Multi-normed spaces, based on non-discrete measures, and their tensor products

TL;DR

This paper extends the concept of multi-normed spaces by replacing discrete measures with arbitrary measures in an index-free framework, and constructs tensor products for these spaces, especially for minimal Lp-amplifications of Lq-spaces.

Contribution

It introduces a non-coordinate approach to multi-normed spaces based on arbitrary measures and constructs explicit tensor products for these spaces, including a transparent form for minimal Lp-amplifications.

Findings

Established tensor product constructions for p-convex amplifications by Lp spaces.

Proved the existence of tensor products in two distinct categories of amplified spaces.

Derived a simple form of the tensor product for minimal Lp-amplifications of Lq-spaces.

Abstract

It was A. Lambert who discovered a new type of structures, situated, in a sense, between normed spaces and (abstract) operator spaces. His definition was based on the notion of amplification a normed space by means of spaces $\ell_2^n$. Afterwards several mathematicians investigated more general structure, "p-multi-normed space", introduced with the help of spaces $\ell_p^n$; $1\le p\le\infty$. In the present paper we pass from $\ell_p$ to $L_p(X,\mu)$ with an arbitrary measure. This happened to be possible in the frame-work of the non-coordinate ("index-free") approach to the notion of amplification, equivalent in the case of a discrete counting measure to the approach in mentioned articles. Two categories arise. One consists of amplifications by means of an arbitrary normed space, and another one consists of p-convex amplifications by means of $L_p(X,\mu)$. Each of them has its own tensor product of its objects whose existence is proved by a respective explicit construction. As a final result, we show that the "p-convex" tensor product has especially transparent form for the so-called minimal $L_p$-amplifications of $L_q$-spaces, where q is the conjugate of p. Namely, tensoring $L_q(Y,\nu)$ and $L_q(Z,\lambda)$, we get $L_q(Y\times Z,\nu\times\lambda)$.

Equations

$$L_q(Y,\nu ){\otimes }_{p\mathbf{L}}{L}_q(Z,\lambda )=L(Y\times Z,\nu \times \lambda ).$$

$$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 34.88863pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\\&\crcr}}}\ignorespaces{\hbox{\kern-27.35pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{{\bf L}E_{\max}\times{\bf L}F\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 45.20055pt\raise 5.86111pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.0pt\hbox{$\scriptstyle{\vartheta_{\infty}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 78.88863pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern-26.53162pt\raise-20.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.74722pt\hbox{$\scriptstyle{\beta_{0}\times{\bf 1}_{{\bf L}F}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-29.5pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 78.88863pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{{\bf L}(E{\otimes}_{pr}F)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 104.54117pt\raise-20.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.75pt\hbox{$\scriptstyle{\beta}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 104.54117pt\raise-30.16669pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern-34.88863pt\raise-40.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{(E{\otimes}_{pr}{\bf L})\times{\bf L}F\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 47.32614pt\raise-34.60834pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.39166pt\hbox{$\scriptstyle{{\cal S}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 82.77753pt\raise-40.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 82.77753pt\raise-40.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{E{\otimes}_{pr}{\bf L}F}$}}}}}}}\ignorespaces}}}}\ignorespaces,$$

$$E\times F\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\Vert R_{\infty }(U)\Vert \le \sum _{k=1}^n\Vert a_k\Vert \Vert {\mathcal{R}}_{\infty }(u_k,v_k)\Vert \le \Vert \mathcal{R}{\Vert }_{\mathbf{L}b}\sum _{k=1}^n\Vert a_k\Vert \Vert u_k\Vert \Vert v_k\Vert .$$

$$U=T\cdot \left[\sum _{k=1}^{n}e{x}_k♢v_k\right].$$

$$\left(\sum _{k=1}^n{a}_k{I}_k^{\star }\right)\cdot \left(\sum _{l=1}^n{I}_l\cdot (u_l♢v_l)\right).$$

$$U+V=\left(a{I}_U^{\star }+b{U}_V^{\star }\right)\cdot \left(\sum _{k=1}^n{I}_U{I}_k^{\prime }\cdot (u_k^1♢v_k^1)+\sum _{l=1}^m{I}_V{I}_l^{\prime \prime }\cdot (u_l^2♢v_l^2)\right).$$

$$\Vert (a{I}_U^{\star }+b{I}_V^{\star })(\xi )\Vert \le \Vert (a{I}_U^{\star }{P}_U(\xi )\Vert +\Vert (b{I}_V^{\star }{P}_V)(\xi )\Vert \le$$

$$\Vert a\Vert \Vert \Vert P_U(\xi )\Vert +\Vert b\Vert \Vert \Vert P_V(\xi )\Vert \le (\Vert a{\Vert }^q+\Vert b{\Vert }^q{)}^{\frac{1}{q}}(\Vert P_U(\xi ){\Vert }^p+\Vert P_V(\xi ){\Vert }^p{)}^{\frac{1}{p}}$$

$$\Vert U+V{\Vert }_{p\mathbf{L}}\le (\Vert a{\Vert }^q+\Vert b{\Vert }^q{)}^{\frac{1}{q}}(\Vert a{\Vert }^q+\Vert b{\Vert }^q{)}^{\frac{1}{p}}=\Vert a{\Vert }^q+\Vert b{\Vert }^q.$$

$$\Vert R_{\infty }(U)\Vert \le \Vert a\Vert {\left(\sum _{k=1}^n\Vert I_k\cdot {\mathcal{R}}_{\infty }(u_k,v_k){\Vert }^p\right)}^{\frac{1}{p}}\le \Vert a\Vert {\left(\sum _{k=1}^n\Vert {\mathcal{R}}_{\infty }(u_k,v_k){\Vert }^p\right)}^{\frac{1}{p}}\le$$

$$\Vert a\Vert {\left(\sum _{k=1}^n\Vert \mathcal{R}{\Vert }_{b\mathbf{L}}^p\Vert u_k{\Vert }^p\Vert v_k{\Vert }^p\right)}^{\frac{1}{p}}=\Vert \mathcal{R}{\Vert }_{b\mathbf{L}}\Vert a\Vert {\left(\sum _{k=1}^n\Vert u_k{\Vert }^p\Vert v_k{\Vert }^p\right)}^{\frac{1}{p}}.$$

$$\Vert A\otimes B(u)\Vert ={\left({\int }_{Z_2}\left[{\int }_{Y_2}{\left|A(\sum _{k=1}^n((B{g}_k)(t)f_k)(s))\right|}^{p}ds\right]dt\right)}^{\frac{1}{p}}\le$$

$$\Vert A\Vert {\left({\int }_{Z_2}{\Vert \sum _{k=1}^n(B{g}_k)(t)f_k\Vert }_{L_p(Y_2)}^{p}dt\right)}^{\frac{1}{p}}=\Vert A\Vert {\left({\int }_{Z_2}\left[{\int }_{Y_2}{\left|\sum _{k=1}^n(B{g}_k)(t)f_k(s)\right|}^{p}ds\right]dt\right)}^{\frac{1}{p}}=$$

$$\Vert A\Vert {\left({\int }_{Y_2}{\Vert \sum _{k=1}^n(B{f}_k(s)g_k)\Vert }_{L_p(Z_2)}^{p}ds\right)}^{\frac{1}{p}}\le \Vert A\Vert \Vert B\Vert {\left({\int }_{Y_2}{\Vert \sum _{k=1}^n(f_k(s)g_k)\Vert }_{L_p(Z_2)}^{p}ds\right)}^{\frac{1}{p}}=$$

$$\Vert A\Vert \Vert B\Vert {\left({\int }_{Y_2\times Z_2}{\left|\sum _{k=1}^n(f_k(s)g_k(t))\right|}^{p}d(s,t)\right)}^{\frac{1}{p}}=\Vert A\Vert \Vert B\Vert \Vert u\Vert .$$

$$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 83.71867pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\\&\crcr}}}\ignorespaces{\hbox{\kern-83.71867pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{(L_{p}(X){\otimes}_{in}L_{q}(Y))\times(L_{p}(X){\otimes}_{in}L_{q}(Z))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 99.1645pt\raise 5.39166pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.39166pt\hbox{$\scriptstyle{{\cal S}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 127.71867pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern-20.5pt\raise-20.1347pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.74722pt\hbox{$\scriptstyle{{\cal I}_{1}\times{\cal I}_{2}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-29.76942pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 127.71867pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{[L_{p}(X){\otimes}^{p}L_{p}(X)]{\otimes}_{in}[L_{q}(Y){\otimes}^{q}L_{q}(Z)]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 208.21791pt\raise-20.1347pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.39166pt\hbox{$\scriptstyle{{\cal J}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 208.21791pt\raise-29.23056pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern-80.07256pt\raise-40.26942pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{{\cal B}(L_{p}(Y),L_{p}(X))\times{\cal B}(L_{p}(Z),L_{p}(X))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 98.58118pt\raise-34.87776pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.39166pt\hbox{$\scriptstyle{{\cal T}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 131.09174pt\raise-40.26942pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 131.09174pt\raise-40.26942pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{{\cal B}(L_{p}(Y){\otimes}^{p}L^{p}(Z),L_{p}(X){\otimes}^{p}L_{p}(X))}$}}}}}}}\ignorespaces}}}}\ignorespaces,$$

Full text

Two categories arise. One consists of amplifications by means of an arbitrary normed space, and another one consists $p$–convex amplifications by means of $L_{p}(X,\mu)$. Each of them has its own tensor product of its objects whose existence is proved by a respective explicit construction. As a final result, we show that the “$p$–convex” tensor product has especially transparent form for the so-called minimal $L_{p}$–amplifications of $L_{q}$–spaces, where $q$ is the conjugate of $p$. Namely, tensoring $L_{q}(Y,\nu)$ and $L_{q}(Z,{\lambda})$, we get $L_{q}(Y\times Z,\nu\times{\lambda})$.

Keywords: ${\bf L}$–space, ${\bf L}$–boundedness, general ${\bf L}$–tensor product, $p$–convex tensor product.

Mathematics Subject Classification (2010): 46L07, 46M05.

1. Introduction

The subject of the present paper is a rather far-reaching (through several steps) generalization of the structure, introduced in the PhD thesis of A. Lambert [1]; his superviser was G. Wittstock, one of the founding fathers of operator space theory. (See also [2]). This structure is, speaking informally, intermediate between the classical structure of a normed space and the structure of an abstract operator space. (The latter is presented in widely known textbooks [3, 4, 5, 6]; see also [7]). Lambert suggested to endow a given linear space $E$ by a sequence of norms on spaces $E^{n};n\in{\mathbb{N}}$, that is on columns of all possible sizes, consisting of vectors from $E$. (Thus, he deals with norms on columns, and not on matrices, as in the theory of operator spaces). These norms must satisfy two axioms, “contractiveness” and “convexity” that were formulated in terms of the spaces $\ell_{2}^{n};n\in{{\mathbb{N}}}$. Lambert called the resulting objects “Operatorfolgenräume”. In the respective rising category Lambert has constructed two tensor products, “maximal” [1, 3.1.1] and “minimal” [1, 3.1.3]; the former one can be considered as a predecessor of the tensor product, introduced in Section 5 of the present paper.

The theory of Lambert had various connections with the classical theory of normed spaces as well as with the theory of operator spaces, shedding in many occasions a new light in their relationship. Later a team of mathematicians, embarking from essentially different problems, related to Banach lattices, came to more general structures. However, it was done again in the frame-work of the “coordinate approach”, based on the consideration of columns of arbitrary size. First, there were Dales and Polyakov [8], soon after joined by Daws, Pham, Ramsden, Laustsen, Oikhberg and Troitsky [9, 10, 11]. These authors created rich and ramified theory, from which we are most interesting in the so-called $p$–multi-normed spaces [11]: those satisfying the analogue of the contractiveness axiom of Lambert, but now in terms of the spaces $\ell_{p}^{n}$ with arbitrary fixed $p;1\leq p\leq\infty$. The “best” of these structures satisfy also the analogue of the convexity axiom, the so-called $p$–convexity. (Lambert has $p=2$).

The present paper pursuers two aims. First, we extend the class of the structures in question, passing from $p$-multi-normed spaces to their “continuous” (non-discrete) versions. Namely, we change, in the capacity of a base space, $\ell_{p}$ to $L_{p}(X,\mu)$ with arbitrary measure. This becomes possible, if we replace the coordinate approach by the so-called non-coordinate approach to what we call an amplification.

In the context of operator spaces the latter approach was known to specialists, and it was systematically presented in [7] (see also [12]). In the context of Lambert spaces it was applied in [13], where several notions and facts from the present paper have their prototypes. The essence of this approach is as follows. Instead of a sequence of norms on all $E^{n}$ we consider a norm on a single space ${\bf L}{\otimes}E$, where ${\bf L}$ is a chosen and fixed “base” space. Such an approach in the case, where ${\bf L}:=L_{p}(X,\mu)$ with discrete counting measure, is equivalent to the coordinate approach, accepted in the above-cited papers. However, it seems that as a whole it provides greater possibilities. In the frame-work of this approach the axiom of contractiveness transforms to the condition on the normed space ${\bf L}{\otimes}E$ to be a contractive module over ${\cal B}({\bf L})$. As to the “non-coordinate” version of the axiom of $p$-convexity, it can be defined under certain assumptions on ${\bf L}{\otimes}E$ that make this space very similar to $L_{p}(X,\mu)$.

Spaces of the form ${\bf L}{\otimes}E$ are called in this paper ${\bf L}$–spaces. Most of all, we are interested in their tensor products. We introduce two essentially different varieties of this notion. The first one, “ ${\otimes}_{\bf L}$ ”, is defined for the case of general base spaces ${\bf L}$ that are endowed with a certain additional structure, a bilinear operator $\diamondsuit:{\bf L}\times{\bf L}\to{\bf L}$, possessing some natural properties. Another kind of tensor product, denoted by “ ${\otimes}_{p{\bf L}}$ ” and called $p$–convex, is constructed for the class of $p$–convex ${\bf L}$-spaces and only in the case, when our base space is $L_{p}(X,\mu)$. Each of these tensor products is defined in terms of universal property for the respective class of bilinear operators, and its existence theorem is proved by displaying its own explicit construction. We present several examples. In particular, we show that the tensor product “${\otimes}_{\bf L}$ ” acquires sufficiently transparent guise for ${\bf L}$–spaces that have the biggest of all possible norms. What is, perhaps, more interesting is that the second tensor product acquires a transparent concrete guise for $L_{p}(X,\mu)$–spaces of the form $L_{q}({\cdot})$, this time endowed with the minimal norm; here $1<p<\infty$ and $q$ is the conjugate to $p$. Namely, up to a $L_{p}(X,\mu)$–isometric isomorphism of $L_{p}(X,\mu)$-  ̵࠭ (which is an analogue of complete isometric isomorphism of operator spaces), we have

[TABLE]

The author is much indebted to N.T.Nemesh for valuable discussions.

2. ${\bf L}$–spaces and ${\bf L}$–quantization.

As usual, we denote by ${\cal B}(E,F)$ the space of all bounded operators between the normed spaces $E$ and $F$, and consider it with the operator norm. We write ${\cal B}(E)$ instead of ${\cal B}(E,E)$. The identity operator on $E$ will be denoted by ${\bf 1}_{E}$. Two projections $P$ and $Q$ on $E$ are called orthogonal, if $PQ=QP=0$.

The symbol ${\otimes}$ is used for the algebraic tensor product of linear spaces and for elementary tensors. The symbols ${\otimes}_{pr}$ and ${\otimes}_{in}$ denote the non–completed projective and injective tensor product of normed spaces, respectively.

Choose and fix (so far arbitrary) normed space ${\bf L}$, which we shall call the b se space. Let us write ${\cal B}$ instead of ${\cal B}({\bf L})$.

In what follows we need the triple notion of the so-called amplification. First, we amplify linear spaces, then linear operators and finally bilinear operators. Note that these amplifications differ from the amplifications from [7], serving in the theory of operator spaces.

The amplification of a given linear space $E$ is the tensor product ${\bf L}{\otimes}E$. Usually we briefly denote it by ${\bf L}E$, and an elementary tensor, say $\xi{\otimes}x;\xi\in{\bf L},x\in E$, by $\xi x$. Note that ${\bf L}E$ is a left module over the algebra ${\cal B}$ with the outer multiplication “ ${\cdot}$ ”, well defined by $a{\cdot}(\xi x):=a(\xi)x$.

Definition 2.1. A semi-norm on ${\bf L}E$ is called ${\bf L}$–seminorm on $E$, if the left ${\cal B}$-module ${\bf L}E$ is contractive, that is if we always have the estimate $\|a{\cdot}u\|\leq\|a\|\|u\|$. The space $E$, endowed by a ${\bf L}$–seminorm, is called seminormed ${\bf L}$–space). If the seminorm in question is actually a norm, we speak, naturally, about a normed ${\bf L}$–space, and in this case we usually omit the word “normed”.

Example 2.2. Let $(X,\mu)$ be a measure space. Our principal example of a base space is $L_{p}(X,\mu)$, where $1\leq p\leq\infty$. (As the main reference in the measure theory, we shall use the textbook [14]). For simplicity we always assume that all our measures have a countable basis. (Thus, the set of atoms is no more than countable). If there is no danger of misunderstanding, we shall speak about a measure space $X$ and the normed space $L_{p}(X)$; in particular, $X\times Y$ denotes the cartesian product of respective measure spaces.

Remark 2.3. As to the former papers, cited above, they consider, after translation into the “index-free” language, the case $X:={\mathbb{N}}$ with the counting measure. In particular, spaces of Lambert are those with ${\bf L}=\ell_{2}$, whereas in spaces of Dales/Polyakov we have ${\bf L}=\ell_{\infty}$ or ${\bf L}=\ell_{1}$. Finally, if ${\bf L}=\ell_{p}$, then the notion of ${\bf L}$–space is equivalent to the notion of $p$–multi-normed space of Dales/Laustsen/Oikberg/Troitsky [11, 2.2]

A seminormed ${\bf L}$–space $E$ becomes seminormed space in the usual sense, if for $x\in E$ we set $\|x\|:=\|\xi x\|$, where $\xi\in{\bf L}$ is an arbitrary vector with $\|\xi\|=1$. Clearly, the result does not depend on a choice of $\xi$. The obtained seminormed space is called underlying space of a given ${\bf L}$-space, and the latter is called an ${\bf L}$–quantization of a former. (We use such a term by analogy with quantizations in operator space theory; see, e.g., [15], [3] or [7]). Obviously, for all $\xi\in{\bf L}$ and $x\in E$ we have $\|\xi x\|=\|\xi\|\|x\|$.

It is easy to verify that the space of scalars, ${\mathbb{C}}$, has the only ${\bf L}$–quantization, given by the identification of ${\bf L}{\mathbb{C}}$ with ${\bf L}$.

Proposition 2.4. Let $E$ be a seminormed ${\bf L}$–space with a normed underlying space. Then the ${\bf L}$–seminorm on ${\bf L}$ is a norm.

$\triangleleft$ Take $u\in{\bf L}E;u\neq 0$ and represent it as $\sum_{k=1}^{n}\xi_{k}x_{k}$, where $\xi_{k}$ are linearly independent, $\|\xi_{1}\|=1$ and $x_{1}\neq 0$. Obviously, there exists $T\in{\cal B}$ with $\|T\xi_{1}\|=1$ and $T\xi_{k}=0$ for $k>1$. Then, according to Definition 2.1, we have $\|T\|\|u\|\geq\|T{\cdot}u\|=\|x_{1}\|>0.\quad\triangleright$

Example 2.5. Every normed space, say $E$, has, generally speaking, a lot of ${\bf L}$–quantizations. We distinguish two of them. The ${\bf L}$–space, denoted by $E_{\max}$, respectively $E_{\min}$, has the ${\bf L}$–norm, obtained by the endowing ${\bf L}E$ with the norm of ${\bf L}{\otimes}_{pr}E$, respectively of ${\bf L}{\otimes}_{in}E$. We denote the norm on the former and on the latter space by $\|{\cdot}\|_{\max}$ and $\|{\cdot}\|_{\min}$, respectively; accordingly, the corresponding ${\bf L}$-quantizations of $E$ will be called maximal and minimal. Clearly, the ${\bf L}$-norm of $E_{\max}$ is the greatest of all ${\bf L}$-norms of ${\bf L}$-quantizations of $E$. The adjective “minimal” will be justified a little bit later.

Example 2.6. We want to introduce an ${\bf L}$–quantization of the “classical” projective tensor product $E{\otimes}_{pr}F$ of two normed spaces, when one of tensor factors, say, to be definite, $F$, is itself an ${\bf L}$–space.

Consider the linear isomorphism $\beta:{\bf L}(E{\otimes}F)\to E{\otimes}_{pr}({\bf L}F):\xi(x{\otimes}y)\mapsto x{\otimes}\xi y$ and introduce a norm on ${\bf L}(E{\otimes}F)$ by setting $\|U\|^{\prime}:=\|\beta(U)\|$. The space $E{\otimes}_{pr}({\bf L}F)$, being a projective tensor product of a normed space and a contractive ${\cal B}$-module, has itself a standard structure of a contractive ${\cal B}$-module. Since $\beta$ is a ${\cal B}$-module morphism, the same is true with ${\bf L}(E{\otimes}F)$. Thus $E{\otimes}F$ becomes an ${\bf L}$–space. Denote the norm of the respective underlying space just by $\|{\cdot}\|$, and the norm on $E{\otimes}_{pr}F$ by $\|{\cdot}\|_{pr}$. We must show that $\|{\cdot}\|=\|{\cdot}\|_{pr}$.

Take an arbitrary $u\in E{\otimes}F$. Since it is clear that $\|{\cdot}\|$ is a cross-norm, we have $\|u\|\leq\|u\|_{pr}$. It remains to show that for every $\xi\in{\bf L},\|\xi\|=1$ we have $\|\xi u\|^{\prime}\geq\|u\|_{pr}$.

Identifying ${\cal B}$-modules ${\bf L}(E{\otimes}F)$ and $E{\otimes}_{pr}({\bf L}F)$ by means of $\beta$, we represent $\xi u$ as $\sum_{k=1}^{n}x_{k}{\otimes}w_{k};x_{k}\in E,w_{k}\in{\bf L}F$. Take a functional $f:{\bf L}\to{\mathbb{C}}$ such that $f(\xi)=1,\|f\|=1$, and the operator $T:{\bf L}\to{\bf L}:\eta\mapsto f(\eta)\xi$; clearly, $\|T\|=1$. It is obvious that $T{\cdot}w_{k}=\xi y_{k}$ for some $y_{k}\in F;k=1,...,n$. Therefore we have $\sum_{k=1}^{n}\|x_{k}\|\|w_{k}\|\geq\sum_{k=1}^{n}\|x_{k}\|\|T{\cdot}w_{k}\|=\sum_{k=1}^{n}\|x_{k}\|\|y_{k}\|$. But $\xi u=T{\cdot}(\xi u)=\sum_{k=1}^{n}x_{k}{\otimes}T{\cdot}w_{k}=\xi(\sum_{k=1}^{n}x_{k}{\otimes}y_{k})$. Consequently, $u=\sum_{k=1}^{n}x_{k}{\otimes}y_{k}$. It follows that $\sum_{k=1}^{n}\|x_{k}\|\|w_{k}\|\geq\|u\|_{pr}$, and this, because of the definition of the projective norm, implies the desired estimate $\|\xi u\|^{\prime}\geq\|u\|_{pr}$.

Remark 2.7. In the textbook [7] the amplification of a space $E$ is defined as ${\cal F}{\otimes}E$, where ${\cal F}$ is the space of bounded finite rank operators on a certain Hilbert space $L$. However, it is worthy to mention that the norm on ${\cal F}{\otimes}E$, making $E$ an (abstract) operator space, is not always a $\bf L$-norm in the sense of Definition 2.1. The simplest counter-example is $E:={\cal B}(L)$ with the standard norm of an operator space. Using the estimates, obtained by Tomiyama [16], one can prove the following assertion: if $\tau:{\cal F}\to{\cal F}$ is the operator, acting as the transpose of corresponding matrices, then for every $C>0$ there exists $u\in{\cal F}E$ such that, notwithstanding $\|\tau\|=1$, we have $\|\tau{\cdot}u\|>C\|u\|$.

3. ${\bf L}$–bounded linear and bilinear operators

Suppose we are given an operator $\varphi:E\to F$ between linear spaces. Denote, for brevity, the operator ${\bf 1}{\otimes}\varphi:{\bf L}E\to{\bf L}F$ (taking $\xi x$ to $\xi\varphi(x)$) by $\varphi_{\infty}$ and call it amplification of $\varphi$. Obviously, $\varphi_{\infty}$ is a morphism of left ${\cal B}$-modules.

Definition 3.1. An operator $\varphi:E\to F$ between seminormed ${\bf L}$–spaces is called ${\bf L}$–bounded, if the operator $\varphi_{\infty}$ is bounded. Then we set $\|\varphi\|_{{\bf L}b}:=\|\varphi_{\infty}\|$. Similarly, in terms of $\varphi_{\infty}$ we define ${\bf L}$–contractive and ${\bf L}$–isometric operator, and also ${\bf L}$–isometric isomorphism.

If $\varphi$ is bounded, being considered between the respective underlying seminormed spaces, we say that it is (just) bounded, and denote its operator seminorm, as usual, by $\|\varphi\|$. Clearly, every $L$–bounded operator $\varphi:E\to F$ is bounded, and $\|\varphi\|\leq\|\varphi\|_{{\bf L}b}$ .

Some operators between ${\bf L}$–spaces, bounded as operators between underlying spaces, are “automatically” ${\bf L}$–bounded. Here is the first phenomenon of that kind.

Proposition 3.2. *Let $E$ be a ${\bf L}$–space. Then every bounded functional $f:E\to{\mathbb{C}}$ is ${\bf L}$–bounded, and $\|f\|_{{\bf L}b}:=\|f\|$. *

$\triangleleft$ Recall that we have $\|f_{\infty}(u)\|=\max\{|\langle f_{\infty}(u),{\alpha}\rangle|;{\alpha}\in{\bf L}^{*},\|{\alpha}\|=1\}$. Set, for every ${\alpha}$, $x_{\alpha}:=({\alpha}{\otimes}{\bf 1}_{E})(u)\in E$; then, representing $u$ as a sum of elementary tensors, we obtain that $\langle f_{\infty}(u),{\alpha}\rangle=f(x_{\alpha})$. Now fix an arbitrary $\eta\in{\bf L};\|\eta\|=1$ and consider the operator $S:{\bf L}\to{\bf L}:\xi\mapsto{\alpha}(\xi)\eta$; obviously, we have $\|S\|=\|{\alpha}\|=1$. But the same representation of $u$ implies $S{\cdot}u=\eta x_{\alpha}$. Therefore we have $\|x_{\alpha}\|=\|\eta x_{\alpha}\|\leq\|S\|\|u\|=\|u\|$. It follows that $|\langle f_{\infty}(u),{\alpha}\rangle|\leq\|f\|\|x_{\alpha}\|=\|f\|\|u\|$, and $\|f_{\infty}\|=\|f\|$. $\triangleright$

As a corollary, for every ${\bf L}$–space $E$ and $u\in{\bf L}$ we have $\|u\|\geq\sup\{\|f_{\infty}(u)\|\}$, where supremum is taken over all $f\in E^{*};\|f\|\leq 1$. But such a supremum is exactly $\|u\|_{\min}$. This justifies the word “minimal” in Example 2.5.

Now we want to define amplifications of bilinear operators. For this we need a certain additional structure, connected with our base space, namely, a fixed bilinear operator, denoted in what follows by $\diamondsuit:{\bf L}\times{\bf L}\to{\bf L}$ and called $\diamondsuit$–operation. Let us write $\xi\diamondsuit\eta$ instead of $\diamondsuit(\xi,\eta)$. We call a $\diamondsuit$–operation metric, if we always have $\|\xi\diamondsuit\eta\|=\|\xi\|\|\eta\|$.

If $E$ is a linear space, $\xi\in{\bf L}$ and $u\in{\bf L}E$, we set $\xi\diamondsuit u:=T_{\xi}{\cdot}u$, where $T_{\xi}\in{\cal B}$ takes $\eta$ to $\xi\diamondsuit\eta$. Thus, this version of a $\diamondsuit$–operation is well defined on elementary tensors by the equality $\xi\diamondsuit\eta x:=(\xi\diamondsuit\eta)x$. Similarly, we introduce $u\diamondsuit\eta\in{\bf L}E$ by the equality $\xi x\diamondsuit\eta:=(\xi\diamondsuit\eta)x$. In the case of a metric $\diamondsuit$–operation we obviously have $T_{\xi}=\|\xi\|S$, where $S$ is an isometry. Therefore, if $E$ is an ${\bf L}$–space, we have

[TABLE]

Example 3.3. Suppose that our base space is $L_{p}(X)$ from Example 2.2. We shall say that two given measure spaces are of the same type, if a) they simultaneously have or do not have continuous ( = non-atomic) part, and b) they have sets of atoms of the same cardinality. As it is well known (see, e.g., [14, Cor. 9.12.18], and also [17, §14] or [18, III.A]), in the case $p\neq 2$ the spaces $L_{p}(X)$ and $L_{p}(Y)$ are isometrically isomorphic if, and only if the respective measure spaces are of the same type. From this we immediately see that in the case $p\neq 2$ the spaces $L_{p}(X)$ and $L_{p}(X\times X)$ are isometrically isomorphic if, and only if our $X$ is either devoid of atoms or has exactly one atom, or has infinite (necessarily countable) set of atoms. Thus, in these three cases the spaces $L_{p}(X\times X)$ and $L_{p}(X)$ are isometrically isomorphic. We choose an arbitrary isometric isomorphism $i:L_{p}(X\times X)\to L_{p}(X)$ and fix it throughout the whole paper. After this we introduce $\diamondsuit:L_{p}(X)\times L_{p}(X)\to L_{p}(X)$ as the composition $i\vartheta_{X}$, where the bilinear operator $\vartheta_{X}:L_{p}(X)\times L_{p}(X)\to L_{p}(X\times X)$ takes a pair $(x,y)$ to the function (of two variables) $x(s)y(t);s,t\in X$. Clearly, such a $\diamondsuit$–operation is metric.

Remark 3.4. Of course, if $p=2$, that is we deal with Hilbert spaces, a metric $\diamondsuit$–operation exists for arbitrary (non-finite) measure spaces. This case was studied in details in [13].

From now on, and up to the end of the paper we assume that our base space is endowed with a metric $\diamondsuit$–operation.

Now let ${\cal R}:E\times F\to G$ be a bilinear operator between linear spaces. Its amplification is the bilinear operator ${\cal R}_{\infty}:{\bf L}E\times{\bf L}F\to{\bf L}G$, associated with the 4-linear operator ${\bf L}\times E\times{\bf L}\times F\to{\bf L}G:(\xi,x,\eta,y)\mapsto(\xi\diamondsuit\eta){\cal R}(x,y)$. Thus, ${\cal R}_{\infty}$ is well defined on elementary tensors by ${\cal R}_{\infty}(\xi x,\eta y)=(\xi\diamondsuit\eta){\cal R}(x,y)$ .

Definition 3.5.. A bilinear operator ${\cal R}$ between ${\bf L}$–spaces is called ${\bf L}$–bounded, respectively, ${\bf L}$–contractive, if its amplification is (just) bounded, respectively, contractive. We put $\|{\cal R}\|_{{\bf L}b}:=\|{\cal R}_{\infty}\|$.

Let ${\cal R}$ be a ${\bf L}$–bounded bilinear operator. Then the equality ${\cal R}_{\infty}(\xi x,\eta y)=(\xi\diamondsuit\eta){\cal R}(x,y)$ implies that ${\cal R}$, being considered between respective underlying spaces is just bounded, and $\|{\cal R}\|\leq\|{\cal R}\|_{{\bf L}b}$. At the same time, similarly to linear operators, sometimes the “classical” boundedness automatically implies the ${\bf L}$–boundedness.

Proposition 3.6. Let $E,F$ be ${\bf L}$-spaces, $f:E\to{\mathbb{C}}$ and $g:F\to{\mathbb{C}}$ bounded functionals. Then the bilinear functional $f\times g:E\times F\to{\mathbb{C}}:(x,y)\mapsto f(x)g(y)$ is ${\bf L}$–bounded, and $\|f\times g\|_{{\bf L}b}=\|f\|\|g\|$.

$\triangleleft$ Since $\|f\times g\|=\|f\|\|g\|$, it suffices to show that $\|f\times g\|_{{\bf L}b}\leq\|f\|\|g\|$. Indeed, combining the obvious formula $(f\times g)_{\infty}(u,v)=f_{\infty}(u)\diamondsuit g_{\infty}(v)$ with Proposition 3.2, we see that $\|(f\times g)_{\infty}(u,v)\|\leq\|f\|\|g\|\|u\|\|v\|$. $\triangleright$

In the following proposition $E$ is a normed space, $F$ is an ${\bf L}$–space. Let us denote the ${\bf L}$–quantization of the space $E{\otimes}_{pr}F$, considered in Example 2.4, by the same symbol $E{\otimes}_{pr}F$; it will not lead to a misunderstanding.

Proposition 3.7. *The canonical bilinear operator $\vartheta:E_{\max}\times F\to E{\otimes}_{p}F:(x,y)\mapsto x{\otimes}y$, considered between the respective ${\bf L}$–spaces, is ${\bf L}$–contractive. *

$\triangleleft$ Consider the diagram

[TABLE]

where the bilinear operator ${\cal S}$ is well defined by taking $(x{\otimes}\xi,v)$ to $x{\otimes}(\xi\diamondsuit v)$, $\beta$ is the isometric operator from Example 2.4, and the “flip” $\beta_{0}$ is its special case (when $F:={\mathbb{C}}$). The diagram is obviously commutative, and ${\cal S}$ is contractive. As a corollary, for $w\in{\bf L}E_{\max}$ and $v\in{\bf L}F$ we have $\|\vartheta_{\infty}(w,v)\|\leq\|w\|\|u\|$. $\triangleright$

4. The general ${\bf L}$–tensor product

Let us fix, throughout this section, two arbitrary chosen ${\bf L}$–spaces $E$ and $F$. Further, let $\mho$ be a subclass of the class of all normed ${\bf L}$-spaces.

Definition 4.1 A pair $(\Theta,\theta)$, consisting of $\Theta\in\mho$ and an ${\bf L}$–contractive bilinear operator $\theta:E\times F\to\Theta$, is called tensor product of $E$ and $F$ relative to $\mho$ if, for every $G\in\mho$ and every ${\bf L}$–bounded bilinear operator ${\cal R}:E\times F\to G$, there exists a unique ${\bf L}$–bounded operator $R:\Theta\to G$ such that the diagram

[TABLE]

is commutative, and moreover $\|R\|_{{\bf L}b}=\|{\cal R}\|_{{\bf L}b}$.

Such a pair is unique in the following sense: if $(\Theta_{k},\theta_{k});k=1,2$ are two pairs, satisfying the given definition for a certain $\mho$, then there is a ${\bf L}$–isometric isomorphism $I:\Theta_{1}\to\Theta_{2}$, such that $I\theta_{1}=\theta_{2}$. This fact is a particular case of a general–categorical observation concerning the uniqueness of an initial object in a category; cf., e.g., [19], [20, Theorem 2.73]. However, the question about the existence of such a pair depends on our luck with the choice of the class $\mho$.

Definition 4.2. The tensor product of $E$ and $F$ relative to the class of all normed ${\bf L}$–spaces is called non-completed general ${\bf L}$–tensor product of our spaces.

We shall prove the existence of such a pair, displaying its explicit construction.

First, we need a sort of “extended” version of the diamond multiplication, this time between elements of amplifications of linear spaces. Namely, for $u\in{\bf L}E,v\in{\bf L}F$ we consider the element $u\diamondsuit v:=\vartheta_{\infty}(u,v)\in{\bf L}(E{\otimes}F)$, where $\vartheta:E\times F\to E{\otimes}F$ is the canonical bilinear operator. In other words, this “diamond operation” is well defined by $\xi x\diamondsuit\eta y:=(\xi\diamondsuit\eta)(x{\otimes}y).$

Proposition 4.3. Every $U\in{\bf L}(E{\otimes}F)$ can be represented as

[TABLE]

*for some natural $n$ and $a_{k}\in{\cal B},u_{k}\in{\bf L}E,v_{k}\in{\bf L}F,k=1,...,n$. *

$\triangleleft$ Since every element of ${\bf L}(E{\otimes}F)$ is the sum of several elements of the form $\xi(x{\otimes}y);\xi\in{\bf L},x\in E,y\in F$, it is sufficient to verify thr assertion on elements of the indicated form. Take an arbitrary vector $\eta\in{\bf L}$ and an arbitrary operator $a\in{\cal B}$, such that $a(\eta\diamondsuit\eta)=\xi$. then we obviously have, $\xi(x{\otimes}y)=a{\cdot}(u\diamondsuit v)$, where $u:=\eta x,v:=\eta y$. $\triangleright$

As a corollary, the operator ${\cal B}{\otimes}_{pr}{\bf L}E{\otimes}_{pr}{\bf L}F\to{\bf L}(E{\otimes}F):(a{\otimes}u{\otimes}v)\mapsto a{\cdot}(u\diamondsuit v)$ is surjective. Therefore the space ${\bf L}(E{\otimes}F)$ can be endowed with the seminorm of the respective quotient space that we denote by $\|{\cdot}\|_{{\bf L}}$. in other words, we have

[TABLE]

where the infimum is taken over all possible representations of $U$ as indicated in (4.1).

Being a quotient module of the module ${\cal B}{\otimes}_{pr}[{\bf L}E{\otimes}_{pr}{\bf L}F]$, which is certainly contractive, the *seminormed ${\cal B}$-module $({\bf L}(E{\otimes}F),\|{\cdot}\|_{{\bf L}})$ is itself contractive. * Therefore $\|{\cdot}\|_{{\bf L}}$ is an ${\bf L}$–seminorm on $E{\otimes}F$. Denote the resulting ${\bf L}$–space by $E{\otimes}_{\bf L}F$.

Observe the obvious estimate

[TABLE]

Since $u\diamondsuit v=\vartheta_{\infty}(u,v)$, we see that $\vartheta$, being considered with values in $E{\otimes}_{{\bf L}}F$, is ${\bf L}$–contractive.

Using that for $\xi\in\bf L;\|\xi\|=1$ we have $\|x{\otimes}y\|=\|(\xi\diamondsuit\xi)x{\otimes}y\|$, we obtain in the underlying space of $E{\otimes}_{\bf L}F$ the estimate

[TABLE]

Proposition 4.4. *Let $G$ be a ${\bf L}$-space, ${\cal R}:E\times F\to G$ an ${\bf L}$–bounded bilinear operator, $R:E{\otimes}_{\bf L}F\to G$ the associated linear operator. Then $R$ is ${\bf L}$–bounded, and $\|{\cal R}\|_{{\bf L}b}=\|R\|_{{\bf L}b}$. *

$\triangleleft$ Take $U\in{\bf L}(E{\otimes}_{{\bf L}}F))$ and represent it according to (4.1). Since $R_{\infty}$ is a ${\cal B}$-module morphism, we have, using the obvious equality $R_{\infty}(u\diamondsuit v)={\cal R}_{\infty}(u,v)$, that $R_{\infty}(U)=\sum_{k=1}^{n}a_{k}{\cdot}{\cal R}_{\infty}(u_{k},v_{k})$. Consequently, we have

[TABLE]

From this, using (4.2), we obtain that $\|R\|_{{\bf L}b}\leq\|{\cal R}\|_{{\bf L}b}$. The inverse inequality follows from (4.3). $\triangleright$

Proposition 4.5. (As a matter of fact),* $\|{\cdot}\|_{{\bf L}}$ is a norm. *

$\triangleleft$ By Proposition 2.4, it is sufficient to show that, for a non-zero elementary tensor $\xi w;w\in E{\otimes}_{{\bf L}}F,\xi\in{{\bf L}};\|\xi\|=1,$ we have $\|\xi w\|_{{\bf L}}\neq 0$. Since $E$ and $F$ are normed spaces, then there exist bounded functionals $f:E\to{\mathbb{C}},g:F\to{\mathbb{C}}$ such that $(f{\otimes}g)w\neq 0$. Set in Proposition 4.4 ${\cal R}:=f\times g:E\times F\to{\mathbb{C}}$. By virtue of Proposition 3.6, ${\cal R}$ is ${\bf L}$–bounded, hence the operator $(f{\otimes}g)_{\infty}:{\bf L}(E{\otimes}_{\bf L}F)\to{\bf L}$ is bounded. At the same time $(f{\otimes}g)_{\infty}(\xi w)=[(f{\otimes}g)(w)]\xi\neq 0$. $\triangleright$

Combining the accumulated facts, we immediately obtain the desired existence theorem:

Theorem 4.6. *The pair $(E{\otimes}_{\bf L}F,\vartheta)$ is a non-completed general ${\bf L}$–tensor product of $E$ and $F$. *

For some concrete tensor factors the introduced tensor product also becomes something concrete:

Theorem 4.7. Let $E$ and $F$ be the spaces from Example 2.6. Suppose that ${\bf L}:=L_{p}(X)$, where $X$ satisfies the conditions, indicated in Example 3.3, and the $\diamondsuit$-operation is taken from the same example. Then there exists an ${\bf L}$–isometric isomorphism $I:E_{\max}{\otimes}_{{\bf L}}F\to E{\otimes}_{pr}F$, acting as the identity operator on the common underlying linear space of our ${\bf L}$–spaces.

$\triangleleft$ Consider the operator $I$, associated with $\vartheta$ (cf. Proposition 3.7). It acts as it is indicated in the formulation and, by virtue of Theorem 4.6, it is ${\bf L}$–contracting. We must only show that $I_{\infty}$ does not decrease norms of vectors.

Take $U\in{\bf L}(E{\otimes}F)$. Identifying ${\bf L}(E{\otimes}F)$ with $E{\otimes}{\bf L}F$, we can represent $U$ as $\sum_{k=1}^{n}x_{k}{\otimes}v_{k};x_{k}\in E,v_{k}\in{\bf L}F$.

Let $p<\infty$ and $q$ its conjugate. Then we choose an arbitrary $e\in{\bf L};\|e\|=1$, denote by $e^{*}\in L_{q}(X)$ a function of norm 1, for which we have $\int_{X}e(s)e^{*}(s)ds=1$, and consider the operator $j:L_{p}(X\times X)\to L_{p}(X)$, taking a function $f(s,t)$ to $g(t):=\int_{X}f(s,t)e^{*}(s)ds$. If $p=\infty$, we set $e(s)\equiv 1$ and introduce $j:L_{p}(X\times X)\to L_{p}(X)$, taking $f(s,t)$ to $g(t):=ess\;sup|f^{t}|\textbf{}$, where $f^{t}(s):=f(s,t)$. Then in both cases we set $T:=ji^{-1}\in{\cal B}$ and see that $\|T\|=1$. Moreover, representing every $v_{k}$ as the sum of elementary tensors from ${\bf L}F$, we easily obtain that

[TABLE]

From this, by virtue of (4.2), we obtain the estimate $\|U\|_{{\bf L}b}\leq\sum_{k=1}^{n}\|x_{k}\|\|v_{k}\|$ and, as a corollary, we have

[TABLE]

where the infimum is taken over all representations of $U$ in the indicated form.

Now look at $I_{\infty}(U)$. It is the same $\sum_{k=1}^{n}x_{k}{\otimes}v_{k}$, only considered in the normed space $E{\otimes}_{pr}{\bf L}F$. It follows that $\|I_{\infty}(U)\|$ is exactly the infimum, indicated in (4.5). Thus, the desired estimate $\|I_{\infty}(U)\|\geq\|U\|_{{\bf L}}$ is obtained. $\triangleright$

Remark 4.8. As an easy corollary of this theorem, we have, up to an ${\bf L}$–isometric isomorphism, that $E_{\max}{\otimes}_{{\bf L}}F_{\max}=[E{\otimes}_{pr}F]_{\max}$. In particular, for a Hilbert space $H$ we have $H_{\max}{\otimes}_{{\bf L}}H_{\max}={\cal N}_{0}(H)_{\max}$, where ${\cal N}_{0}(H)$ is the space of finite rank operators on $H$, equipped with the trace class norm.

5. $p$–convexity and $p$–convex tensor product

Now we need one more, apart from $\diamondsuit$–operation, additional structure in our base space. Namely, we say that ${\bf L}$ is a stratified space, if a certain family ${\cal P}$ of projections of norm 1 (or 0), acting on ${\bf L}$, is distinguished, and it is such that $P,Q\in{\cal P}$ implies $PQ=QP\in{\cal P}$, and if $P,Q\in{\cal P}$ are orthogonal, then $P+Q\in{\cal P}$. Projections that belong to ${\cal P}$ will be called proper.

Example 5.1. Let ${\bf L}:=L_{p}(X)$. If $X^{\prime}$ is a measurable subset in $X$, we denote by $P_{X^{\prime}}\in{\cal B}(L_{p}(X))$ the projection, acting as $f\mapsto f\chi$, where $\chi$ is the characteristic function of $X^{\prime}$. Of course, if the measure of $X^{\prime}$ is positive, we have $\|P_{X^{\prime}}\|=1$. We shall identify the image of this projection with $L_{p}(X^{\prime})$. Clearly, projections of this sort are orthogonal if, and only if the respective sets have intersection of measure 0. It is obvious that the family of projections of the indicated form satisfies the conditions, formulated above. Speaking about $L_{p}(X)$ as of a stratified space, we shall always mean this particular family.

In what follows, if numbers ${\lambda}_{k}\geq 0;k=1,...,n$ are given, we shall understand the expression $(\sum_{k=1}^{n}{\lambda}_{k}^{p})^{\frac{1}{p}}$ as $\max\{{\lambda}_{1},...,{\lambda}_{n}\}$ in the case $p=\infty$.

Let us distinguish, for convenience of future references, the following obvious observation.

Proposition 5.2. Let $X$ and $Y$ be two measure spaces, $S_{k}\in{\cal B}(L_{p}(X),L_{p}(Y));\\ k=1,\dots,n$, $X_{k}^{\prime}$ $Y_{k}^{\prime}$ – two families of pairwise non-intersect measurable subsets in $X$ $Y$, respectively. Then we have $\|\sum_{k=1}^{n}P_{Y_{k}^{\prime}}S_{k}P_{X_{k}^{\prime}}\|\leq\max\{\|S_{k}\|;k=1,\dots,n\}$. $\triangleleft\quad\triangleright$

Let $u$ be an element of a certain seminormed ${\bf L}$–space. We call a projection $P\in{\cal B}$ a *support * of $u$, if $P{\cdot}u=u$.

Definition 5.3. Let ${\bf L}$ be a stratified space, $1\leq p\leq\infty$. hen a seminormed ${\bf L}$–space $E$ is called $p$–convex, if for every $u,v\in{\bf L}E$ with orthogonal proper supports, we have $\|u+v\|\leq(\|u\|^{p}+\|v\|^{p})^{\frac{1}{p}}$. As an immediate corollary, for $u_{1},...,u_{n}\in{\bf L}E$ with pairwise orthogonal supports from ${\cal P}$, we have $\|\sum_{k=1}^{n}u_{k}\|\leq(\|\sum_{k=1}^{n}\|u_{k}\|^{p})^{\frac{1}{p}}$.

For the special case ${\bf L}:=\ell_{p}$ this definition is equivalent to the definion of a $p$–convex $p$–multi-normed space in [11]. Also it worthy to mention, in this connection, the theory of $p$–operator spaces of Daws [21]; see also earlier papers of Pisier [22] and Le Merdy [23].

As to the base space ${\bf L}$ itself, it is called $p$–convex, if it is $p$–convex after its identification with the ${\bf L}$–space ${\mathbb{C}}$. Needless to say, $L_{p}(X)$, as a base space, is $p$–convex.

Clearly, for every stratified ${\bf L}$ all seminormed ${\bf L}$–spaces are 1-convex. Also it is obvious that every $p$–convex space is $r$–convex for each $r;1\leq r<p$.

Proposition 5.4. If ${\bf L}$ is $p$–convex, in particular, if ${\bf L}:=L_{p}(X)$, then every ${\bf L}$–space $E$ with the minimal quantization is -convex.

$\triangleleft$ If $u,v\in{\bf L}E$ have orthogonal supports, then for every $f\in E^{*}$ elements $f_{\infty}(u),f_{\infty}(v)$ have orthogonal supports in ${\bf L}$. Therefore we have $\|f_{\infty}(u+v)\|^{p}\leq\|f_{\infty}(u)\|^{p}+\|f_{\infty}(v)\|^{p}$. It remains to take the relevant supremum in the right part of the inequality over all $f\in E^{*};\|f\|=1$, and then do the same with the left part. $\triangleright$

At the same time in the case $p>1$ and ${\bf L}:=L_{p}(X)$ the maximal quantization of a normed space is not, generally speaking, $p$–convex. The space $E:=\ell_{1}$ serves as the simplest counter-example.

One can construct $p$–convex spaces, embarking from other $p$–convex spaces. For example, it is not difficult to show that, for $q_{k}$–convex ${\bf L}$–spaces $E_{k};k=1,2$ and $p\leq\min\{q_{1},q_{2}\}$, the ${\bf L}$–space $E_{1}\oplus E_{2}$, being considered with the ${\bf L}$–norm of the $\ell_{p}$–sum of normed spaces ${\bf L}E_{1}$ and ${\bf L}E_{2}$, is $p$–convex.

Now suppose that we are given a space ${\bf L}$, which is stratified and has a $\diamondsuit$–operation. Fix two ${\bf L}$–spaces $E$ and $F$.

Definition 5.5. The tensor product of $E$ and $F$ relative to the class of all $p$–convex ${\bf L}$–spaces is called non-completed $p$–convex ${\bf L}$–tensor product of our spaces.

Unfortunately, at the moment we can prove the existence of such a tensor product only under rather burdensome additional assumptions on $\diamondsuit$ and especially on ${\cal P}$. In fact, all our examples of triples $({\bf L},{\cal P},\diamondsuit)$, satisfying these conditions, are, in a sense, too close to triples, arising in the case of base spaces $L_{p}(X)$, moreover with (however mild) assumptions on $X$. Therefore we decided not to bother the reader with the list of these conditions. Instead,

from now on up to the end of the paper we consider, in capacity of base spaces, only spaces $L_{p}(X)$, where $X$ is a measure space that either has no atoms or has infinite set of atoms.

We call a measure space of the indicated sort convenient. The family of proper projections, acting on $L_{p}(X)$, is defined according to Example 5.1, and a $\diamondsuit$–operation according to Example 3.3. Note the the case of a single atom, permitted in the latter example, now is forbidden; otherwise (here we open our cards) the future Proposition 5.6 fails to be true.

We call an isometric operator on $L_{p}(X)$ proper, if its image coincides with the image of some proper projection or, equvalently, its image is $L_{p}(X^{\prime})$ for some measurable subset $X^{\prime}$ of $X$. We call two proper isometries disjoint, if the intersection of their images is $\{0\}$ or, equvalently, the corresponding projections are orthogonal. Since $X$ is convenient, it contains an infinite family of pairwise disjoint measurable subsets of the same type as $X$ (cf. Example 3.3). Here is an immediate corollary.

**Proposition 5.6. ** There exists an infinite family of pairwise disjoint proper isometries, acting on $L_{p}(X)$. $\triangleleft\quad\triangleright$

(By the way, in the case of general ${\bf L}$ it would be sufficient to find two disjoint proper isometries, say $I_{1}$ and $I_{2}$. Then the isometries $I_{1},I_{1}I_{2},I_{1}I_{2}^{2},I_{1}I_{2}^{3},...$ would fit).

If $I\in{\cal B}$ is a proper isometry with image $L_{p}(X^{\prime})$, we denote by $I^{\star}\in{\cal B}$ the operator $I^{-1}P_{X^{\prime}}$, that is the coisometric operator ( = quotient map), acting as $I^{-1}$ on $L_{p}(X^{\prime})$ and vanishing on the complementary subspace $L_{p}(X\setminus X^{\prime})$. It is clear that, if $I_{k};k=1,...,n$ are pairwise disjoint proper isometries, we have

[TABLE]

We proceed to the explicit construction of the $p$-convex tensor product.

Proposition 5.7. If $E,F$ are linear spaces, then every $U\in L(E{\otimes}F)$ can be represented as

[TABLE]

*where $a\in{\cal B}$, and $I_{k}$ are pairwise disjoint proper isometries. *

$\triangleleft$ Represent $U$ as in (4.1). By virtue of Proposition 5.6, there exist $n$ proper pairwise disjoint isometries $I_{k}$. Consider in ${\bf L}(E{\otimes}F)$ the element

[TABLE]

It follows from (5.1) that it is exactly $U$. $\triangleright$

From now on we assume that we are given two arbitrary (not necessarily $p$–convex!) ${\bf L}$–spaces $E$ and $F$. Assign to every $U\in L(E{\otimes}F)$ the number

[TABLE]

where the infimum is taken over all possible representations of $U$ in the form (5.2).

We distinguish the obvious

Proposition 5.8. *For every $U\in{\bf L}(E{\otimes}F)$ and $a\in{\cal B}$ we have $\|a{\cdot}U\|_{p{\bf L}}\leq\|a\|\|U\|_{p{\bf L}}$. * $\triangleleft\quad\triangleright$

What is less obvious, it is

Proposition 5.9. *The function $U\mapsto\|U\|_{{p{\bf L}}}$ is a seminorm on ${{\bf L}}(E{\otimes}F)$. *

$\triangleleft$ Suppose that $U=a{\cdot}\sum_{k=1}^{n}I^{\prime}_{k}{\cdot}(u^{1}_{k}\diamondsuit v^{1}_{k})$ and $V=b{\cdot}\sum_{l=1}^{m}I^{\prime\prime}_{l}{\cdot}(u^{2}_{l}\diamondsuit v^{2}_{l})$, where $I^{\prime}_{k}$ are proper pairwise disjoint isometries, and the same is true for $I^{\prime\prime}_{l}$.

Choose, in an arbitrary way, one more pair of disjoint isometries $I_{U},I_{V}\in{\cal B}$ and observe that

[TABLE]

Evidently, the compositions $I_{U}I^{\prime}_{k}$ and $I_{V}I^{\prime\prime}_{l}$ are proper isometries and, being considered all together, they are pairwise disjoint. Therefore, by virtue of (5.3) and the previous proposition, we have

[TABLE]

Our nearest aim is to obtain the estimate $\|aI_{U}^{\star}+bI_{V}^{\star}\|\leq(\|a\|^{q}+\|b\|^{q})^{\frac{1}{q}}$, where $q$ is the conjugate number to $p$. (As usual, we hold that 1 and $\infty$ are mutually conjugate).

Take $\xi\in{\bf L},\|\xi\|\leq 1$ and denote the proper projections, corresponding to our isometries, by $P_{U}=I_{U}I_{U}^{\star}$ and $P_{V}=I_{V}I_{V}^{\star}$. Then, taking into account the Gölder inequality, we obtain that

[TABLE]

[TABLE]

Since our projections are orthogonal, the scond factor is equal to $((\|P_{U}(\xi)+P_{V}(\xi)\|)^{p})^{\frac{1}{p}}$, and therefore it does not exceed $\|P_{U}+P_{V}\|=1$. Our desired estimate follows.

Obviously, we can obtain representations of our $U$, by multiplying $a$ on a certain constant and dividing all $u_{k}^{1}$ on the same constant; in a similar way we can deal with $V$. Consequently, we have a right to assume in the case $1<p<\infty$ that $\|a\|^{q}=\sum_{k=1}^{n}\|u^{1}_{k}\|^{p}\|v^{1}_{k}\|^{p}$ and $\|b\|^{q}=\sum_{l=1}^{m}\|u^{2}_{l}\|^{p}\|v^{2}_{l}\|^{p}$. Also we can assume in the case $p=1$ that $\|a\|=\|b\|=1$, and in the case $p=\infty$ that $\max\{\|u^{1}_{k}\|\|v^{1}_{k}\|;k=1,...,n\}=\max\{\|u^{2}_{l}\|\|v^{2}_{l}\|;l=1,...,m\}$.

Consequently, in the case $1<p<\infty$ we have

[TABLE]

But $q-1=q/p$, hence $\|a\|^{q}=\|a\|(\sum_{k=1}^{n}\|u^{1}_{k}\|^{p}\|v^{1}_{k}\|^{p})^{\frac{1}{p}}$. Together with the similar equality for $\|b\|^{q}$, this provides the estimate

[TABLE]

It is easy to verify that the same estimate is valid in the remaining cases.

Thus, in all cases, taking the infimum from (5.3), we obtain the triangle inequality: $\|U+V\|_{p{\bf L}}\leq\|U\|_{p{\bf L}}+\|V\|_{p{\bf L}}$.

The property of seminorms, concerning the scalar multiplication, is immediate. $\triangleright$

We see that $\|{\cdot}\|_{\bf L}$ is an ${\bf L}$–seminorm on $E{\otimes}F$. Denote the resulting ${\bf L}$–seminormed space by $E{\otimes}_{p{{\bf L}}}F$.

Proposition 5.10. *The introduced space is $p$–convex. *

$\triangleleft$ Let $U,V\in{\bf L}(E{\otimes}F)$ have orthogonal supports $P_{1}$ and $P_{2}$. Choose their arbitrary representations in the same form, as in Proposition 5.9, and take corresponding $I_{U}$ and $I_{V}$. The estimate (5.4) appears. Obviously, we have a right to assume that $a=P_{1}a,b=P_{2}b$, and also that $\|a\|=\|b\|=1$. Thus, in the notations $P_{U}:=I_{U}I_{U}^{\star}$ and $P_{V}:=I_{V}I_{V}^{\star}$ we have $aI^{\prime\star}=P_{1}aI^{\prime\star}P_{U}$ $bI^{\prime\prime\star}=P_{2}bI^{\prime\prime\star}P_{V}$. From this, by virtue of Proposition 5.2, we have $\|aI^{\prime\star}+bI^{\prime\prime\star}\|\leq\max\{\|aI^{\prime\star}\|,\|bI^{\prime\prime\star}\|\}\leq\max\{\|a\|,\|b\|\}=1$, hence $\|U+V\|_{p{\bf L}}\leq\left(\sum_{k=1}^{n}\|u^{1}_{k}\|^{p}\|v^{1}_{k}\|^{p}+\sum_{l=1}^{m}\|u^{2}_{l}\|^{p}\|v^{2}_{l}\|^{p}\right)^{\frac{1}{p}}$. It remains to take the corresponding infima. $\triangleright$

Similarly to the case of the tensor product “ ${\otimes}_{\bf L}$ ”, we have the estimate

[TABLE]

It follows that the canonical bilinear operator $\vartheta:E\times F\to E{\otimes}_{p{{\bf L}}}F$ (just as $\vartheta:E\times F\to E{\otimes}_{{\bf L}}F$ in Section 4) is ${{\bf L}}$–contractive.

The same argument that provides the estimate (4.4), shows that in the underlying seminormed space of $E{\otimes}_{p{\bf L}}F$ we have the estimate

[TABLE]

Proposition 5.11. *Let $G$ be a $p$–convex ${\bf L}$–space, ${\cal R}:E\times F\to G$ an ${\bf L}$–bounded bilinear operator. Then the associated linear operator $R:E{\otimes}_{p{\bf L}}F\to G$ is ${\bf L}$–bounded, and $\|{\cal R}\|_{{\bf L}b}\|=\|R\|_{{\bf L}b}$. *

Take $U\in{\bf L}(E{\otimes}_{l}F))$ and its representation as in (5.2). Since $R_{\infty}$ is a ${\cal B}$–module morphism, we see that $R_{\infty}(U)=a{\cdot}(\sum_{k=1}^{n}I_{k}{\cdot}{\cal R}_{\infty}(u_{k},v_{k}))$. Look at the elements $I_{k}{\cdot}{\cal R}_{\infty}(u_{k},v_{k})\in{\bf L}G$. They have pairwise orthogonal supports, namely $I_{k}I_{k}^{\star}$, and $G$ is $p$–convex. From this we obtain that

[TABLE]

[TABLE]

Therefore $\|R\|_{{b{\bf L}}}\leq\|{\cal R}\|_{{b{\bf L}}}$. The inverse inequality follows from (5.6). $\triangleright$

Proposition 5.12. (As a matter of fact),* $\|{\cdot}\|_{p{\bf L}}$ is a norm. *

$\triangleleft$ Needless to say that ${\mathbb{C}}$ is a $p$–convex ${\bf L}$–space, hence we have a right to use Proposition 5.11. Therefore the proof of Proposition 4.5 goes up to obvious modifications. $\triangleright$

Combining the relevant propositions, we obtain the following existence theorem:

Theorem 5.13. *The pair $(E{\otimes}_{p{\bf L}}F,\vartheta)$ is a non-completed $p$–convex tensor product of ${\bf L}$–spaces $E$ and $F$. *

Such a theorem, in our opinion, can be considered as a far-reaching generalization of certain results of Lambert concerning the maximal tensor product of his “Operatorfolgenräume”; see [1, pp. 73-78]. In this connection we recall the papers of Blecher/Paulsen [24] and Effros/Ruan [25] about projective tensor products of operator spaces: they are at the source of all these constructions.

Remark 5.14. We do not discuss here the “non-discrete” version of another, the so-called minimal tensor product, that was introduced by Lambert (in the frame-work of the “coordinate” approach) for 2–convex $l_{2}$–spaces in [1, 3.1.3].

6. $p$–convex tensor product of spaces $L_{q}({\cdot})$

In conclusion, we shall present an example, when the $p$–convex tensor product acquires especially transparent form. It happens that in the case $1<p<\infty$ one should take, in the role of “the best” tensor factors, the spaces $L_{q}({\cdot});q:=p/(p-1)$ with the minimal quantization, discussed in Example 2.5.

We remember that our base space ${\bf L}$ is $L_{p}(X)$ for convenient $X$. Moreover, throughout this section, all ${\bf L}$–spaces are supposed to be endowed with the minimal quantization.

Let $Y$ and $Z$ be two measure spaces. Consider the linear operator

$J:L_{p}(Y){\otimes}L_{p}(Z)\to L_{p}(Y\times Z)$, well defined by the rule $x{\otimes}y\mapsto x(s)y(t);\\ s\in Y,t\in Z$. It is easy to see that it is injective, and its image consists of functions of the form $\sum_{k=1}^{n}f_{k}(s)g_{k}(t);f_{k}\in L_{p}(Y),g_{k}\in L_{p}(Z)$. We see that this image is a normed subspace in $L_{p}(Y\times Z)$, which is dense provided $p<\infty$. Denote it by $L_{p}(Y){\otimes}^{p}L_{p}(Z)$. Obviously, we can identify this space with the tensor product $L_{p}(Y){\otimes}L_{p}(Z)$, endowed with the respective induced norm.

Proposition 6.1. Let $A:L_{p}(Y_{1})\to L_{p}(Y_{2})$ and $B:L_{p}(Z_{1})\to L_{p}(Z_{2})$ be two bounded operators. Then the operator $A{\otimes}B:L_{p}(Y_{1}){\otimes}^{p}L_{p}(Z_{1})\to L_{p}(Y_{2}){\otimes}^{p}L_{p}(Z_{2})$ is also bounded, and $\|A{\otimes}B\|\leq\|A\|\|B\|$.

$\triangleleft$ Every $u\in L_{p}(Y_{1}){\otimes}^{p}L_{p}(Z_{1})$ is a function of the form $\sum_{k=1}^{n}f_{k}(s)g_{k}(t):\\ f_{k}\in L_{p}(Y_{1}),g_{k}\in L_{p}(Z_{1})$. If $p<\infty$, then, by virtue of Fubini Theorem we have

[TABLE]

[TABLE]

[TABLE]

[TABLE]

If $p=\infty$, then a similar argument works, only instead of Fubini Theorem we apply, for functions $h\in L_{\infty}(Y,Z)$, $h_{s}\in L_{\infty}(Z):h_{s}(t):=h(s,t)$ and $h^{t}\in L_{\infty}(Y):h^{t}(s):=h(s,t)$, the equality $ess\;sup|h|=ess\;sup[ess\;sup|h_{s}|]=ess\;sup[ess\;sup|h^{t}|]$. $\triangleright$

We recall that the norm in the injective tensor product $E{\otimes}_{in}F$ of two normed spaces can be expressed in terms of an isometric operator from $E{\otimes}_{in}F$ into ${\cal B}(E^{\prime},F)$, where $E^{\prime}$ is an arbitrary subspace in $E^{*}$ such that for every $x\in E$ we have $\|x\|=\sup\{|f(x);f\in E^{\prime},\|f\|=1\}$. (For example, $E^{\prime}$ can be a dense subspace in $E^{*}$ or a predual to $E$, if such a predual exists). See, e.g., [26, pp. 62-63]) or [27, §4], [28, pp. 45-46] (and also, of course, [29]). In the particular case of spaces $L_{p}({\cdot})$, the relevant assertion acquires the following form. Denote by $\langle f,g\rangle$ the classical duality $(f,g)\mapsto\int_{Y}f(t)g(t)dt$ between the spaces $L_{p}(Y)$ and $L_{q}(Y)$, and denote by $\langle\langle u,v\rangle\rangle$ the duality between the spaces $L_{p}(Y){\otimes}^{p}L_{q}(Z)$ and $L_{q}(Y){\otimes}^{q}L_{q}(Z)$, well defined on elementary tensor with the help of the equality $\langle\langle y_{1}{\otimes}z_{1},y_{2}{\otimes}z_{2}\rangle\rangle=\langle y_{1},y_{2}\rangle\langle z_{1},z_{2}\rangle$. In the following proposition $Y,Z,Y_{1},...$ are arbitrary measure spaces.

Proposition 6.2. (i) There exists an isometric operator ${\cal I}:L_{p}(Y){\otimes}_{in}L_{q}(Z)\to{\cal B}(L_{p}(Z),L_{p}(Y))$, (in particular, ${\cal I}:{\bf L}L_{q}(Z)\to{\cal B}(L_{p}(Z),{\bf L})$), well defined by taking $y{\otimes}z$ to the operator, acting by the rule $z^{\prime}\mapsto\langle z^{\prime},z\rangle y$.

(ii) If, in addition, $1<p<\infty$, (or, equivalently, $1<q<\infty$), then there exists an isometric operator ${\cal J}:[L_{p}(Y_{1}){\otimes}^{p}L_{p}(Z_{1})]{\otimes}_{in}[L_{q}(Y_{2}){\otimes}^{q}L_{q}(Z_{2})]\to{\cal B}(L_{p}(Y_{2}){\otimes}^{p}L_{p}(Z_{2}),L_{p}(Y_{1}){\otimes}^{p}L_{p}(Z_{1})$, well defined by taking $u{\otimes}v$ to the operator, acting by the rule $v^{\prime}\mapsto\langle\langle v^{\prime},v\rangle\rangle u$.

$\triangleleft$ (i). It is valid because $L_{p}(Z)$ is the dual (or the predual provided $p=1$) to $L_{q}(Z)$.

(ii). It is valid because in the case $1<p<\infty$ the dual space of $(L_{q}(Y_{2}){\otimes}^{q}L_{q}(Z_{2}))$ coincides with $L_{p}(Y_{2}\times Z_{2})$, and the latter space contains $L_{p}(Y_{2}){\otimes}^{p}L_{p}(Z_{2})$ as a dense subspace. $\triangleright$

Proposition 6.3. Let $Y$ and $Z$ be measure spaces, $1<p<\infty$. Then the bilinear operator ${\cal R}:L_{q}(Y)\times L_{q}(Z)\to L_{q}(Y){\otimes}^{q}L_{q}(Z)\subset L_{q}(Y\times Z)$, taking a pair $(y,z)$ to $y{\otimes}z=y(s)z(t);s\in Y,t\in Z$, is ${\bf L}$–contractive.

$\triangleleft$ Consider the bilinear operator ${\cal S}:{\bf L}L_{q}(Y)\times{\bf L}L_{q}(Z)=(L_{p}(X){\otimes}_{in}L_{q}(Y))\times(L_{p}(X){\otimes}_{in}L_{q}(Z))\to[L_{p}(X){\otimes}^{p}L_{p}(X)]{\otimes}_{in}[L_{q}(Y){\otimes}^{q}L_{q}(Z)]$, well defined on elementary tensors by the rule $(\xi{\otimes}y,\eta{\otimes}z)\mapsto(\xi{\otimes}\eta){\otimes}(y{\otimes}z)$. Take $u\in L_{p}(X){\otimes}L_{q}(Y),\\ v\in L_{p}(X){\otimes}L_{q}(Z)$. It is clear that ${\cal R}_{\infty}=(i_{0}{\otimes}{\bf 1}){\cal S}$, where $i_{0}:L_{p}(X){\otimes}^{p}L_{p}(X)\to L_{p}(X)$ is the respective restriction of the isometric isomorphism $i$, introduced in Example 3.3, and ${\bf 1}$ is the identity operator on $L_{q}(Y){\otimes}^{q}L_{q}(Z)$. Since we deal with the injective tensor product, $i_{0}{\otimes}{\bf 1}$ is an isometry together with $i_{0}$ and ${\bf 1}$ (“the injtctive property”, see, e.g., [27, §4]). Therefore it is sufficient to show that ${\cal S}$ is contractive.

Consider the diagram

[TABLE]

where ${\cal T}$ is the bilinear operator, taking a pair $(A,B)$ to $A{\otimes}B$, and ${\cal I}_{k};k=1,2$ and ${\cal J}$ are the respective special cases of isometric operators ${\cal I}$ and ${\cal J}$ from Proposition 6.2. As it is easy to verify, this diagram is commutative. But by virtue of Proposition 6.1 ${\cal T}$ is contractive. Hence, ${\cal S}$ has the same property. $\triangleright$

Theorem 6.4. Let $Y$ and $Z$ be measure spaces, and $1<p<\infty$. Then we have $L_{q}(Y){\otimes}_{p{\bf L}}L_{q}(Z)=L_{q}(Y){\otimes}^{q}L_{q}(Z)$. (Recall that the right side of the equality denotes the dense subspace in $L_{q}(Y\times Z)$, consisting of degenerate funnctions). The equality means that there exists the ${\bf L}$–isometric isomorphism, well defined by the rule $x{\otimes}y\mapsto x(s)y(t)$ (in other words, it takes $x{\otimes}y$ to the same elementary tensor, only considered in $L_{q}(Y){\otimes}^{q}L_{q}(Z)$).

$\triangleleft$ Since all participating spaces are $p$–convex (Proposition 5.4), the Existence Theorem 5.13, together with Proposition 6.3, provides the ${\bf L}$–contractive operator $R:L_{q}(Y){\otimes}_{p{\bf L}}L_{q}(Z)\to L_{q}(Y){\otimes}^{q}L_{q}(Z)$, acting as it is indicated in the formulation. We must only show that its amplification does not decrease norms.

Of course, every $L_{q}({\cdot});q<\infty$ contains the dense subspace $L_{q}^{0}({\cdot})$, consisting of linear combinations of characteristic functions of subsets with finite measure, and we have right to assume that these subset are pairwise disjoint. Therefore, because of the estimate (5.7), it is sufficient to show that $R_{\infty}$ does not decrease norms of (finite) sums of elementary tensors of the form $\xi(y{\otimes}z)$, $\xi\in L_{p}(X),y\in L_{q}^{0}(Y),z\in L_{q}^{0}(Z)$.

It is easy to see that every mentioned sum can be represented as $U:=\sum_{k,l}\xi_{k,l}y_{k}{\otimes}z_{l}$, where $y_{k}$ and $z_{l}$ are functions of norm 1, proportional to the characteristic functions of pairwise non-intersecting subsets $Y_{k}\subseteq Y$ and $Z_{l}\subseteq Z$, respectively.

Look at $R_{\infty}(U)$. It is, of course, a certain sum $\sum_{k,l}\xi_{k,l}e_{k,l}$, where $e_{k,l}\in L_{q}(Y\times Z)$ are functions of norm 1, proportional to characteristic functions of subsets $Y_{k}\times Z_{l}\subseteq Y\times Z$; in other words, $e_{k,l}=y_{k}{\otimes}z_{l}\in L_{q}(Y){\otimes}^{q}L_{q}(Z)\subset L_{q}(Y\times Z)$. Therefore, by virtue of Proposition 6.2(i), $\|R_{\infty}(U)\|$ is the norm of the operator $S:L_{p}(Y\times Z)\to L_{p}(X)$, taking $h(s,t)$ to $\sum_{k,l}\langle h,e_{k,l}\rangle\xi_{k,l}$.

Now return to our initial $U$. Obviously, there exist pairwise non-intersecting subsets $X_{k}^{1};k=1,...,n$, respectively, $X_{l}^{2};l=1,...,m$ of finite measure.

Denote by $\eta_{k}$ the functions of norm 1 in ${\bf L}:=L_{p}(X)$, proportional to the characteristic functions of subsets $X_{k}^{1}$, and set $u:=\sum_{k=1}^{n}\eta_{k}y_{k}\in{\bf L}L_{q}(Y)$. By virtue of the same Proposition 6.2, $\|u\|$ is the norm of the operator $S_{u}:L_{p}(Y)\to L_{p}(X)$, taking $g$ to $\sum_{k=1}^{n}\langle g,y_{k}\rangle\eta_{k}$. Since we obviously have $S_{u}=\sum_{k=1}^{n}P_{X_{k}^{1}}S_{u,k}P_{Y_{k}}$, where $S_{u,k}:g\mapsto\langle g,y_{k}\rangle\eta_{k}$, we obtain that $\|S_{u}\|=\max\{\|S_{u,k}\|;k=1,...,n\}$ (Proposition 5.2). But we have $\|S_{u,k}(g)\|\leq\|g\|\|y_{k}\|\|\eta_{k}\|$; therefore $\|S_{u,k}\|\leq 1$, hence $\|S_{u}\|\leq 1$. Thus, we have $\|u\|\leq 1$. Using the same argument, we set $v:=\sum_{l=1}^{m}\zeta_{l}z_{l}\in{\bf L}L_{q}(Z)$ for similarly chosen $\zeta_{l}$ and we see that $\|v\|\leq 1$. (As a matter of fact, we have $\|u\|=\|v\|=1$, but we do not need it now).

It is clear that $u\diamondsuit v=\sum_{k,l}(\eta_{k}\diamondsuit\zeta_{l})y_{k}{\otimes}z_{l}$. Therefore it follows from the definition of the ${\bf L}$–norm on $L_{q}(Y){\otimes}_{p{\bf L}}L_{q}(Z)$ (see (5.3)) that the theorem will be proved, if we shall find an operator $T\in{\cal B}:={\cal B}(L_{p}(X))$ such that $T(\eta_{k}\diamondsuit\zeta_{l})=\xi_{k,l}$ (hence, $U=T{\cdot}(u\diamondsuit v)$) and such that $\|T\|\leq\|S\|$.

Choose a contractive operator $i_{k,l}:L_{p}(X_{k}^{1}\times X_{l}^{2})\to L_{p}(Y_{k}\times Z_{l})$, taking the constant function $\eta_{k}(s)\zeta_{l}(t);s\in X_{k}^{1},t\in X_{l}^{2}$ to a certain constant function $e^{*}_{k,t}$ of the same norm 1. (For example, one can choose $i_{k,l}:=gf$, where $f$ is a functional of norm 1, taking the first of mentioned constants to $\equiv 1$, and $g$ is the operator, taking $\equiv 1$ to $e^{*}_{k,t}$). After this, identifying each $L_{p}(Y_{k}\times Z_{l})$ and $L_{p}(X_{k}^{1}\times X_{l}^{2})$ with the respective subspace in $L_{p}(Y\times Z)$ or, according to the sense, in $L_{p}(X\times X)$, we introduce the operator $D:=S(\sum_{k,l}Q_{k,l}i_{k,l}P_{k,l}):L_{p}(X\times X)\to L_{p}(X)$, where $P_{k,l}$ is the natural projection of $L_{p}(X\times X)$ on $L_{p}(X_{k}^{1}\times X_{l}^{2})$, and $Q_{k,l}$ is the natural projection of $L_{p}(Y\times Z)$ on $L_{p}(Y_{k}\times Z_{l})$. We see that the operators $\sum_{k,l}Q_{k,l}i_{k,l}P_{k,l}$ and $i_{k,l}$ take $\eta_{k}(s)\zeta_{l}(t)$ to the same constant $e^{*}_{k,t}$, and the obvious equality $\langle e^{*}_{k,l},e_{k,l}\rangle=1$ implies that $S(e^{*}_{k,l})=\xi_{k,l}$. Therefore $D$ takes $\eta_{k}(s)\zeta_{l}(t)$ to $\xi_{k,l}$.

Further, by virtue of Proposition 5.2 we obtain that $\|\sum Q_{k,l}i_{k,l}P_{k,l}\|\leq\max_{k,l}\{\|i_{k,l}P_{k,l}\|\}=1$. It follows that $\|D\|\leq\|S\|$.

Finally, let us recall the isometric isomorphism $i:L_{p}(X\times X)\to L_{p}(X)$ (see Example 3.3). This map, in particular, takes every function of the form $\eta(s)\zeta(t)$ (identified with $\eta{\otimes}\zeta\in L_{p}(X){\otimes}^{p}L_{p}(X)$) to $\eta\diamondsuit\zeta$. From this we see that the operator $T:=Di^{-1}:L_{p}(X)\to L_{p}(X)$ is exactly what we need. $\triangleright$

Remark 6.5. Throughout all paper, we did not assume that our normed spaces are complete. However, principal notions and facts have, as a rule, “complete” versions. ${\bf L}$–space is called complete (or Banach), if its underlying normed space is complete. As in the “classical” context, every ${\bf L}$–space has a completion; its definition and properties, as well as the existence theorem repeat with obvious modifications what was said in [7, Ch. 4] for the case of operator spaces. (See also similar argument in [13] for the case, where ${\bf L}$ is a Hilbert space). Moreover, both tensor products, introduced above, have their “Banach” versions; one must only to consider in Definition 4.2, in the capacity of $\mho$, the class of complete ${\bf L}$-spaces, whereas in Definition 5.5 one must consider the class of complete $p$–convex ${\bf L}$–spaces. Then, in particular, we obtain the following version of Theorem 6.5: if ${\bf L}:=L_{p}(X)$ and $1<p<\infty$, then completed $p$–convex tensor product of ${\bf L}$-spaces $L_{q}(Y)$ and $L_{q}(Z)$ is $L_{q}(Y\times Z)$.