (jU\Phi_K)^{**}\Psi = \lim\,\tr A^{**}_n\Psi= \lim\,\tr \Psi A_n.$ Now, it follows from (3) and (2) that $ \wt\ffi(jU\Phi_K)=\tr U\circ z.$ Finally, we get from (1): $ \ffi(U)= \tr U\circ z.$ Thus, the functional $ \ffi$ is defined by an element of $ X^*\wh\ot_{p'} Y.$ Inversely, if $ z\in X^*\wh\ot_{p'} Y,$ put $ \ffi(U)=\tr U\circ z$ for $ U\in \R$ (the trace is defined since $ U\circ z\in X^*\wh\ot X).$ We have to show that the linear functional $ \ffi$ is bounded on a neighborhood $ \omega_{K,\e}$ of zero in $ \tau_p.$ For this, we need the following fact, which proof is rather standard (see [3], Lemma 1.2): %!!! %\proclaim {\bf 1.2. Lemma} {\it If $ z\in X^*\wh\ot_q Y,$ then $ z\in X^*\wh\ot_q Y_K,$ where $ K=\ove{\Gamma(K)}$ is a compact in $ Y.$ }\ %\endproclaim\rm %\demo{\it Proof of the lemma %}\ %Let %$ z=\sum x'_n\ot y_n,$ $ \{c_n\}\in c_0$ %and %$ \sum \|x'_n\|^q\,c_n^{-1}<+\infty. $ %Consider the operator %$ A_1:l_q\to Y,$ $ A_1\{a_n\}= \sum a_n c_n y_n.$ %Since this operator is compact, one can find a compact %$ K\sbs Y$ %and an operator %$ A_2: l_q\to Y_K,$ %for which %$ A_1=\Phi_K A_2.$ %Put %$ z_0=\sum c_n^{-1}\,x'_n\ot e_n$\, ($ e_n$ %are orths in %$ l_q).$ %Then %$ \ove{z}:=(\bold 1\ot A_2)(z_0)\in X^*\wh\ot_q Y_K,$ %and %$ \Phi_K(\ove{z})=z.$ %$\quad\blacksquare$\enddemo %!!! %Let us continue the proof of the theorem. Now, let $ K$ be a compact subset of $ Y,$ for which $ z\in X^*\wh\ot_{p'} Y_K.$ If $ U\in \omega_{K,1},$ then $ \pi_p(U\Phi_K)<1$ and $ |\tr U\Phi_K\circ z|\lee \|z\|_{X^*\wh\ot_{p'} Y_K}\cdot$ %\, $\pi_p(U\Phi_K)\lee C.$ %$\quad\blacksquare$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \vskip 0.1cm {\bf Corollary 1.}\it \, $ (\R,\tau_p)'=(\R,\sigma)',$ where $ \sigma=\sigma(\R, X^*\wh\ot_{p'} Y).$ Thus, the closures of convex subsets of the space $ \Pi_p(Y, X)$ in $ \tau_p$ and in $ \sigma$ are the same. %$\quad\blacksquare$ \rm \vskip 0.1cm {\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 \rm {\it Proof} If the map $ j$ is one-to-one then the annihilator $ j^{-1}(0)^\perp$ of its kernel in the space, dual to $ X^*\wh\ot_{p'}Y,$ coincides with $ \Pi_p(Y, X^{**}).$ On the other hand, in any case $ j^{-1}(0)^\perp= \ove{Y^*\ot X}^{\,*}$ (the closure in $ {}^*$\!-weak topology of the space $ \Pi_p(Y, X^{**}));$ by Corollary from Proposition 1, $$ \Pi_p(Y,X)\cap \ove{Y^*\ot X}^{\,*}= \ove{Y^*\ot X}^{\,\tau_p}. $$ Therefore, $ \Pi_p(Y,X)= \ove{Y^*\ot X}^{\,\tau_p}.$ \vskip 0.1cm For a reflexive space $ X,$ the dual space to $ X^*\wh\ot_{p'} Y$ is equal to $ \Pi_p(Y,X).$ Consequently, it follows from the last two statements \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 Recall a definition of the {\it approximation property $AP_p$ of order}\, $p, p\in (0,\infty]$ (see, e.g., [2] and [7]): %!!! a Banach space $X$ has the $AP_p$ if for every Banach space $Y$ (equivalently, for every reflexive Banach space $Y,$ see [2] or [5]) one has the equality $Y^*\wh\ot_p X = \N_p(Y,X).$ It follows now from Proposition 1 and Corollary 2: \vskip 0.1cm {\bf Corollary 3.}\it \, For $p\in[1,\infty]$ and for every Banach space $ X$ the following are equivalent: $1)$ $X$ has the $AP_p;$ %for each space %$ Y$ %the canonical mapping %$Y^*\wh\ot_p X \to \N_p(Y,X)$ %is one-to-one; $2)$ for each Banach space $ Y$\ $ \ove{X^*\ot_{p'} Y}^{\,\tau_{p'}}= \Pi_{p'}(X,Y).$ $3)$ for each reflexive Banach space $ Y$\ $ \ove{X^*\ot_{p'} Y}^{\,\tau_{p'}}= \Pi_{p'}(X,Y).$ $3')$ for each reflexive Banach space $ Y$\ $ \ove{X^*\ot_{p'} Y}^{\,\lambda_{p'}}= \Pi_{p'}(X,Y).$ \vskip 0.1cm \rm As shown in [2], [4] and [5] , for each $p, p\in [1,\infty], p\neq2,$ there exist reflexive Banach spaces without the $AP_p.$ Thus, we get from the last consequence \vskip 0.1cm {\bf Corollary 4.}\it \, For $1\le p\le \infty, p\neq2,$ there are two reflexive Banach spaces $X$ and $Y$ such that the natural map $Y^*\wh\ot_p X \to \N_p(Y,X)$ is not one-to-one and $ \ove{X^*\ot_{p'} Y}^{\,\tau_{p'}}\neq \Pi_{p'}(X,Y).$ \vskip 0.1cm \rm Since, as we said, $\tau_q$ equals to the topology $\lambda_q$ from [1], we get firstly Theorem 4.11 [1] (the case $p>2$ below) and secondly the negative answer to the question in the end of the paper [1]. Or, generally, we have: \vskip 0.1cm {\bf Theorem.}\it \, Let $1\le q\neq2 \le\infty.$ Then there is a $($reflexive$)$ Banach space that fails the approximation property of type $q$ of $[1].$ \rm %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%% ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ \bigskip \bigskip \medskip \begin{thebibliography}{0} \bigskip \medskip \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{2} 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{3} O.I. Reinov: \textit{Approximation of operators in Banach spaces}, Application of functional analysis in the approximation theory (KGU, Kalinin) (1985), 128-142. \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 non-p-nuclear operators with p-nuclear second adjoints}, Doklady AN SSSR, 256(1981), 43--47. \bibitem{6} A. Pietsch: Operator ideals, North-Holland, Deutscher Verlag der Wiss., Berlin, 1978. \bibitem{7} P. Saphar: \textit{Produits tensoriels d'espaces de Banach et classes d'applications lineaires}, Studia Math. 38(1970), 71--100. %%%%%%%%%%%%%% \end{thebibliography} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \end{document} %%%%%%%% %%%%%%%% %%%%%%%% %%%%%%%% \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 \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 %%%%%%%%