# Free and projective generalized multinormed spaces

A. Ya. Helemskii  Faculty of Mechanics and Mathematics, Moscow State
(Lomonosov) University, Moscow, Leninskie Gory, Russia 119991
<EMAIL_ADDRESS>and  T. Oikhberg  Dept. of Mathematics, University of
Illinois, Urbana IL 61801, USA<EMAIL_ADDRESS>
###### Abstract.

The paper investigates free and projective ${\bf L}$-spaces, where ${\bf L}$
is a given normed space. These spaces form a far-reaching generalization of
known $p$-multinormed spaces; in particular, if ${\bf L}=L_{p}(X)$, the ${\bf
L}$-spaces can be considered as $p$-multinormed spaces, based on arbitrary
$\sigma$-finite measure spaces $X$ (for “canonical” $p$-multinormed spaces,
$X=\mathbb{N}$ with the counting measure). We first describe a “naturally
appearing” functor, based on paving ${\bf L}$ with contractively complemented
finite dimensional subspaces. This finite dimensionality is essential; it
permits us to describe a free ${\bf L}$-space for this functor. As a
corollary, we obtain a wide variety of projective ${\bf L}$-spaces. For “nice”
${\bf L}$ (such as the space of simple $p$-integrable functions on a measure
space), we obtain a full description of projective ${\bf L}$-spaces.

###### Key words and phrases:

${\bf L}$–space, ${\bf L}$–contractive operator, projective ${\bf L}$–space,
$\odot$-free ${\bf L}$–space, well composed ${\bf L}$–space.

###### 2020 Mathematics Subject Classification:

46L07, 46M05

The first author was supported by the Russian Foundation for Basic Research
(Grant No. 19-01-00447). The second author was supported by the NSF award
1912897.

## Introduction

In recent years a new substantial concept appeared in functional analysis.
This is the notion of $p$-multinormed space [2], a natural extension of
multinormed and dual multinormed spaces of Dales and Polyakov (where $p$ is 1
or $\infty$) and, on the other hand, of “operator sequence spaces” of Lambert
[11] (with $p=2$, and convexity is assumed). These spaces happened to be
intimately connected with the theory of Banach lattices, that is with the
important part of a wide circle of questions concerning the concept of
positivity. More recently, a connection between $2$-multinormed spaces and
sectorial operators was discovered and investigated by Kalton, Lorist, and
Weis in [10].

After the paper [2], the $p$-multinormed spaces, in their turn, were
generalized, in the frame-work of the so-called non-coordinate approach, to
what can be considered as “$p$-multinormed spaces based on arbitrary measures”
[6]. Taking ${\bf L}$ is a normed space (for instance, $L_{p}(X)$ or
$L_{p}^{0}(X)$, where $X$ is a $\sigma$-finite measure; $L_{p}^{0}(X)$ is the
span of simple functions in $L_{p}(X)$), we define an ${\bf L}$-space $E$ as a
linear space for which ${\bf L}\otimes E$ is equipped with a norm, satisfying
certain natural conditions (specified in Section 1, where relevant definitions
and basic facts are collected). Specializing to $X:=\mathbb{N}$ with the
counting measure, we recover $p$-multinormed spaces.

The aim of this paper is to describe projective (= “homologically best”) ${\bf
L}$-spaces. We consider two versions of this notion. The first one is the
metric projectivity which is, roughly speaking, a suitable concretization of
the general categorical projectivity. The second one is the extreme
projectivity, taking into account specific features of functional-analytic
structures, based on the presence of norm. Extremely projective Banach spaces
were characterized by Grothendick [5]; much later Blecher [1] described (in
different terms) extremely projective operator spaces. Metric projectivity of
Banach spaces appeared, under different names, in old papers of Semadeni [17]
and Graven [4], and then, for operator spaces, in [7].

Category theory provides us with a valuable tool for studying projective
objects: freeness. Our goal is to (i) introduce an appropriate notion of
freeness, and (ii) to characterize the free objects. Then projective objects
can be described as well, since they are exactly the retracts of free objects.
This approach has been applied to quite a few categories in functional
analysis, and not only for the metric projectivity, but for the extreme
projectivity as well (in the latter case one needs to replace retracts by the
so-called near-retracts). For application to operator spaces and modules see
[7, 8]; general matricially normed spaces are treated in [9].

In Section 2 of the present paper we introduce a reasonable notion of a free
${\bf L}$–space. Its definition depends on our choice of a family
$L^{\nu};\nu\in\Lambda$ ($\Lambda$ is an index set) of contractively
complemented subspaces which “paves” ${\bf L}$. We show that free ${\bf
L}$–spaces do exist if, and only if, all our subspaces $L^{\nu}$ are finite-
dimensional (Theorem 2.15). If the latter condition is fulfilled, we obtain a
collection of projective ${\bf L}$–spaces. In particular, this is the case
when ${\bf L}$ is $L_{p}(X)$ or $L_{p}^{0}(X)$, and every $L^{\nu}$ is a
linear span of characteristic functions of several measurable subsets of $X$.

Addressing freeness, we show that the desirable free ${\bf L}$-spaces are the
so-called $\oplus_{1}$-sums of all possible families of some special ${\bf
L}$-space, which, in its turn, is the $\oplus_{1}$-sum of all minimal ${\bf
L}$-spaces $(L^{\nu})^{*};\nu\in\Lambda$ (Propositions 2.12 and 2.11). The
words “minimal ${\bf L}$–space” mean that we consider in ${\bf
L}\otimes(L^{\nu})^{*}$ the injective tensor norm. One of the importants steps
in the proof consists of establishing a bijection, for a given ${\bf L}$-space
$E$, between the unit ball of ${\bf L}\otimes E$ and a specific set of
operators from $(L^{\nu})^{*}$ into $E$.

In Section 3 we turn to the study of projectivity. We call an ${\bf L}$-space
“well composed” if it is an $\oplus_{1}$-sum of a family of ${\bf L}$-spaces,
such that each of them is minimal ${\bf L}$–space $L^{*}$ for some finite-
dimensional contractively complemented subspace $L$ of ${\bf L}$ (summands may
be different). Then every retract, respectively near-retract of a well
composed ${\bf L}$-space is metrically, respectively extremely projective.
Moreover, if ${\bf L}$, like in the case of $L^{0}_{p}(X)$, is the set-
theoretical union of its finite-dimensional contractively complemented
subspaces, we obtain a full description of relevant ${\bf L}$-spaces: for such
an ${\bf L}$ a given ${\bf L}$-space is metrically, respectively extremely
projective if, and only if, it is a retract, respectively near-retract of some
well composed ${\bf L}$-space (Theorem 3.7). As a particular case, we obtain
(Theorem 3.10) a description of metrically and extremely projective
$p$-multinormed spaces. The same theorem also provides a “quotient
representation” of $p$-multinormed spaces. This generalizes and sharpens some
of the results, obtained by the second author in [14].

##  1\. ${\bf L}$–spaces and ${\bf L}$–bounded operators

First we fix the notation. If $G$ and $E$ are linear spaces, we denote by
${\mathcal{L}}(G,E)$ the set of all linear operators from $G$ to $E$. If, in
addition, $G$ and $E$ are normed, denote by ${\mathcal{B}}(G,E)$ the space of
all bounded operators between them, endowed with the operator norm, and write
${\mathcal{B}}(E)$ instead of ${\mathcal{B}}(E,E)$. The identity map on a set
$M$ will be denoted by ${\bf 1}_{M}$. The symbol $\otimes$ is used for the
algebraic tensor product of linear spaces and linear operators, and also for
elementary tensors. The closed and open unit ball in a normed space $E$ is
denoted by $B(E)$ and $B^{0}(E)$, respectively.

Choose and fix (so far arbitrary) normed space ${\bf L}$, which we shall call
base space.

For an arbitrary base space, ${\bf L}\otimes E$ is a left module over the
algebra ${\mathcal{B}}({\bf L})$ with the outer multiplication “ $\cdot$ ”,
well defined by $a\cdot(\xi\otimes x):=a\xi\otimes x$.

######  Definition 1.1.

Suppose $E$ is a linear space. A norm on ${\bf L}\otimes E$ is called ${\bf
L}$–norm on $E$, if ${\bf L}\otimes E$ is contractive as a left
${\mathcal{B}}(L)$-module ${\bf L}\otimes E$: the inequality $\|a\cdot
u\|\leq\|a\|\|u\|$ holds for any $a\in{\mathcal{B}}(L)$ and $u\in{\bf
L}\otimes E$. This will be referred as contractibility property. The space
$E$, endowed by an ${\bf L}$–norm, is called ${\bf L}$–space.

######  Remark 1.2.

Any ${\bf L}$-space $E$ can be equipped with a norm: for $e\in E$, define
$\|e\|:=\|\xi\otimes e\|$, where $\xi\in{\bf L}$ has norm one. Note that, if
$\xi,\eta\in{\bf L}$ have norm one, then, by Hahn-Banach Theorem, there exists
a norm one $a\in{\mathcal{B}}({\bf L})$ so that $a\xi=\eta$; thus, $\|e\|$ is
well-defined. With this norm on $E$, the norm on ${\bf L}\otimes E$ becomes a
cross-norm – that is, $\|\xi\otimes e\|=\|\xi\|\|e\|$ for any $\xi\in{\bf L}$
and $e\in E$.

Our principal examples of base spaces ${\bf L}$ are $L_{p}(X)$, where $X$ is a
measure space (always assumed to be $\sigma$-finite), $p\in[1,\infty]$, and
its subspace $L^{0}_{p}(X)$, consisting of simple functions. As any bounded
operator $u:L^{0}_{p}(X)\to L^{0}_{p}(X)$ extends to a bounded operator
$\widetilde{u}:L_{p}(X)\to L_{p}(X)$ (of the same norm), any $L_{p}(X)$-space
is also an $L^{0}_{p}(X)$-space. We claim that the converse is also true.
Indeed, suppose $E$ is an $L^{0}_{p}(X)$-space; denote the corresponding norm
on $L^{0}_{p}(X)\otimes E$ by $\|\cdot\|_{0}$. Let $\Lambda$ be the net of all
finite $\sigma$-subalgebras of $X$. For $\nu\in\Lambda$, let $L^{\nu}\subset
L^{0}_{p}(X)$ be the set of all $\nu$-measurable functions – that is, of
functions $\sum_{i}\alpha_{i}\chi_{A_{i}}$, with scalars $\alpha_{i}$, and
$A_{i}\in\nu$. Any such $L^{\nu}$ is a range of a (contractive) conditional
expectation, denoted by $Q^{\nu}$. For $u\in L_{p}(X)\otimes E$, let
$\|u\|:=\sup_{\nu\in\Lambda}\|Q^{\nu}\cdot u\|_{0}$. This quantity is well
defined: if $u=\sum_{k}\xi_{k}\otimes x_{k}$, then
$\|u\|\leq\sum_{k}\|\xi_{k}\|\|x_{k}\|$. It is easy to see that $\|\cdot\|$ is
a norm which coincides with $\|\cdot\|_{0}$ on $L^{0}_{p}(X)\otimes E$.

Further, $\|\cdot\|$ is a $L_{p}(X)$-norm – that is, the inequality $\|a\cdot
u\|\leq\|u\|$ holds for any contraction $a\in{\mathcal{B}}(L_{p}(X))$, and any
$u=\sum_{i=k}^{n}\xi_{k}\otimes x_{k}\in L_{p}(X)\otimes E$. To establish
this, fix $\varepsilon>0$, and find $\mu\in\Lambda$ so that
$\sum_{k=1}^{n}\|\xi_{k}\|\|Q^{\mu}x_{k}-x_{k}\|<\varepsilon$ (this is
possible, due to the density of simple functions in $L_{p}(X)$). For any
$\nu\in\Lambda$,

$Q^{\nu}(a\cdot
u)=\sum_{k=1}^{n}Q^{\nu}a\big{(}Q^{\mu}\big{)}^{2}\xi_{k}\otimes
x_{k}+\sum_{k=1}^{n}Q^{\nu}a\big{(}\xi_{k}-Q^{\mu}\xi_{k}\big{)}\otimes x_{k}$

(recall that $\big{(}Q^{\mu}\big{)}^{2}=Q^{\mu}$). As $\|\cdot\|_{0}$ is an
$L^{0}_{p}(X)$-norm, we have

$\Big{\|}\sum_{k=1}^{n}Q^{\nu}a\big{(}Q^{\mu}\big{)}^{2}\xi_{k}\otimes
x_{k}\Big{\|}_{0}\leq\big{\|}Q^{\nu}aQ^{\mu}\big{\|}\Big{\|}\sum_{k=1}^{n}Q^{\mu}\xi_{k}\otimes
x_{k}\Big{\|}_{0}\leq\|u\|.$

From the triangle inequality and Remark 1.2,

$\Big{\|}\sum_{k=1}^{n}Q^{\nu}a\big{(}\xi_{k}-Q^{\mu}\xi_{k}\big{)}\otimes
x_{k}\Big{\|}_{0}\leq\sum_{k=1}^{n}\big{\|}Q^{\nu}a\big{\|}\big{\|}\xi_{k}-Q^{\mu}\xi_{k}\big{\|}\|x_{k}\|\leq\varepsilon,$

hence, by the triangle inequality, $\|Q^{\nu}a\cdot
u\|_{0}\leq\|u\|+\varepsilon$ for any $\nu\in\Lambda$. Taking the supremum
over $\nu$, and recalling that $\varepsilon$ can be arbitrarily small, we
conclude $\|a\cdot u\|\leq\|u\|$.

For ${\bf L}$ as above, ${\bf L}$-spaces generalize $p$-multinormed spaces,
investigated in e.g. [2]. Indeed, suppose $X:={\mathbb{N}}$ with the counting
measure. Then $L_{p}^{0}(X)$ is $\ell_{p}^{0}$ – the space of finite
sequences, with the norm inherited from $\ell_{p}$. It is easy to see that any
$\ell_{p}^{0}$-norm on $E$ corresponds to a sequence of cross-norms
$\|\cdot\|_{n}$ on spaces $\ell_{p}^{n}\otimes E$, with the property that
$\|a\cdot u\|_{n}\leq\|a\|\|u\|_{m}$, for any $u\in\ell_{p}^{m}\otimes E$, and
$u:\ell_{p}^{n}\to\ell_{p}^{m}$. The preceding paragraph establishes that any
$\ell_{p}^{0}$-space is an $\ell_{p}$-space, and vice versa.

Every linear subspace $F$ of an ${\bf L}$-space $E$ will be considered as an
${\bf L}$-space with the ${\bf L}$-norm, induced by the embedding of ${\bf
L}\otimes F$ into ${\bf L}\otimes E$.

If an operator $\varphi:G\to E$ between linear spaces is given, we shall use,
for the operator ${\bf 1}_{\bf L}\otimes\varphi:={\bf L}\otimes G\to{\bf
L}\otimes E$, the brief notation $\varphi_{\infty}$. Obviously, this is a
morphism of left ${\mathcal{B}}({\bf L})$-modules.

The following definition is inspired by various “quantizations” (found, for
instance, in [3, 2, 7]).

######  Definition 1.3.

An operator $\varphi:G\to E$ between ${\bf L}$–spaces is called ${\bf
L}$–bounded if $\varphi_{\infty}$ is bounded. In a similar way we define the
notions of an ${\bf L}$–contractive operator, ${\bf L}$-coisometric operator (
= ${\bf L}$-quotient mapping) ${\bf L}$-strictly coisometric operator ( =
${\bf L}$-exact quotient mapping), and so on. In particular, our $\varphi$ is
${\bf L}$-strictly coisometric, respectively ${\bf L}$-coisometric when
$\varphi_{\infty}$ maps $B({\bf L}\otimes G)$ onto $B({\bf L}\otimes E)$,
respectively $B^{0}({\bf L}\otimes G)$ onto $B^{0}({\bf L}\otimes E)$.

The operator norm of $\varphi_{\infty}$ is denoted by $\|\varphi\|_{{\bf
L}b}$. The set of ${\bf L}$–bounded, respectively ${\bf L}$–contractive
operators between ${\bf L}$-spaces $G$ and $E$ is denoted by
${\mathcal{CB}}(G,E)$, respectively ${\mathcal{CC}}(G,E)$.

Let $P,G,E$ be linear spaces, $\tau:G\to E$, $\varphi:P\to E$ operators. We
recall that an operator $\psi:P\to E$ is called a lifting of $\varphi$ across
$\tau$, if it makes the diagram

(1.1)
$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\scriptstyle{\varphi}$$\textstyle{E}$

commutative.

######  Definition 1.4.

An ${\bf L}$–space $P$ is called metrically projective, if for every ${\bf
L}$–spaces $G,E$ every ${\bf L}$-strictly-coisometric operator $\tau:G\to E$
and every ${\bf L}$-bounded operator $\varphi:P\to E$ there exists a lifting
$\psi:P\to G$ of $\varphi$ across $\tau$, such that $\|\psi\|_{{\bf
L}b}=\|\varphi\|_{{\bf L}b}$.

An ${\bf L}$–space $P$ is called extremely projective, if for $G,E$ and
$\varphi$ as before, every ${\bf L}$-coisometric operator $\tau:G\to E$ and
every $\varepsilon>0$ there exists a lifting $\psi:P\to G$ of $\varphi$ across
$\tau$, such that $\|\psi\|_{{\bf L}b}<\|\varphi\|_{{\bf L}b}+\varepsilon$.

We shall use these definitions in an equivalent form, given by the result
below.

######  Proposition 1.5.

$(i)$ $P$ is metrically projective if, and only if for every ${\bf
L}$-strictly-coisometric $\tau:G\to E$ and an ${\bf L}$-contractive
$\varphi:P\to E$ there exists an ${\bf L}$-contractive lifting of $\varphi$
across $\tau$.

$(ii)$ $P$ is extremely projective if, and only if for every ${\bf
L}$-coisometric $\tau:G\to E$ and an ${\bf L}$-contractive $\varphi:P\to E$
with $\|\varphi\|_{{\bf L}b}<1$ there exists an ${\bf L}$-contractive lifting
of $\varphi$ across $\tau$. ${\square}$

######  Definition 1.6.

An ${\bf L}$-contractive operator is called retraction, if it has a right
inverse ${\bf L}$-contractive operator (which, of course, must be an ${\bf
L}$-isometry). Further, an ${\bf L}$-contractive operator is a near-
retraction, if, for every $\varepsilon>0$, it has a right inverse ${\bf
L}$-bounded operator $\rho$ such that $\|\rho\|_{{\bf L}b}<1+\varepsilon$. An
${\bf L}$-space $E$ is called retract (near-retract) of an ${\bf L}$-space $G$
if there is a retraction (respectively, near retraction) from $G$ onto $E$.

In more geometric terms, $E$ is a retract of $G$ if and only if it is a so-
called ${\bf L}$-direct summand of $G$111Not to be confused with $L$-summands
from the theory of $M$-ideals., that is $E$ is ${\bf L}$-isometrically
isomorphic to an ${\bf L}$-subspace $F$ of $G$ such that there is an ${\bf
L}$-contractive projection of $G$ onto $F$. Following [1], we say that $E$ is
an almost ${\bf L}$-direct summand of $G$ when for every $\varepsilon>0$ there
is a subspace $F$ of $E$ such that there exists a projection $Q$ of $G$ onto
$F$ with $\|Q\|_{{\bf L}b}<1+\varepsilon$ and there exists an ${\bf
L}$-topological isomorphism $I:E\to F$ with $\|I\|_{{\bf L}b},\|I^{-1}\|_{{\bf
L}b}<1+\varepsilon$. Equivalently, for any $\varepsilon>0$ there exist a
contractive operator from $G$ onto $E$, which has a right inverse $\rho$ with
$\|\rho\|_{{\bf L}b}<1+\varepsilon$. Consequently, any near-retract of $G$
must be an almost ${\bf L}$-direct summand.

######  Proposition 1.7.

$(i)$ A retract of a metrically projective ${\bf L}$-space is itself
metrically projective.

$(ii)$ A near-retract of an extremely projective ${\bf L}$-space is itself
extremely projective.

###### Proof.

We restrict ourselves to the p. (ii), since p. (i) is simpler. Suppose that
$P_{0}$ is extremely projective, $P$ is a near-restract of $P_{0}$, while
$\tau$, $\varphi$, and $G$ are as in Proposition 1.5(ii). Fix
$\lambda\in(\|\varphi\|_{{\bf L}b},1)$, and find an ${\bf L}$-contractive map
$\sigma:P_{0}\to P$, with a right inverse $\rho$ satisfying $\|\rho\|_{{\bf
L}b}<1/\lambda$. As $\|\varphi\sigma\|_{{\bf L}b}\leq\|\varphi\|_{{\bf
L}b}<\lambda$, the extreme projectivity of $P_{0}$ implies the existence of
$\psi_{0}:P_{0}\to G$ with $\tau\psi_{0}=\varphi\sigma$, and
$\|\psi_{0}\|_{{\bf L}b}<\lambda$. Let $\psi=\psi_{0}\rho$; then
$\tau\psi=\varphi\sigma\rho=\varphi$ (that is, $\psi$ lifts $\varphi$).
Further, $\|\psi\|_{{\bf L}b}\leq\|\psi_{0}\|_{{\bf L}b}\|\rho\|_{{\bf
L}b}<1$. ∎

######  Remark 1.8.

The proof above shows that any almost ${\bf L}$-direct summand of an extremely
projective ${\bf L}$-space is extremely projective.

##  2\. The functor $\odot$, and $\odot$-free ${\bf L}$–spaces

Denote by ${\bf L}{\bf Nor_{1}}$ the category with ${\bf L}$-spaces as objects
and ${\bf L}$-contractive operators as morphisms. As usually, the category of
sets and maps is denoted by ${\bf Set}$.

Let ${\square}:{\bf L}{\bf Nor_{1}}\to{\bf Set}$ be (so far) an arbitrary
functor.

######  Definition 2.1.

An ${\bf L}$-contractive operator $\tau$ is called ${\square}$-admissible, if
the map ${\square}(\tau)$ is surjective (i.e. it is a retraction in ${\bf
Set}$).

######  Example 2.2.

Consider the functor $\bigcirc:{\bf L}{\bf Nor_{1}}\to{\bf Set}$, taking $E$
to $B({\bf L}\otimes E)$, and taking the ${\bf L}$-contractive operator
$\varphi:E\to G$ to the restriction of $\varphi_{\infty}$ to $B(L\otimes E)$.
We see that an ${\bf L}$-contractive operator $\tau$ is $\bigcirc$-admissible
if, and only if it is ${\bf L}$-strictly coisometric.

We shall be mostly concerned with a different functor, better suited for
describing projective ${\bf L}$-spaces. In what follows, we suppose that we
are given a family $L^{\nu};\nu\in\Lambda$, where $\Lambda$ is some index set,
of contractively complemented subspaces of ${\bf L}$.

######  Example 2.3.

We are mainly concerned with ${\bf L}=L^{0}_{p}(X)$. Take $\Lambda$ to be the
family of all finite sets of pairwise disjoint measurable subsets in $X$; let
$L^{\nu};\nu\in\Lambda$ the span of characteristic functions of sets in $\nu$;
$Q^{\nu}$ is the corresponding conditional expectation.

Note that we do not have much latitude in selecting the spaces $L^{\nu}$.
Indeed, any contractive projection on $L^{0}_{p}(X)$ extends to that on
$L_{p}(X)$ by continuity. Contractive projections on $L_{p}(X)$ are known to
be “modified conditional expectations” (see e.g. [15, Theorem 4.3]). Further,
by [18], contractively complemented subspaces of $L_{p}(X)$ are themselves
$L_{p}$ spaces.

Denote, for brevity, $B_{E}:=B({\bf L}\otimes E),B^{0}_{E}:=B^{0}({\bf
L}\otimes E),B^{\nu}_{E}:=B(L^{\nu}\otimes
E),B^{0\nu}_{E}:=B^{0}(L^{\nu}\otimes E)$ and consider, for every ${\bf
L}$-space $E$, the set ${\bf
B}_{E}:=\textsf{X}\\{B^{\nu}_{E};\nu\in\Lambda\\}$, that is the Cartesian
product of the closed unit balls of the normed spaces
$B^{\nu}_{E};\nu\in\Lambda$. Thus, its elements can be represented as ‘rows’
$(...,v_{\nu},...)_{\nu\in\Lambda}$, with $v_{\nu}\in B^{\nu}_{E}$. Let us
introduce the functor

$\odot:{\bf L}{\bf Nor_{1}}\to{\bf Set},$

taking $E$ to ${\bf B}_{E}$ and an ${\bf L}$-contractive operator
$\varphi:G\to E$ to the map
$\odot(\varphi):\odot(G)\to\odot(E),(...,v_{\nu},...)\mapsto(...,\varphi_{\infty}(v_{\nu}),...)$.
(This map is well defined since, for every $v\in B_{G}^{\nu}$, we have
$\varphi_{\infty}(v)=\varphi_{\infty}(Q^{\nu}\cdot
v)=Q^{\nu}\cdot\varphi_{\infty}(v)\in B_{E}^{\nu})$.

Similarly, we define the space ${\bf
B}^{0}_{E}:=\textsf{X}\\{B^{0\nu}_{E};\nu\in\Lambda\\}$ and introduce the
functor $\odot^{0}:{\bf L}{\bf Nor_{1}}\to{\bf Set}$, taking $E$ to ${\bf
B}^{0}_{E}$ and an ${\bf L}$-contractive operator $\varphi:G\to E$ to the map
$\odot^{0}(\varphi):\odot^{0}(G)\to\odot^{0}(E)$,

######  Proposition 2.4.

Any ${\bf L}$-strictly coisometric operator is $\odot$-admissible, whereas any
${\bf L}$-coisometric operator is $\odot^{0}$-admissible.

###### Proof.

If $\tau:G\to E$ is ${\bf L}$-strictly coisometric, let us take $u\in
B^{\nu}_{E}$. Then $u=\tau_{\infty}(v)$ for some $v\in B_{G}$. Hence
$\tau_{\infty}(Q^{\nu}\cdot v)=Q^{\nu}\cdot\tau_{\infty}v=Q^{\nu}\cdot
u=u,Q^{\nu}\cdot v\in L^{\nu}\otimes G$ and $\|Q^{\nu}\cdot
v\|\leq\|Q^{\nu}\|\|v\|\leq 1$. The proof in the ‘${\bf L}$\- coisometric’
case is similar. ∎

In certain cases, the converse of Proposition 2.4 holds. We say that ${\bf L}$
is properly presented (relative to the family $L^{\nu}:\nu\in\Lambda$ of
complemented subspaces) if for any $N\in\mathbb{N}$, and
$f_{1},\ldots,f_{N}\in{\bf L}$, there exists $\nu\in\Lambda$ with
$f_{1},\ldots,f_{N}\in L^{\nu}$. Proper presentation occurs, for instance, in
the setting of Example 2.3. For $f_{1},\ldots,f_{N}\in{\bf L}=L^{0}_{p}(X)$,
let $\nu$ be the (finite) $\sigma$-algebra of subsets of $X$, generated by
$f_{1},\ldots,f_{N}$. Clearly $f_{1},\ldots,f_{N}\in L^{\nu}$.

######  Proposition 2.5.

Suppose ${\bf L}$ is properly presented relative to the family
$L^{\nu}:\nu\in\Lambda$, and let $\odot$ and $\odot^{0}$ be the functors
arising from this family. Suppose, further, that $G$ and $E$ are ${\bf
L}$-spaces, and $\phi:G\to E$ is a linear operator. Then:

  1. (1)

$\|\phi\|_{{\bf L}b}\leq 1$ if and only if $\odot\phi$ $($equivalently,
$\odot^{0}\phi)$ is a well-defined map on ${\bf Set}$.

  2. (2)

$\phi$ is $\odot-$, respectively $\odot^{0}$–admissible if, and only if, it is
${\bf L}$-strictly coisometric, respectively ${\bf L}$-coisometric.
${\square}$

The following definition is a particular case of a well known general-
categorical definition (see, e.,g., Definition 6 in[9]). In what follows,
${\square}:{\bf L}{\bf Nor_{1}}\to{\bf Set}$ is an arbitrary functor, $M$ a
set.

######  Definition 2.6.

An ${\bf L}$-space ${\bf F}^{\square}(M)$ is called a ${\square}$-free space
with the base $M$, if, for every ${\bf L}$-space $E$ there exists a bijection

${\bf I}_{E}:{\bf Set}(M,{\square}E)\to{\mathcal{CC}}({\bf F}^{\square}(M),E)$

between the respective sets of morphisms, and these bijections have the so-
called natural property. The latter means that for every
$\varphi\in{\mathcal{CC}}(G,E)$ we have the commutative diagram

(2.1) $\textstyle{{\bf
Set}(M,{\square}G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\bf
I}_{G}}$$\scriptstyle{({\square}\varphi)^{*}}$$\textstyle{{\mathcal{CC}}({\bf
F}^{\square}(M),G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi^{*}}$$\textstyle{{{\bf
Set}(M,{\square}E)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{{\bf
I}_{E}}}$$\textstyle{{\mathcal{CC}}({\bf F}^{\square}(M),E).)}$

where $\varphi^{*}$ acts by composition as $\psi\mapsto\varphi\circ\psi$ and
$({\square}\varphi)^{*}$ as $\rho\mapsto({\square}\varphi)\circ\rho$.

A simple “diagram-chasing” argument shows that the free space of $M$ is
unique.

Henceforth, we shall write ${\bf F}(M)$ for ${\bf F}^{\square}(M)$, if there
is no confusion about the functor ${\square}$.

The following proposition is well known in the general categorical context of
adjoint functors [13].

######  Proposition 2.7.

Suppose that for a given ${\bf L}$-space $E$ there exists a ${\square}$-free
space ${\bf F}({\square}E)$ with the base ${\square}E$. Then there exist a
${\square}$-admissible operator $\pi:{\bf F}({\square}E)\to E$.

###### Proof.

To simplify the notation, denote ${\bf F}({\square}E)$ by $G$. We show that
$\pi:={\bf I}_{E}({\bf 1}_{{\square}E})$ is admissible, by establishing that
${\square}\pi\circ\psi={\bf 1}_{{\square}E}$, where $\psi={\bf
I}_{G}^{-1}({\bf 1}_{G})$. To this end, observe that, due to the natural
property (2.1),

${\bf I}_{E}\big{(}{\square}\pi\circ\psi\big{)}=\pi\circ{\bf I}_{G}\big{(}{\bf
I}_{G}^{-1}({\bf 1}_{G})\big{)}=\pi\circ{\bf 1}_{G}=\pi,$

hence

${\square}\pi\circ\psi={\bf I}_{E}{\bf
I}_{E}^{-1}\big{(}{\square}\pi\circ\psi\big{)}={\bf I}_{E}^{-1}\pi={\bf
1}_{{\square}E}.\qed$

The main result of the rest of this section – Theorem 2.15 – determines the
necessary and sufficient condition for the existence of $\odot$-free ${\bf
L}$-spaces.

First we establish that only the case of finite dimensional spaces $L^{\nu}$
is interesting.

######  Proposition 2.8.

Suppose that, for some set $M$, there exists a $\odot$–free ${\bf L}$–space
with the base $M$. Then all subspaces $L^{\nu}$ are finite-dimensional.

###### Proof.

Let ${\bf F}$ be our $\odot$–free ${\bf L}$–space. Taking, in the capacity of
$E$, the same ${\bf F}$, we obtain a bijection ${\bf I}_{\bf F}:{\bf
Set}(M,\odot{\bf F})\to{\mathcal{CC}}({\bf F},{\bf F})$. Let $f={\bf I}_{\bf
F}^{-1}({\bf 1}_{\bf F})$. For any $t\in M$, $f(t)$ is a “row,” say
$(...,u_{\nu},...);\nu\in\Lambda$, where $u_{\nu}\in B^{\nu}_{\bf F}$ for all
$\nu$.

Fix $\nu\in\Lambda$, and represent $u_{\nu}\in L^{\nu}\otimes{\bf F}$ as
$\sum_{k=1}^{n}\xi_{k}\otimes x_{k}$ for some $\xi_{k}\in L^{\nu},x_{k}\in{\bf
F};k=1,...,n$. Take an arbitrary non-zero $\eta\in L^{\nu}$ and $x\in{\bf F}$
such that $\|\eta\otimes x\|\leq 1$. Consider an arbitrary function
$g:M\to{\bf B}_{\bf F}$ where $g(t)\in{\bf B}_{\bf F}$ is a ‘row’ with
$\eta\otimes x$ on the ‘$\nu$-th’ place. Then $\varphi:={\bf I}_{\bf F}(g)$ is
an ${\bf L}$–contractive operator, acting on ${\bf F}$. Therefore, by the
“natural property”, we have

${\bf I}_{\bf F}(\odot\varphi\circ f)=\varphi{\bf I}_{\bf F}(f)=\varphi{\bf
I}_{\bf F}{\bf I}_{\bf F}^{-1}({\bf 1}_{\bf F})=\varphi{\bf 1}_{\bf
F}=\varphi={\bf I}_{\bf F}(g),$

hence $\odot\varphi\circ f=g$. This, in particular, means that the ‘row’
$[\odot\varphi\circ f](t)$ coincides with the ‘row’ $g(t)$. Looking at the
‘$\nu$-th’ place in these ‘rows’, we see that
$\varphi_{\infty}(u_{\nu})=\eta\otimes x$, that is $\eta\otimes
x=\sum_{k=1}^{n}\xi_{k}\otimes\varphi(x_{k})$. This implies that $\eta\in\rm
span\,\\{\xi_{k}:1\leq k\leq n\\}$. As $\eta\in L^{\nu}$ is arbitrary, the
space $L^{\nu}$ coincides with the latter span. ∎

To deal with the functor $\odot$, we shall use a certain family of simpler
functors. Fix, for a time, $\nu\in\Lambda$ and consider the functor
$\odot_{\nu}$, taking $E$ to $B^{\nu}_{E}$ and taking an ${\bf L}$-contractive
operator $\varphi:G\to E$ to the respective restriction
$\odot(\varphi):\odot(G)\to\odot(E)$ of the operator $\varphi_{\infty}$. Thus,
for every $\varphi\in{\mathcal{CC}}(E,F)$ and ${\bf
u}=(...,u_{\nu},...)\in{\bf B}_{G}$ we have

(2.2) $(\odot\varphi({\bf u}))_{\nu}=\odot_{\nu}\varphi(u_{\nu}).$

Fix a projection $Q^{\nu}=Q:{\bf L}\to L^{\nu}$ of norm 1. Equip the dual
space $(L^{\nu})^{*}$ with the so-called minimal ${\bf L}$-norm. That is, we
supply ${\bf L}\otimes(L^{\nu})^{*}$ with the norm of the injective tensor
product of respective normed spaces (see, e.g., [16]). For an ${\bf L}$-space
$E$ we introduce the operator

${\bf I}^{\nu}_{E}:{\bf L}\otimes E\to{\mathcal{L}}((L^{\nu})^{*},E),$

assigning to $u\in{\bf L}\otimes E$ the operator, taking $g\in(L^{\nu})^{*}$
to $(g\otimes{\bf 1}_{E})(Q\cdot u)$.

In Propositions 2.9, 2.10, and 2.11 below we suppose that the space $L^{\nu}$
is finite-dimensional. In this case we take an Auerbach basis in $L^{\nu}$,
that is a basis $e_{k}^{\nu}=e_{k};k=1,...,n_{\nu}$ in that space together
with a basis $e_{k}^{\nu*}=e_{k}^{*}$ in $(L^{\nu})^{*}$ such that
$e^{*}_{k}(e_{l})=\delta_{kl}$, and $\|e_{k}\|=1=\|e_{k}^{*}\|$ for any $k$
(see, e.g., [12, p. 12]). We fix a distinguished element

(2.3) $w^{\nu}=w:=\sum_{k}e_{k}\otimes e_{k}^{*}\in{\bf
L}\otimes(L^{\nu})^{*}.$

As before, $Q^{\nu}$ (or $Q$ for short) is a fixed projection from ${\bf L}$
onto $L^{\nu}$.

######  Proposition 2.9.

For every $u\in{\bf L}\otimes E$ the operator ${\bf I}^{\nu}_{E}(u)$ is ${\bf
L}$–bounded, and we have $\|{\bf I}^{\nu}_{E}(u)\|_{{\bf L}b}=\|Q\cdot u\|$.

###### Proof.

Our task is to show that, for any $u\in{\bf L}\otimes E$, we have $\|{\bf
I}_{E}^{\nu}(u)_{\infty}\|=\|Q\cdot u\|$. Define the linear map $J:{\bf
L}\otimes(L^{\nu})^{*}\to{\mathcal{B}}(L^{\nu},{\bf L})$ by letting
$J(\xi\otimes g)$ ($\xi\in{\bf L},g\in(L^{\nu})^{*}$ be the operator
$\eta\mapsto g(\eta)\xi$ ($\eta\in L^{\nu}$). As ${\bf L}\otimes(L^{\nu})^{*}$
is endowed with the injective tensor product norm, $J$ is an isometry (see,
e.g., [16]). Further, $J(w)(e_{l})=e_{l}$ for all $l=1,...,n$. Therefore
$J(w)$ is just the canonical embedding of $L^{\nu}$ into ${\bf L}$, and hence
$\|w\|=1$.

For $u:=\xi\otimes x$ we have ${\bf
I}^{\nu}_{E}(u)(e_{k}^{*})=e_{k}^{*}(Q\xi)x$. By linearity, ${\bf
I}^{\nu}_{E}(u)_{\infty}(w)=\sum_{k}e_{k}\otimes e_{k}^{*}(Q\xi)x$. The
equality $\eta=\sum_{k}e_{k}^{*}(\eta)e_{k}$ is true for any $\eta\in
L^{\nu}$, and therefore,

${\bf I}^{\nu}_{E}(u)_{\infty}(w)=\sum_{k}e_{k}\otimes
e_{k}^{*}(Q\xi)x=Q\xi\otimes x=Q\cdot u$

holds for any elementary tensor product $u$. By linearity, ${\bf
I}^{\nu}_{E}(u)_{\infty}(w)=Q\cdot u$ for any $u\in{\bf L}\otimes E$. To show
$\|{\bf I}^{\nu}(u)_{\infty}\|\geq\|Q\cdot u\|$, recall that $\|w\|\leq 1$.

To prove the converse, observe first that $J(v)\cdot(Q\cdot u)={\bf
I}^{\nu}_{E}(u)_{\infty}(v)$ holds for any $u\in{\bf L}\otimes E$ and
$v\in{\bf L}\otimes(L_{\nu})^{*}$. By linearity, it suffices to consider
elementary tensors $u=\xi\otimes x$ and $v=\eta\otimes f$. It is easy to check
that then, $J(v)\cdot(Q\cdot u)=f(Q\xi)\eta\otimes x$, whereas ${\bf
I}^{\nu}_{E}(u)_{\infty}(v)=\eta\otimes f(Q\xi)x$. Therefore, taking into
account the contractibility property and our choice of norm on ${\bf
L}\otimes(L^{\nu})^{*}$, we have

$\|{\bf I}^{\nu}_{E}(u)_{\infty}(v)\|\leq\|J(v)\cdot(Q\cdot
u)\|\leq\|J(v)\|\|Q\cdot u\|=\|Q\cdot u\|\|v\|$

Taking the supremum over all $v$ of norm not exceeding $1$, we conclude that
$\|{\bf I}^{\nu}_{E}(u)_{\infty}\|\leq\|Q\cdot u\|$. ∎

######  Proposition 2.10.

The operator ${\bf I}^{\nu}_{E}$, being restricted to $L^{\nu}\otimes E$, is
an ${\bf L}$-isometric isomorphism between the latter space and
$({\mathcal{CB}}((L^{\nu})^{*},E))_{{\bf L}b}$.

###### Proof.

Since elements of $L^{\nu}\otimes E$ are exactly those $u\in{\bf L}\otimes E$
with $u=Q\cdot u$, Proposition 2.9 implies that our restriction of ${\bf
I}^{\nu}_{E}$ is ${\bf L}$-isometric. It is easy to check that the inverse map
is given by ${\mathcal{CB}}((L^{\nu})^{*},E))\to L^{\nu}\otimes
E:\varphi\mapsto\varphi_{\infty}(w)$; this establishes the bijectivity. ∎

From now on let us agree to identify every map from a singleton to some set
with its image in this set.

######  Proposition 2.11.

The space $(L^{\nu})^{*}$, supplied with the minimal ${\bf L}$-norm, is the
$\odot_{\nu}$-free ${\bf L}$-space whose base is a singleton.

###### Proof.

Proposition 2.10 shows that ${\bf I}^{\nu}_{E}$ implements a bijection between
$B^{\nu}_{E}$ and the set ${\mathcal{CC}}((L^{\nu})^{*},E)$. As to the natural
property, it is easy to check that for every $u\in B^{\nu}_{G}$ and
$\varphi\in{\mathcal{CC}}(G,E)$ both maps $\varphi^{*}({\bf I}^{\nu}_{G}(u))$
and ${\bf I}^{\nu}_{E}(\odot_{\nu}\varphi)^{*}(u)$ (cf. diagram (2.1)) take
$f\in(L^{\nu})^{*}$ to $(f\otimes\varphi)u$. ∎

For the rest of this paper, we shall use the following construction. For a
family $(E_{i})_{i\in\Delta}$ of ${\bf L}$-spaces, consider their algebraic
sum $\oplus_{i}E_{i}$ and make it an ${\bf L}$-space in the following way: we
identify ${\bf L}\otimes(\oplus_{i}E_{i})$ with $\oplus_{i}({\bf L}\otimes
E_{i})$ and consider in the latter space the norm of the (non-completed)
$l_{1}$-sum of its direct summands (that is, for an element
$u\in\oplus_{i}({\bf L}\otimes E_{i});u=(...,u_{i},...);u_{i}\in{\bf L}\otimes
E_{i}$ we set $\|u\|_{1}:=\sum_{i}\|u_{i}\|)$. Obviously, we obtain an ${\bf
L}$-space, called the $\oplus_{1}$-sum of our spaces and denoted by
$(\oplus_{i}E_{i})_{1}$. Together with the latter, we consider the family of
operators ${\bf
in}_{i}:E_{i}\to(\oplus_{i}E_{i})_{1}:x\mapsto(...0,0,x_{i},0,0,..);x_{i}=x$;
$i\in\Delta$. These are, of course, ${\bf L}$-isometric and hence ${\bf
L}$-contractive.

Now suppose that for some ${\bf L}$-space $E$ we are given a family of ${\bf
L}$-contractive operators $\varphi_{i}:E_{i}\to E;i\in\Delta$. Consider the
operator $\oplus_{i}\varphi_{i}:(\oplus_{i}E_{i})_{1}\to
E:x=(...,x_{i},...)\mapsto\sum_{i}\varphi_{i}(x_{i})$, well defined by
$(\oplus_{i}\varphi_{i}){\bf{in}}_{i}=\varphi_{i}$. It is easy to see that
$\oplus_{i}\varphi_{i}$ is ${\bf L}$-contractive, and the map
$\\{\varphi_{i};i\in\Delta\\}\mapsto\oplus_{i}\varphi_{i}$ is a bijection
between the set of all families $\\{\varphi_{i}\in{\mathcal{CC}}(E_{i},E)\\}$
and the set ${\mathcal{CC}}(\oplus_{i}E_{i},E)$.

In the categorical language our construction means that the space
$(\oplus_{i}E_{i})_{1}$, together with the family ${\bf in}_{i}$, is the
coproduct of objects $E_{i}$ in ${\bf L}{\bf Nor_{1}}$.

It is easy to see that for given $\varphi_{i}$ and a ${\bf L}$-contractive
operator $\psi:E\to F$, where $F$ is another ${\bf L}$-space, we have

(2.4) $\psi(\oplus_{i}\varphi_{i})=\oplus_{i}(\psi\varphi_{i})$

Moreover, we obviously have

(2.5) $\|\oplus_{i}\varphi_{i}\|_{{\bf L}b}=\sup_{i}\|\varphi_{i}\|_{{\bf
L}b}$

From now on, we shall denote the $\odot_{\nu}$-free space of a singleton,
constructed in Proposition 2.11, by ${\bf F}^{\nu}(\star)$

######  Proposition 2.12.

$($i$)$ ${\bf F}(\star):=(\oplus_{\nu}{\bf F}^{\nu}(\star))_{1}$ is a
$\odot$-free ${\bf L}$-space with the base $\\{\star\\}$.

$($ii$)$ For any set $M$, ${\bf F}(M):=(\oplus_{t\in M}{\bf F}(t))_{1}$ is a
$\odot$-free ${\bf L}$-space with $M$ as its base. Here, ${\bf F}(t)$ is ${\bf
F}(\star)$ for $t:=\star$.

In the categorical language p. (ii) means that ${\bf F}(M)$ is the coproduct
of $card(M)$ copies of ${\bf F}(\star)$.

###### Proof.

(i) Identify ${\bf Set}(\\{\star\\},B^{\nu}_{E})$ with $B^{\nu}_{E}$.
Proposition 2.9 shows that, for any ${\bf L}$-space $E$, the map ${\bf
I}^{\nu}_{E}:{\bf Set}(\\{\star\\},B^{\nu}_{E})\to{\mathcal{CC}}({\bf
F}^{\nu}(\star),E)$ is bijective.

For ${\bf u}=(...,u_{\nu},...)\in{\bf B}_{E}=\odot(E)$ define ${\bf
I}^{\star}_{E}({\bf u}):=\oplus_{\nu}({\bf
I}^{\nu}_{E})(u_{\nu})\in{\mathcal{CC}}({\bf F}(\star),E)$. Since ${\bf
I}_{E}^{\nu}$ is a bijection for every $\nu\in\Lambda$, the same is obviously
true for ${\bf I}^{\star}_{E}$. To establish the natural property, fix ${\bf
L}$-spaces $E$ and $G$, and $\varphi\in{\mathcal{CC}}(G,E)$. Pick ${\bf
u}=(...,u_{\nu},...)\in{\bf B}_{G}$. The natural property of bijections ${\bf
I}^{\nu}_{E}$, together with (2.2) and (2.4), yields

$\displaystyle\varphi{\bf I}^{\star}_{G}({\bf u})$
$\displaystyle=\varphi[\oplus_{\nu}{\bf
I}^{\nu}_{G}(u_{\nu})]=\oplus_{\nu}[\varphi({\bf I}^{\nu}_{G}(u_{\nu}))]$
$\displaystyle=\oplus_{\nu}{\bf
I}^{\nu}_{E}[(\odot_{\nu}\varphi)(u_{\nu})]=\oplus_{\nu}{\bf
I}^{\nu}_{E}[(\odot\varphi)(u)]_{\nu}={\bf I}_{E}^{\star}(\odot\varphi({\bf
u})).$

(ii) Let $E$ be as before. Suppose a function $f:M\to\odot(E)$, or, what is
the same, $f:M\to{\bf B}_{E}$, is given. Set ${\bf I}_{E}(f):=\oplus_{t\in
M}{\bf I}_{E}^{t}(f(t))\in{\mathcal{CC}}({\bf F}(M),E)$, where ${\bf
I}_{E}^{t}$ ($t\in M$) are copies of ${\bf I}_{E}^{\star}$. The map ${\bf
I}_{E}:{\bf Set}(M,\odot(E))\to{\mathcal{CC}}({\bf F}(M),E)$ is bijective,
since all “summands” ${\bf I}_{E}^{t}$ are. The relevant natural property
follows, for $\varphi\in{\mathcal{CC}}(G,E)$ and $f:M\to{\bf B}_{G}$, from the
equalities

$\varphi{\bf I}_{G}(f)=\varphi[\oplus_{t}{\bf
I}_{G}^{t}(f(t))]=\oplus_{t}[\varphi{\bf I}_{G}^{t}(f(t))]=\oplus_{t}{\bf
I}_{E}^{t}(\odot\varphi)(f(t))={\bf I}_{E}(\odot\varphi(f)).\qed$

######  Remark 2.13.

Actually, Proposition 2.12 is but a particular case of a certain general-
categorical assertion. Namely, let ${\mathcal{K}}$ be some category with
coproduct. We shall denote the coproduct of a family $Y_{i};i\in\Delta$ of
objects in ${\mathcal{K}}$ by $\oplus_{i}Y_{i}$. Further, consider a family of
functors ${\square}_{\mu}:{\mathcal{K}}\to{\bf Set}$, where $\mu$ runs some
index set $\Lambda^{\prime}$. Introduce the “combined” functor
${\square}:{\mathcal{K}}\to{\bf Set}$, taking $Y$ to the Cartesian product of
sets ${\square}_{\mu}(Y);\mu\in\Lambda^{\prime}$. Further, ${\square}$ takes a
morphism $\varphi:Y\to Z$ to the map, transforming a “row”
$(...,u_{\mu},...);u_{\mu}\in{\square}_{\mu}(Y)$ to
$(...,{\square}_{\mu}\varphi(u_{\mu}),...)$. Now suppose that for every $\mu$
there exists a ${\square}_{\mu}$-free object in ${\mathcal{K}}$ with a
singleton, say $\\{\star\\}$, as its base. Then, if ${\bf F}^{\nu}(\star)$ is
the space constructed above, then:

(i) ${\bf F}(\star):=(\oplus_{\mu}{\bf F}^{\mu}(\star))$ is a ${\square}$-free
object in ${\mathcal{K}}$ with the base $\\{\star\\}$.

(ii) For every set $M$ there exists a ${\square}$-free object in
${\mathcal{K}}$ with $M$ as its base, and it is ${\bf F}(M):=(\oplus_{t\in
M}{\bf F}(t))$, where ${\bf F}(t)$ is ${\bf F}(\star)$ for $t:=\star$.

The proof proceeds as in Proposition 2.12, with obvious modifications.

######  Remark 2.14.

Consider the category ${\bf L}{\bf Nor}$ of ${\bf L}$-spaces and all ${\bf
L}$-bounded operators. Define the functor $\overline{\odot}:{\bf L}{\bf
Nor}\to{\bf Set}$, taking $E$ to $L^{\nu}\otimes E$. Modifying the proof of
Proposition 2.11, one shows that $(L^{\nu})^{*}$ is a free ${\bf L}$-space
with respect to $\overline{\odot}$. However, this observation does not produce
a full analogue of Proposition 2.12(ii): the category ${\bf L}{\bf Nor}$,
unlike ${\bf L}{\bf Nor_{1}}$, has no coproducts of infinite families of ${\bf
L}$-spaces.

Combining Propositions 2.11, 2.12 and 2.8, we immediately obtain

######  Theorem 2.15.

For every set $M$ there exists a $\odot$-free ${\bf L}$–space with the base
$M$ if, and only if, all spaces $L^{\nu}$ are finite-dimensional.

######  Remark 2.16.

In particular, we see that for the functor $\bigcirc$ from Example 2.2
$\bigcirc$-free ${\bf L}$–spaces exist only when ${\bf L}$ is finite-
dimensional. In this setting, our family of subspaces $(L^{\nu})$ consists of
only one space, namely ${\bf L}$ itself.

For $f:M\to{\bf B}_{E}:t\mapsto{\bf u}(t):=(...,u_{\nu}(t),...)$ we define
$|||f|||:=\sup_{t}\sup_{\nu}\|u_{\nu}(t)\|$ and, for $\lambda\in[0,1]$, we set
$\lambda f:M\to{\bf B}_{E}:t\mapsto\lambda{\bf u}(t)$, where $(\lambda{\bf
u})(t):=(...,\lambda u_{\nu}(t),...)$. Of course, for our $f$ and
$\lambda\in[0,1]$ we have $|||\lambda f|||=\lambda|||f|||$.

######  Proposition 2.17.

If ${\bf F}(M)$ is a $\odot$-free ${\bf L}$-space with the base $M$, then for
every ${\bf L}$-space $E$, we have $\|{\bf I}_{E}(f)\|_{{\bf L}b}=|||f|||$;
here, ${\bf I}_{E}:{\bf Set}(M,\odot(E))\to{\mathcal{CC}}({\bf F}(M),E)$ is
the bijection appearing in Definition 2.6.

Thus, our bijections are, in a sense, “isometric” maps, and to prove this we
do not have to use a concrete construction of ${\bf F}(M)$.

###### Proof.

First, for $\lambda\in[0,1]$, applying natural property to
$\varphi:=\lambda{\bf 1}_{E}$, and observing that $\odot(\lambda{\bf
1}_{E})(f)=\lambda f$, we have ${\bf I}_{E}(\lambda f)=\lambda{\bf I}_{E}(f)$.

Representing every $f:M\to{\bf B}_{E}$ as $\lambda f_{0}$ with
$|||f_{0}|||=1$, we see that it suffices to show that the desired equality
holds provided $|||f|||=1$. If it would be not so, then for some
$\lambda\in(0,1)$, we (still) would have $\lambda^{-1}{\bf
I}_{E}(f)\in{\mathcal{CC}}({\bf F}(M),E)$. There exists $g:M\to{\bf B}_{E}$
with ${\bf I}_{E}(g)=\lambda^{-1}{\bf I}_{E}(f)$. Then we have ${\bf
I}_{E}(\lambda(g))={\bf I}_{E}(f)$. Hence $f=\lambda g$ and
$|||f|||=\lambda|||g|||<1$, a contradiction. ∎

######  Remark 2.18.

In Proposition 2.7, we constructed the “canonical” morphism $\pi:{\bf
F}^{\square}({\square}E)\to E$. For the special case when ${\square}$ is our
functor $\odot$, we can describe $\pi$ in more detail. By Proposition 2.12,
the free ${\bf L}$-space ${\bf F}(\odot E)$ is
$\oplus_{\nu,u}(L^{\nu,u})^{*}$, where $({\nu,u})$ runs all pairs
$(\nu\in\Lambda,u\in\odot E)$ and $(L^{\nu,u})^{*}$ is a copy of
$(L^{\nu})^{*}$ with the minimal ${\bf L}$-norm. In other words, it is the
coproduct in ${\bf LNor_{1}}$ of the family $\\{(L^{\nu,u})^{*}\\}$. Here, $u$
is the collection of elements $u_{\mu}\in B(L^{\mu}\otimes E)$, with $\mu$
running over $\Lambda$. For such $(\nu,u)$, define
$\varphi_{\nu,u}:(L^{\nu,u})^{*}\to E$ as ${\bf
I}^{\nu}_{E}(u_{\nu}):(L^{\nu})^{*}\to E$ (recall that $L^{\nu,u}=L^{\nu}$).
Following Proposition 2.7, define $\pi:{\bf F}(\odot E)\to E$ to be the
coproduct $\oplus_{\nu,u}\varphi_{\nu,u}$, We show directly that $\pi$ is
$\odot$-admissible, that is, the map $\odot\pi:\odot{\bf F}(\odot E)\to\odot
E$ is surjective.

An elements $v\in\odot{\bf F}(\odot E)$ can be represented as “rows”
$(...,v_{\nu},...)$ where $v_{\nu}\in
B(\oplus_{\mu,u}L^{\nu}\otimes(L^{\mu,u})^{*})$ (with indices
$\mu\in\Lambda,u\in\odot E$); $\oplus$ refers to the $\ell_{1}$ sum (coproduct
in ${\bf Nor_{1}}$, as described in the paragraph preceding Proposition 2.12).
Then $\odot\pi(v)=(\ldots,z_{\nu},\ldots)$, where

$z_{\nu}=\sum_{m\in\Lambda,u\in\odot E}\big{(}{\bf
1}_{L^{\nu}}\otimes\oplus_{\mu,u}\phi_{\mu,u}\big{)}(v_{\nu})\in
L^{\nu}\otimes E.$

More specifically, we can write
$v_{\nu}=\big{(}\alpha_{\mu,u}^{\nu}\big{)}_{\mu\in\Lambda,u\in\odot E}$, with
$\alpha_{\mu,u}^{\nu}\in L^{\nu}\otimes(L^{\mu,u})^{*}$. Then

$z_{\nu}=\sum_{m\in\Lambda,u\in\odot E}\big{(}{\bf
1}_{L^{\nu}}\otimes\phi_{\mu,u}\big{)}\alpha_{\mu,u}^{\nu}.$

Now suppose we are given $z=(\ldots,z_{\nu},\ldots)\in\odot E$ (with
$z_{\nu}\in B(L^{\nu}\otimes E)$ for $\nu\in\Lambda$). Our goal is to find
$v\in\odot{\bf F}(\odot E)$ so that $\odot\pi(v)=z$. For $\nu\in\Lambda$
define by setting
$v_{\nu}=\big{(}\alpha_{\mu,u}^{\nu}\big{)}_{\mu\in\Lambda,u\in\odot E}$,
where $\alpha_{\nu,z_{\nu}}^{\nu}=w^{\nu}$ ($w^{\nu}$ was defined in (2.3)),
and $\alpha_{\mu,u}^{\nu}=0$ if $(\mu,u)\neq(\nu,z_{n})$. Let
$v=(\ldots,v_{\nu},\ldots)$.

By the proof of Proposition 2.9, $\|w\|=1$, hence $v\in\odot{\bf F}(\odot E)$.
It is easy to verify that $\big{(}{\bf
1}_{L^{\nu}}\otimes\phi_{\nu,u}\big{)}w^{\nu}=u$ for any $u\in
L^{\nu}\otimes(L^{\nu,u})^{*}$, hence also for $u=z_{n}$. Therefore,
$\odot\pi(v)=z$.

##  3\. Applications to the projectivity

Suppose, for a moment, that we have an arbitrary base space and an arbitrary
functor ${\square}:{\bf L}{\bf Nor_{1}}\to{\bf Set}$.

######  Definition 3.1.

An ${\bf L}$-space $P$ is called ${\square}$-projective, if for every ${\bf
L}$–spaces $G$ and $E$, every ${\square}$-admissible operator $\tau:G\to E$
and every ${\bf L}$-contractive operator $\varphi:P\to E$ there exists an
${\bf L}$-contractive lifting of $\varphi$ across $\tau$. Our $P$ is called
asymptotically ${\square}$–projective, if for every $G,E$ and $\tau$ as
before, and $\varphi:P\to E$ with $\|\varphi\|_{{\bf L}b}<1$ there exists an
${\bf L}$–contractive lifting $\psi$ of $\varphi$ across $\tau$.

######  Proposition 3.2.

Every retract of a ${\square}$-projective ${\bf L}$-space is itself a
${\square}$-projective ${\bf L}$-space. Every near-retract of an
asymptotically ${\square}$-projective ${\bf L}$–space is itself an
asymptotically ${\square}$-projective ${\bf L}$–space.

###### Proof.

Proceed as in Proposition 1.7, with obvious modifications. ∎

######  Proposition 3.3.

$(i)$ For every $M$, the ${\square}$-free ${\bf L}$-space ${\bf
F}^{\square}(M)$ $($if it does exist$)$ is ${\square}$-projective.

$(ii)$ A retract, respectively near-retract of a ${\square}$-free ${\bf
L}$-space is ${\square}$–projective, respectively asymptotically
${\square}$–projective.

$(iii)$ If every set is a base of some ${\square}$-free ${\bf L}$-space, then
every ${\square}$–projective ${\bf L}$-space is a retract, and every
asymptotically ${\square}$–projective ${\bf L}$-space a near-retract of some
${\square}$-free ${\bf L}$-space.

###### Proof.

(i) It is a particular case of a known categorical statement. Suppose $\tau$
is admissible; find a map $\rho:{\square}E\to{\square}G$ so that
$({\square}\tau)\rho={\bf 1}_{{\square}E}$. For $\varphi:{\bf
F}^{\square}(M)\to E$ let $\psi={\bf I}_{G}(\rho\circ{\bf
I}_{E}^{-1}(\varphi))$, then $\tau\psi=\varphi$.

(ii) follows from p. (i), Proposition 3.2, and the obvious observation that
${\square}$–projective ${\bf L}$-spaces are asymptotically
${\square}$–projective.

(iii) Let $P$ be a ${\square}$-projective ${\bf L}$-space. Proposition 2.7
provides a ${\square}$–admissible operator $\pi:{\bf F}({\square}P)\to P$. By
the projectivity of $P$, $\varphi={\bf 1}_{P}$ has an ${\bf L}$-contractive
lifting across $\pi$, which we denote by $\psi$. In other words, $\psi$ is an
${\bf L}$-contractive right inverse of $\pi$ – that is, $P$ is a retract of
${\bf F}({\square}P)$.

If $P$ is only asymptotically projective, the preceding proof needs to be
modified. For any $\varepsilon\in(0,1)$, $\varphi:=(1+\varepsilon)^{-1}{\bf
1}_{P}$ has an ${\bf L}$-contractive lifting $\psi$ across $\pi$. Then
$(1+\varepsilon)\psi$ is a right inverse of $\pi$, hence $\pi$ is a near
retraction. ∎

Return to our special functors $\odot,\odot^{0}:{\bf L}{\bf Nor_{1}}\to{\bf
Set}$, corresponding to a (so far arbitrary) family of contractively
complemented subspaces $L^{\nu}$ of ${\bf L}$. Propositions 2.4 and 2.5 imply:

######  Proposition 3.4.

In the above notation, any $\odot$-projective $(\odot^{0}$-projective$)$ ${\bf
L}$-space is metrically projective $($respectively, extremely projective$)$.
If ${\bf L}$ is properly presented relative to the family
$L^{\nu}:\nu\in\Lambda$, then the converse impications also hold.

Next we establish a connection between projectivity and freeness. Part (ii) of
Theorem 3.5 below can be deduced from the general theory of asymptotic rigged
categories, introduced in [7, 8]. For the sake of clarity, we present a direct
approach.

######  Theorem 3.5.

Let $M$ be an arbitrary set, and ${\bf F}(M)$ the $\odot$–free ${\bf L}$-space
with the base $M$ $($cf. Theorem 2.15$)$. Then:

(i) ${\bf F}(M)$ is $\odot$-projective and metrically projective.

(ii) ${\bf F}(M)$ is asymptotically $\odot^{0}$-projective and extremely
projective.

###### Proof.

$(i)$ follows from Propositions 3.3(i) and 3.4 combined.

$(ii)$ Let $G$ and $E$ be ${\bf L}$-spaces, $\varphi:{\bf F}(M)\to E$ a
completely contractive operator with $\|\varphi\|_{{\bf L}b}=:\theta<1$, and
$\tau:G\to E$ an $\odot^{0}$-admissible operator. According to Propositions
1.5 and 3.4, our task is to find an ${\bf L}$-contractive operator $\psi:{\bf
F}(M)\to G$, making the diagram (1.1) commutative (that is, $\tau\psi=\phi$).

By the condition on $\tau$, for every $\nu\in\Lambda$ the restriction of
$\tau_{\infty}$ to $L^{\nu}\otimes G$ is a coisometric operator. Define the
map $\rho_{\nu}:B^{\nu}_{E}\to B^{\nu}_{G}$: if $u\in B^{\nu}_{E}$ satisfies
$\|u\|\leq\theta$, find $\rho_{\nu}(u)$ so that
$\tau_{\infty}\rho_{\nu}(u)=u$. If $\|u\|>\theta$, let $\rho_{\nu}(u)$ be an
arbitrary element of $B^{\nu}_{G}$. Now consider the map $\rho:{\bf
B}_{E}\to{\bf B}_{G}$, taking ${\bf u}=(...,u_{\nu},...)$ to ${\bf
v}=(...,\rho_{\nu}(u_{\nu}),...)$. We see, in the notation of Proposition
2.17, that if ${\bf u}:M\to{\bf B}_{E}$ satisfies $|||{\bf u}|||\leq\theta$,
then

$(\odot\tau)\rho({\bf u}(t))={\bf u}(t).$

Now replace ${\bf u}$ with ${\bf I}_{E}^{-1}(\varphi):M\to{\bf B}_{E}$, where
${\bf I}_{E}:{\bf Set}(M,{\bf B}_{E})\to{\mathcal{CC}}({\bf F}(M),E)$ is the
bijection from Definition 2.6. Taking into account the aforementioned
proposition and (2.5), for every $t\in M$ we obtain

$(\odot\tau)\rho[{\bf I}^{-1}_{E}(\varphi)]={\bf I}_{E}^{-1}(\varphi).$

Finally, set $\psi:={\bf I}_{G}(\rho[{\bf I}_{E}^{-1}(\varphi)])$. As our
bijections are natural, we conclude that $\tau\psi={\bf
I}_{E}[(\odot\tau)\rho[{\bf I}_{E}^{-1}(\varphi)]]=\varphi$. ∎

Suppose now that $L^{\nu};\nu\in\Lambda$ is the family of all contractively
complemented finite dimensional subspaces of ${\bf L}$, and ${\bf L}$ is
properly presented relative to $(L^{\nu})$ (this is the case, for instance,
when ${\bf L}=L_{p}^{0}(X)$, and the family $(L^{\nu})$ is as described in in
Example 2.3). Let $\odot$ be the functor corresponding to
$L^{\nu};\nu\in\Lambda$. Proposition 3.3(i,iii), considered for the case
${\square}:=\odot$, together with Proposition 3.4 gives

######  Proposition 3.6.

In the above notation, every metrically projective ${\bf L}$–space is a
retract, and every extremely projective ${\bf L}$-space a near-retract of some
$\odot$–free ${\bf L}$–space. ${\square}$

Now, combining this proposition with the explicit construction of the
$\odot$-free ${\bf L}$–space ${\bf F}(M)$, provided by Propositions 2.11 and
2.12, we obtain the following description of projective ${\bf L}$-spaces.

We say that an ${\bf L}$-space is well composed if it is the $\ell_{1}$ sum
$\oplus_{\mu\in\mathcal{M}}{\rm{MIN}}(Z_{\mu}^{*})$. Here $Z_{\mu}$ is a
finite dimensional contractively complemented subspace of ${\bf L}$, and
${\rm{MIN}}(Z_{\mu}^{*})$ refers to the minimal ${\bf L}$ structure of its
dual – that is, we equip ${\bf L}\otimes Z_{\mu}^{*}$ with the injective
tensor norm. Proposition 2.12 shows that $\odot$-free spaces are well
composed; moreover, for every $\mu$ there exists $\nu\in\Lambda$ so that
$Z_{\mu}=L^{\nu}$, and the cardinality of
$\\{\mu\in\mathcal{M}:Z_{\mu}=L^{\mu}\\}$ is independent of $\nu$.

######  Theorem 3.7.

$(i)$ For every ${\bf L}$ a retract, respectively near-retract of a well
composed ${\bf L}$–space is metrically, respectively extremely projective.

$(ii)$ If ${\bf L}$ is properly presented, then every metrically, respectively
extremely projective ${\bf L}$–space is a retract, respectively near-retract
of some well composed ${\bf L}$–space.

$(iii)$ If ${\bf L}$ is properly presented, then every ${\bf L}$–space is a
strictly coisometric image of a well-composed ${\bf L}$–space.

###### Proof.

(i) We provide a proof for metric projectivity, as extreme projectivity is
handled similarly. Suppose $\tau:G\to E$ is a strict ${\bf L}$-coisometry. It
suffices to show that, if $L$ is a finite dimensional subspace of ${\bf L}$,
complemented via a contractive projection $Q$, then any ${\bf L}$-contractive
operator $\varphi:L^{*}\to E$ admits an ${\bf L}$-contractive lifting
$\psi:L^{*}\to G$, with $\tau\psi=\varphi$.

For an ${\bf L}$-space $U$, define the map ${\bf I}_{U}:{\bf L}\otimes
U\to{\mathcal{L}}(L^{*},U)$ in a manner similar to ${\bf I}_{U}^{\nu}$,
introduced before Proposition 2.9). For $u\in{\bf L}\otimes U$, ${\bf I}_{U}u$
is the operator $L^{*}\ni g\mapsto(g\otimes{\bf 1}_{U})(Q\cdot u)$.
Proposition 2.9 shows that $\|u\|\geq\|{\bf I}_{U}u\|_{{\bf L}b}$ fr any
$u\in{\bf L}\otimes E$; moreover, ${\bf I}_{U}$ implements a bijective
isometry from $L\otimes U$ (regarded as a subspace of ${\bf L}\otimes U$) onto
${\mathcal{CB}}(L^{*},U)$. Abusing the notation slightly, we talk about ${\bf
I}_{U}^{-1}:{\mathcal{CB}}(L^{*},U)\to L\otimes U$.

Now fix $\varphi\in{\mathcal{CC}}(L^{*},E)$. By the coisometric property of
$\tau_{\infty}$, there exists $u\in B({\bf L}\otimes G)$ so that
$\tau_{\infty}u={\bf I}_{E}^{-1}\varphi$. Clearly $Q\cdot u$ has the same
properties. Then $\psi:={\bf I}_{G}(Q\cdot u)$ is the desired lifting.

(ii) is a direct corollary of Proposition 3.6.

(iii) Suppose $E$ is an ${\bf L}$-space. Use Theorem 2.15 to find the free
object ${\bf F}(\odot E)$. By Proposition 2.7, there exists an
$\odot$-admissible map from ${\bf F}(\odot E)$ onto $E$. By Propositions 2.5,
such a map is ${\bf L}$-strongly coisometric. ∎

Next we specialize to the case when ${\bf L}$ is either $L_{p}(X)$ or
$L_{p}^{0}(X)$. In the discussion preceding Definition 1.3, we established
that the classes of $L_{p}(X)$-spaces and $L_{p}^{0}(X)$-spaces are identical
(that is, an $L_{p}(X)$-space must be an $L_{p}^{0}(X)$-space, and vice versa;
there is a bijective correspondence between norms as well). We begin our
analysis by stating a lemma, which links the notions of ${\bf L}$-(strict)
coisometricity, and being a (near) retract, for bases ${\bf L}=L_{p}(X)$ and
${\bf L}=L_{p}^{0}(X)$.

######  Lemma 3.8.

Suppose $E$ and $G$ are $L_{p}^{0}(X)$-spaces, or, equivalently,
$L_{p}(X)$-spaces.

$(i)$ If $\tau:G\to E$ is an $L_{p}(X)$-coisometric $(L_{p}(X)$-strictly
coisometric$)$ map, then it is also $L_{p}^{0}(X)$-coisometric
$($respectively, $L_{p}^{0}(X)$-strictly coisometric$)$.

$(ii)$ If $\tau:G\to E$ is $L_{p}^{0}(X)$-coisometric, then it is also
$L_{p}(X)$-coisometric.

$(iii)$ A map $\tau:G\to E$ is a retraction (near-retraction) in the category
of $L_{p}(X)$-spaces if and only if it is a retraction (near-retraction) in
the category of $L_{p}^{0}(X)$-spaces.

Remark 3.11 below shows that part (ii) of the lemma cannot be improved.
Specifically, it provides an example of an $L_{p}^{0}(X)$-strictly coisometric
map $\tau:G\to E$, which is not $L_{p}(X)$-strictly coisometric.

###### Proof.

(i) We deal with strict coisometries (the case of coisometries is similar).
Fix $u=\sum_{k=1}^{N}\xi_{k}\otimes u_{k}\in L_{p}^{0}(X)\otimes E\subset
L_{p}(X)\otimes E$. Due to $\tau$ being $L_{p}(X)$-strictly coisometric, there
exists $v\in L_{p}(X)\otimes G$ so that $\tau_{\infty}v=u$, and

$\|v\|_{L_{p}(X)\otimes G}=\|u\|_{L_{p}(X)\otimes
E}=\|u\|_{L_{p}^{0}(X)\otimes E}.$

Find $\nu\in\Lambda$ so that $\xi_{k}\in L^{\nu}$ for all $k$, and let
$v^{\prime}=Q^{\nu}v$. Clearly $v^{\prime}\in L^{\nu}\otimes G\subset
L_{p}^{0}(X)\otimes G$, $\tau_{\infty}v^{\prime}=u$, and

$\|v^{\prime}\|_{L_{p}^{0}(X)\otimes G}=\|v^{\prime}\|_{L_{p}(X)\otimes
G}\leq\|v\|=\|u\|,$

establishing the $L_{p}^{0}(X)$-strict coisometricity of $\tau$.

(ii) Consider $u=\sum_{k=1}^{N}\xi_{k}\otimes x_{k}\in L_{p}(X)\otimes E$. Fix
$\varepsilon>0$; our goal is to find $v\in L_{p}(X)\otimes G$ so that
$\|v\|<\|u\|+\varepsilon$, and $\tau_{\infty}v=u$. To this end, find
$\nu\in\Lambda$ so that
$\sum_{k}\|\xi_{k}-Q^{\nu}\xi_{k}\|\|x_{k}\|<\varepsilon/2$. Further, find
$w\in L_{p}^{0}(X)\otimes G$ so that
$\tau_{\infty}w=\sum_{k}Q^{\nu}\xi_{k}\otimes x_{k}$, and
$\|w\|<\|u\|+\varepsilon/2$. For each $k$ find $y_{k}\in G$ so that $\tau
y_{k}=x_{k}$, and $\sum_{k}\|\xi_{k}-Q^{\nu}\xi_{k}\|\|y_{k}\|<\varepsilon/2$.
Then $v=w+\sum_{k}\big{(}\xi_{k}-Q^{\nu}\xi_{k}\big{)}\otimes y_{k}$ is the
desired lifting of $u$.

(iii) This statement follows from the fact that, for any map $\varphi$ between
$L_{p}(X)$-spaces (or, equivalently, $L_{p}^{0}(X)$-spaces),
$\|\varphi\|_{L_{p}(X)b}=\|\varphi\|_{L_{p}^{0}(X)b}$. ∎

Note that $L_{p}^{0}(X)$ is properly presented, and the spaces $L^{\nu}$
involved in this proper presentation are isometric copies of
$\ell_{p}^{n_{\nu}}$. This immediately leads to:

######  Lemma 3.9.

An $L_{p}^{0}(X)$-space is well-composed if and only if it is an $\ell_{1}$
sum $\oplus_{\mu\in\mathcal{M}}{\rm{MIN}}\big{(}\ell_{q}^{n_{\mu}}\big{)}$,
for some family $(n_{\mu})_{\mu\in{\mathcal{M}}}$; here, $1/p+1/q=1$.
${\square}$

Having laid the groundwork, we can now characterize projective $L_{p}(X)$ and
$L_{p}^{0}(X)$ spaces.

######  Theorem 3.10.

$(i)$ An $L_{p}^{0}(X)$-space is metrically $($extremely$)$ projective if and
only if it is a retract $($respectively, near retract$)$ of a well-composed
$L_{p}^{0}(X)$-space.

$(ii)$ An $L_{p}(X)$-space is extremely projective if and only if it is a near
retract of a well-composed $L_{p}^{0}(X)$-space.

$(iii)$ Any $L_{p}(X)$-space which is a retract of a well-composed
$L_{p}^{0}(X)$-space is metrically projective.

$(iv)$ Any $L_{p}^{0}(X)$-space $(L_{p}(X)$-space$)$ is an image of a well-
composed $L_{p}^{0}(X)$-space under an $L_{p}^{0}(X)$-strict coisometry
$($respectively, $L_{p}(X)$-coisometry$)$.

Proposition 3.12 below shows that, in part (iv), an $L_{p}(X)$-space need not
be a strictly coisometric image of a well-composed space.

###### Proof.

(i) follows from Theorem 3.7(i,ii).

(ii) By Lemma 3.8(i,ii), $P$ is extremely projective as an $L_{p}(X)$-space if
and only if it is extremely projective as an $L_{p}^{0}(X)$-space. By part (i)
of this theorem, the last condition is equivalent to being a near-retract of a
well-composed $L_{p}^{0}(X)$-space. By Lemma 3.8(iii), the notions of being a
retract or near-retract in the categories of $L_{p}(X)$-spaces and
$L_{p}^{0}(X)$-spaces coincide.

(iii) Suppose $E,G,P$ are $L_{p}(X)$-spaces, $P$ is a retract of a well-
composed $L_{p}^{0}(X)$-space, $\varphi:P\to E$ is an $L_{p}(X)$-contraction,
and $\tau:G\to E$ is an $L_{p}(X)$-strict coisometry. By Lemma 3.8(i), $\tau$
is an $L_{p}^{0}(X)$-strict coisometry as well. Combining part (i) of this
theorem with Lemma 3.8(iii), we conclude that $\varphi$ has an
$L_{p}^{0}(X)$-contractive lifting $\psi:P\to G$. To complete the proof, note
that $\psi$ must be $L_{p}(X)$-contractive as well.

In (iv), the first statement is a consequence of Theorem 3.7(iii). The second
one follows by observing that any $L_{p}^{0}(X)$-strict coisometry is an
$L_{p}(X)$-coisometry. ∎

######  Remark 3.11.

We do not know whether a metrically projective $L_{p}(X)$-space is necessarily
metrically projective in the category of $L_{p}^{0}(X)$-spaces. The following
example shows that an $L_{p}^{0}(X)$-strict coisometry may fail to be an
$L_{p}(X)$-strict coisometry.

Let $I=[0,2\pi)$. Equip $G=\ell_{1}(I)$ and $E=\ell_{2}^{2}$ with an
$L_{1}$-structure arising from $L_{1}(G)$ and $L_{1}(E)$ respectively (we use
$L_{1}$ and $L_{1}^{0}$ for $L_{1}(0,1)$ and $L_{1}^{0}(0,1)$). These are
indeed $L_{1}$-spaces, by [2, Sections 1.8-9]. Denote the canonical bases of
$G$ and $E$ by $g_{t}$ ($t\in I$) and $e_{1},e_{2}$, respectively.

Define $\tau:G\to E:g_{t}\mapsto\cos te_{1}+\sin te_{2}$, and note it is a
strict coisometry on the Banach space level. Indeed, any $x\in E$ can be
written as $\|x\|(\cos te_{1}+\sin te_{2})$, with some $t\in I$. Then
$x=\tau(\|x\|g_{t})$.

Next show that $\tau$ is $L_{1}^{0}$-strictly coisometric. Indeed, any element
$u\in L_{1}^{0}\otimes E$ can be written as $u=\sum_{k}{\bf 1}_{A_{k}}\otimes
x_{k}$, with $x_{k}\in E$, and $A_{k}$ being disjoint measurable subsets of
$(0,1)$. Then $\|u\|=\sum_{k}\lambda(A_{k})\|x_{k}\|$, where $\lambda$ is the
Lebesgue measure. For each $k$ find $y_{k}\in G$ with $\|x_{k}\|=\|y_{k}\|$,
$\tau y_{k}=x_{k}$. Let $v=\sum_{k}{\bf 1}_{A_{k}}\otimes x_{k}$, and note
that $\tau_{\infty}v=u$, and $\|v\|=\|u\|$.

To show that $\tau$ is not $L_{1}$-strictly coisometric, note first that, if
$\|y\|=1$, and $\tau y=\cos te_{1}+\sin te_{2}$ (for some $t\in I$), then
$y=g_{t}$. Indeed, write $y=\sum_{s\in I}\alpha_{s}g_{s}$, where
$\sum_{s}|\alpha_{s}|=1$ (hence $\alpha_{s}=0$ for at most countably many
values of $s$). Then

$1=\langle\cos te_{1}+\sin te_{2},\tau y\rangle+\sum_{s}\alpha s\langle\cos
te_{1}+\sin te_{2},\cos se_{1}+\sin
se_{2}\rangle=\sum_{s}\alpha_{s}\cos(t-s).$

The only way for this equality to hold is to have $\alpha_{t}=1$,
$\alpha_{s}=0$ for $s\neq t$.

Now consider $u=\xi_{1}\otimes e_{1}+\xi_{2}\otimes e_{2}$, where
$\xi_{1}(s)=\cos 2\pi s$ and $\xi_{2}(s)=\sin 2\pi s$ for $s\in(0,1)$. Thus,
$u(s)=\cos(2\pi s)e_{1}+\sin(2\pi s)e_{2}$ for any $s$. Consequently,
$\|u\|_{L_{1}(E)}=\int_{0}^{1}\|u(s)\|\,ds=1$. We shall show that $u$ has no
norm $1$ lifting.

Suppose, for the sake of contradiction, that there exists
$v=\sum_{k=1}^{N}\xi_{k}\otimes y_{k}\in L_{1}\otimes G$ so that
$\tau_{\infty}v=u$, and $\|v\|=1$. In the coordinate form, we must have that
$\|\sum_{k}\xi_{k}(s)y_{k}\|=1$, and
$\tau\big{(}\sum_{k}\xi_{k}(s)y_{k}\big{)}=u(s)=\cos(2\pi s)e_{1}+\sin(2\pi
s)e_{2}$, for any $s\in{\mathcal{S}}$, where ${\mathcal{S}}$ has measure $1$.
Consequently, $\sum_{k}\xi_{k}(s)y_{k}=g_{2\pi s}$, for any
$s\in{\mathcal{S}}$. This, however, is impossible, as each $y_{k}$ has
countable support in $I$.

We close this paper by an example indicating that the category of
$L_{p}(X)$-spaces may not be the “right” one.

######  Proposition 3.12.

There exists an $L_{1}(0,1)$-space which cannot be represented as an
$L_{1}(0,1)$-strictly coisometric image of a well-composed
$L_{1}^{0}(0,1)$-space.

###### Proof.

Denote by $E$ the space $C({\mathbb{T}})$, where ${\mathbb{T}}$ is the unit
circle, equipped with its minimal $L_{1}(0,1)$ structure – that is, the norm
on $L_{1}(0,1)\otimes C({\mathbb{T}})\simeq C({\mathbb{T}},L_{1}(0,1))$ comes
from the injective tensor product. Consider normalized independent Gaussian
random variables $g_{1},g_{2}\in L_{1}(0,1)$, and the coordinate functions
$f_{1},f_{2}\in C({\mathbb{T}})$. Let $u=g_{1}\otimes f_{1}+g_{2}\otimes
f_{2}$. Then $\|u\|=1$. In fact, $u$ corresponds to the weak∗ to norm
continuous operator

$\tilde{u}:L_{\infty}(0,1)\to C({\mathbb{T}}):h\mapsto\Big{(}\int
hg_{1}\Big{)}f_{1}+\Big{(}\int hg_{2}\Big{)}f_{2}.$

One can write $\tilde{u}=jq$; here $j:\ell_{2}^{2}\to
C({\mathbb{T}}):\delta_{i}\mapsto f_{i}$ ($\delta_{1},\delta_{2}$ form the
canonical basis in $\ell_{2}^{2}$) is an isometric embedding, and $q$ is the
strictly coisometric adjoint of $q_{*}:\ell_{2}^{2}\to L_{1}:\delta_{i}\mapsto
g_{i}$. Consequently, $\tilde{u}$ maps the closed unit ball of $L_{\infty}$
onto the closed unit ball of $j(\ell_{2}^{2})\subset L_{1}$.

Suppose, for the sake of contradiction, that there exists an
$L_{1}(0,1)$-strict coisometry $\tau$ from an $\ell_{1}$ sum
$\oplus{\rm{MIN}}(\ell_{\infty}^{n_{i}})$ onto $C({\mathbb{T}})$. Find $v\in
L_{1}(0,1)\otimes\oplus{\rm{MIN}}(\ell_{\infty}^{n_{i}})$ so that $\|v\|=1$,
and ${\bf 1}_{L_{1}}\otimes\tau(v)=u$. As $\oplus$ denotes the algebraic sum,
we can write $v$ as a finite sum $\oplus_{i\in I}v_{i}$, with $v_{i}\in
L_{1}(0,1)\otimes\ell_{\infty}^{n_{i}}$, and $\|v\|=\sum_{i}\|v_{i}\|$. Based
on $v$, we construct an operator $\tilde{v}$ from $L_{\infty}(0,1)$ into the
finite dimensional space $Z=\big{(}\oplus_{i\in
I}\ell_{\infty}^{n_{i}}\big{)}_{1}$. The equality $u=({\bf
1}_{L_{1}}\otimes\tau)v$ translates into $\tilde{u}=\tau|_{Z}\tilde{v}$.

As noted above, for every $\eta\in\mathrm{ran\,}\tilde{u}$ we can find $\xi\in
L_{\infty}(0,1)$ so that $\|\xi\|=\|\eta\|$, and $\tilde{u}\xi=\eta$. Now let
$Y=\mathrm{ran\,}\tilde{v}\subset Z$; then,
$\tau(Y)\subset\mathrm{ran\,}\tilde{u}$, and for every
$\eta\in\mathrm{ran\,}\tilde{u}$ we can find $y\in Y$ so that
$\|y\|=\|\eta\|$, and $\tau y=\eta$. In other words, $\tau$ acts as a strict
coisometry from $Y$ onto $\mathrm{ran\,}\tilde{u}\simeq\ell_{2}^{2}$. However,
the unit ball of $Z$ is a polytope, hence the same is true for $Y$. As a
circle is not a cross-section of a polytope, no coisometry from $Y$ onto a
copy of $\ell_{2}^{2}$ exists. ∎

## References

  * [1] D. P. Blecher, The standard dual of an operator space, Pacific J. Math., 153 (1992), 15–30.
  * [2] H. G. Dales, N. J. Laustsen, T. Oikhberg, V. G. Troitsky, Multi-norms and Banach lattices, Dissertationes Math. (Rozprawy Mat.) 524, 115 pp. (2017)
  * [3] E. G. Effros, Z.-J. Ruan, Operator Spaces, Clarendon Press, Oxford, 2000.
  * [4] A. W. M. Graven, Injective and Projective Banach modules, Indagationes Mathematicae, 82 (1979) 253–272.
  * [5] A. Grothendieck, Une caractérisation vectorielle-métrique des espaces $L^{1}$, Canad. J. Math. 7 (1955) 552–561.
  * [6] A. Ya. Helemskii, Multinormed spaces, based on non–discrete measures, and their tensor products, Izvestiya RAN: Ser. Mat. 82:2 (2018) 23–25.
  * [7] A. Ya. Helemskii, Metric freeness and projectivity for classical and quantum normed modules, Sb. Math, 204 (2013), 1056–1083.
  * [8] A. Ya. Helemskii, Projectivity for operator modules: approach based on freedom, Rev. Roum. Math. Pures Appl, 53:2 (2014) 219–236.
  * [9] A. Ya. Helemskii, Projective and free matricially normed spaces. In Banach algebras and applications, De Gruyter, Berlin, 2020, 151–164.
  * [10] N. Kalton, E. Lorist, and L. Weis, Euclidean structures and operator theory in Banach spaces. Mem. Amer. Math. Soc., to appear, https://arxiv.org/abs/1912.09347.
  * [11] A. Lambert, Operatorfolgenräume, Dissertation, Saarbrüken (2002).
  * [12] J. Lindenstrauss, L. Tzafriri, Classical Banach Spaces I and II. Sequence Spaces; Function Spaces. Springer-Verlag, Berlin, 1996.
  * [13] S. Mac Lane, Categories for the Working Mathematician. Springer-Verlag, Berlin, 1971.
  * [14] T. Oikhberg, Injectivity and projectivity in $p$-multinormed spaces. Positivity 22 (2018), 1023–1037.
  * [15] B. Randrianantoanina. Norm-one projections in Banach spaces. Taiwanese J. Math. 5 (2001), 35–95.
  * [16] R. A. Ryan, Introduction to Tensor Products of Banach Saces. Springer-Verlag, Berlin, 2002.
  * [17] Z. Semadeni, Projectivity, injectivity and duality, Rozprawy Mat. 35 (1963) 1–47.
  * [18] L. Tzafriri. Remarks on contractive projections in $L_{p}$-spaces. Israel J. Math. 7 (1969) 9–15.