A note on universal operators between separable Banach spaces

Abstract

We compare two types of universal operators constructed relatively recently by Cabello Sánchez, and the authors. The first operator ${\Omega }$ acts on the Gurariĭ space, while the second one ${\mathbf {P}}_{{{\mathbb {S}}}}$ has values in a fixed separable Banach space ${{\mathbb {S}}}$. We show that if ${{\mathbb {S}}}$ is the Gurariĭ space, then both operators are isometric. We also prove that, for a fixed space ${{\mathbb {S}}}$, the operator ${\mathbf {P}}_{{{\mathbb {S}}}}$ is isometrically unique. Finally, we show that ${\Omega }$ is generic in the sense of a natural infinite game.

1 Universal operators

The purpose of this note is to discuss two constructions of universal operators between separable Banach spaces. We are interested in isometric universality. Namely, an operator U is universal if its restrictions to closed subspaces are, up to linear isometries, all linear operators of norm not exceeding $\Vert U\Vert$. To be more precise, a bounded linear operator $U:V \rightarrow W$ acting between separable Banach spaces is universal if for every linear operator $T:X \rightarrow Y$ with X, Y separable and $\Vert T\Vert \le \Vert U\Vert$, there exist linear isometric embeddings $i:X \rightarrow V$, $j:Y \rightarrow W$ such that the diagram

is commutative, that is, $U \circ \, i = j \circ T$. Such an operator has been relatively recently constructed by the authors [5]. Another recent work [2], due to Cabello Sánchez and the present authors, contains in particular a construction of a linear operator that is universal in a different sense. Namely, let us say that a bounded linear operator $U:V \rightarrow W$ is left-universal (for operators into W) if for every linear operator $T:X \rightarrow W$ with X separable and $\Vert T\Vert \le \Vert U\Vert$ there exists a linear isometric embedding $i:X \rightarrow V$ for which the diagram

is commutative, that is, $U \circ i = T$. Note that if W is separable, then U is right-invertible (i.e. a projection), as one can take the identity of W in place of T. Clearly, if W is isometrically universal in the class of all separable Banach spaces then a left-universal operator with values into W is universal. The left-universal operator U constructed in [2] had been later essentially used (with a suitable space W) for finding an isometrically universal graded Fréchet space [1]. There exist other concepts of universality in operator theory, see the introduction of [5] for more details and references.

Let us note the following simple facts related to universal operators.

Proposition 1.1

Let $U:V \rightarrow W$ be a bounded linear operator acting between separable Banach spaces.

• (1) If U is universal then both V and W are isometrically universal among the class of separable Banach spaces.

• (2) Assume U is left-universal. Then $\ker U$ is isometrically universal among the class of separable Banach spaces. Furthermore, U is right-invertible, that is, there exists an isometric embedding $e:W \rightarrow V$ such that $U \circ e = {\text {id}_{W}}$.

• (3) Assume U is (left-)universal. Then ${\lambda }U$ is (left-)universal for every ${\lambda }>0$.

Proof

(1) Fix a separable Banach space X. Taking the zero operator $T:X \rightarrow 0$, we see that V contains an isometric copy of X. Taking the identity ${\text {id}_{X}}$, we see that W contains an isometric copy of X.

(2) The same argument as above, using the zero operators, shows that $\ker U$ is isometrically universal. Taking the identity ${\text {id}_{W}}$, we obtain the required isometric embedding $e:W \rightarrow V$.

(3) Assume U is universal, fix ${\lambda }>0$ and fix $T:X \rightarrow Y$ with $\Vert T\Vert \le {\lambda }\Vert U\Vert$. Then $\Vert {\lambda }^{-1}T\Vert \le \Vert U\Vert$, therefore there are isometric embeddings $i:X \rightarrow V$, $j:Y \rightarrow W$ such that $U \circ i = j \circ ({\lambda }^{-1} T)$. Finally, $({\lambda }U) \circ i = j \circ T$. If U is left-universal, the argument is the same, the only difference is that $j = {\text {id}_{W}}$. $\square$

By (3) above, we may restrict attention to non-expansive operators. It turns out that there is an easy way of constructing left-universal operators, once we have in hand an isometrically universal space. The argument below was pointed out to us by Przemysław Wojtaszczyk.

Example 1.2

Let V be an isometrically universal Banach space and let W be an arbitrary Banach space. Consider $V \oplus W$ with the maximum norm and let

$$\begin{aligned} \pi :V \oplus W \rightarrow W \end{aligned}$$

be the canonical projection. Given a non-expansive operator $T:X \rightarrow W$ with X separable, choose an isometric embedding $e:X \rightarrow V$ and define $j:X \rightarrow V \oplus W$ by $j(x) = (e(x), T(x))$. Then j is an isometric embedding and $\pi \circ j = T$, showing that $\pi$ is left-universal. Of course, if additionally W is isometrically universal, then $\pi$ is a universal operator.

Perhaps the most well known universal Banach space is ${{\mathcal {C}}}([0,1])$, the space of all continuous (real or complex) valued functions on the unit interval, endowed with the maximum norm. In view of the example above, there exists a universal operator from ${{\mathcal {C}}}([0,1]) \oplus {{\mathcal {C}}}([0,1])$ onto ${{\mathcal {C}}}([0,1])$. This leads to (at least potentially) many other universal operators, namely:

Proposition 1.3

Let V, W be isometrically universal separable Banach spaces. Then there exists a universal operator from V into W.

Proof

Fix a universal operator $\pi :E \rightarrow F$ (for instance, $E = {{\mathcal {C}}}([0,1]) \oplus {{\mathcal {C}}}([0,1])$ and $F = {{\mathcal {C}}}([0,1])$) and fix a linear isometric embedding $e:E \rightarrow V$. Using the amalgamation property for linear operators (see, e.g. [4, Lemma 1.2] where it is proved for almost isometries, however exactly the same construction works for arbitrary operators), we find a separable Banach space $V'$, a linear isometric embedding $e':F \rightarrow V'$, and a non-expansive linear operator $\Omega :V \rightarrow V'$ for which the diagram

is commutative. As W is isometrically universal, we may additionally assume that $V' = W$, replacing $\Omega$ by $i \circ \Omega$ and $e'$ by $i \circ e'$, where i is a fixed isometric embedding of $V'$ into W. It is evident that now $\Omega$ is a universal operator, because of the universality of $\pi$. $\square$

As a consequence, there exists a universal operator on ${{\mathcal {C}}}([0,1])$. We do not know whether there exists a left-universal operator on ${{\mathcal {C}}}([0,1])$. The situation changes when replacing [0, 1] with the Cantor set $2^{{\mathbb {N}}}$. The space ${{\mathcal {C}}}(2^{{\mathbb {N}}})$ is linearly isomorphic (but not isometric) to ${{\mathcal {C}}}([0,1])$ and it is isometrically universal, too. Furthermore, ${{\mathcal {C}}}(2^{{\mathbb {N}}}) \oplus {{\mathcal {C}}}(2^{{\mathbb {N}}})$ with the maximum norm is linearly isometric to ${{\mathcal {C}}}(2^{{\mathbb {N}}})$, because the disjoint sum of two copies of the Cantor set is homeomorphic to the Cantor set. Thus, Example 1.2 provides a left-universal operator on ${{\mathcal {C}}}(2^{{\mathbb {N}}})$.

Another, not so well known, universal Banach space is the Gurariĭ space. This is the unique, up to a linear isometry, separable Banach space ${{\mathbb {G}}}$ satisfying the following condition:

• For every $\varepsilon >0$, for every finite-dimensional spaces $X_0 \subseteq X$, for every linear isometric embedding $f_0:X_0 \rightarrow {{\mathbb {G}}}$ there exists a linear $\varepsilon$-isometric embedding $f:X \rightarrow {{\mathbb {G}}}$ such that $f \restriction X_0 = f_0$.

By an $\varepsilon$-isometric embedding (briefly: $\varepsilon$-embedding) we mean a linear operator f satisfying

$$\begin{aligned} (1-\varepsilon )\Vert x\Vert \le \Vert f(x)\Vert \le (1+\varepsilon )\Vert x\Vert \end{aligned}$$

for every x in the domain of f. The space ${{\mathbb {G}}}$ was constructed by Gurariĭ [6]; its uniqueness was proved by Lusky [9].

The universal operator constructed in [5] has a special property that actually makes it unique, up to linear isometries. Below we quote the precise result.

Theorem 1.4

([5]) There exists a non-expansive linear operator ${\Omega }:{{\mathbb {G}}} \rightarrow {{\mathbb {G}}}$ with the following property:

• (G) Given $\varepsilon >0$, given a non-expansive operator $T:X \rightarrow Y$ between finite-dimensional spaces, given $X_0 \subseteq X$, $Y_0 \subseteq Y$ and isometric embeddings $i:X_0 \rightarrow {{\mathbb {G}}}$, $j:Y_0 \rightarrow {{\mathbb {G}}}$ such that ${\Omega }\circ i = j \circ (T\restriction X_0)$, there exist $\varepsilon$-embeddings $i':X \rightarrow {{\mathbb {G}}}$, $j':Y \rightarrow {{\mathbb {G}}}$ satisfying

  $$\begin{aligned} \Vert i'\restriction X_0 - i\Vert \le \varepsilon , \quad \Vert j'\restriction Y_0 - j\Vert \le \varepsilon , \qquad \text {and}\qquad \Vert {\Omega }\circ i' - j' \circ T\Vert \le \varepsilon . \end{aligned}$$

Furthermore, ${\Omega }$ is a universal operator and property (G) specifies it uniquely, up to a linear isometry.

According to [5], we shall call condition (G) the Gurariĭ property. What makes this operator of particular interest is perhaps its almost homogeneity:

Theorem 1.5

([5]) Given finite-dimensional subspaces $X_0,X_1,Y_0,Y_1$ of ${{\mathbb {G}}}$, given linear isometries $i:X_0 \rightarrow X_1$, $j:Y_0 \rightarrow Y_1$ such that ${\Omega }\circ i = j \circ {\Omega }$, for every $\varepsilon >0$ there exist bijective linear isometries $I:{{\mathbb {G}}} \rightarrow {{\mathbb {G}}}$, $J:{{\mathbb {G}}} \rightarrow {{\mathbb {G}}}$ satisfying ${\Omega }\circ I = J \circ {\Omega }$ and $\Vert I \restriction X_0 - i\Vert < \varepsilon$, $\Vert J \restriction Y_0 - j\Vert < \varepsilon$.

We now describe the left-universal operators constructed in [2]. Fix a separable Banach space ${{\mathbb {S}}}$.

Theorem 1.6

([2, Section 6]) There exists a non-expansive linear operator ${\mathbf {P}}_{{{\mathbb {S}}}}:V_{{\mathbb {S}}} \rightarrow {{\mathbb {S}}}$ with $V_{{\mathbb {S}}}$ a separable Banach space, satisfying the following condition:

$(\ddagger )$:

For every finite-dimensional spaces $X_0 \subseteq X$, for every non-expansive linear operator $T:X \rightarrow {{\mathbb {S}}}$, for every linear isometric embedding $e:X_0 \rightarrow V_{{\mathbb {S}}}$ such that  ${\mathbf {P}}_{{{\mathbb {S}}}}\circ e \!=\! T \restriction X_0$, for every $\varepsilon >0$ there exists an $\varepsilon$-embedding $f:X \rightarrow V_{{\mathbb {S}}}$ satisfying

$$\begin{aligned} \Vert f \restriction X_0 - e\Vert \le \varepsilon \qquad \text {and} \qquad \Vert {\mathbf {P}}_{{{\mathbb {S}}}}\circ f - T\Vert \le \varepsilon . \end{aligned}$$

Furthermore, ${\mathbf {P}}_{{{\mathbb {S}}}}$ is left-universal for operators into ${{\mathbb {S}}}$.

We shall say that an operator P has the left-Gurariĭ property if it satisfies ($\ddagger$) in place of ${\mathbf {P}}_{{{\mathbb {S}}}}$. Of course, unlike the Gurariĭ property, the left-Gurariĭ property involves a parameter ${{\mathbb {S}}}$, namely, the common range of the operators. The left-universality of ${\mathbf {P}}_{{{\mathbb {S}}}}$ implies that it is right-invertible, i.e., there is an isometric embedding $J:{{\mathbb {S}}} \rightarrow V_{{\mathbb {S}}}$ such that ${\mathbf {P}}_{{{\mathbb {S}}}}\circ J = {\text {id}_{{{\mathbb {S}}}}}$. The operator $J \circ {\mathbf {P}}_{{{\mathbb {S}}}}$ is a non-expansive projection of $V_{{\mathbb {S}}}$ onto an isometric copy of ${{\mathbb {S}}}$.

Actually, the operator ${\mathbf {P}}_{{{\mathbb {S}}}}$ was constructed in [2] in case where ${{\mathbb {S}}}$ had some additional property, needed only for determining the domain of ${\mathbf {P}}_{{{\mathbb {S}}}}$. Moreover, [2] deals with p-Banach spaces, where $p \in (0, 1]$, however $p=1$ gives exactly the result stated above. Operators ${\mathbf {P}}_{{{\mathbb {S}}}}$ have the following property which can be called almost left-homogeneity.

Theorem 1.7

Given finite-dimensional subspaces $X_0, X_1$ of $V_{{\mathbb {S}}}$, a linear isometry $h:X_0 \rightarrow X_1$ such that ${\mathbf {P}}_{{{\mathbb {S}}}} \circ h = {\mathbf {P}}_{{{\mathbb {S}}}} \restriction X_0$, for every $\varepsilon >0$ there exists a bijective linear isometry $H:V_{{\mathbb {S}}} \rightarrow {{\mathbb {S}}}$ satisfying ${\mathbf {P}}_{{{\mathbb {S}}}} \circ H = {\mathbf {P}}_{{{\mathbb {S}}}}$ and $\Vert H \restriction X_0 - h\Vert < \varepsilon$.

Two operators U, V are isometric if there are bijective linear isometries i, j such that $U \circ j = i \circ V$. In this note we present a proof that condition $(\ddagger )$ determines ${\mathbf {P}}_{{{\mathbb {S}}}}$ uniquely, up to linear isometries. The arguments will also provide a proof of Theorem 1.7. Furthermore, we show that ${\Omega }= {\mathbf {P}}_{{{\mathbb {G}}}}$ and that ${\Omega }$ is a generic operator in the space of all non-expansive operators on the Gurariĭ space into itself, in the sense of a natural variant of the Banach-Mazur game.

2 Properties of ${\Omega }$ and ${\mathbf {P}}_{{{\mathbb {S}}}}$

Let us recall the following easy fact concerning finite-dimensional normed spaces (cf. [4, Thm. 2.7] or [1, Claim 2.3]). It actually says that the strong operator topology is equivalent to the norm topology in the space of linear operators with a fixed finite-dimensional domain.

Lemma 2.1

Let A be a vector basis of a finite-dimensional normed space E. For every $\varepsilon >0$ there exists $\delta >0$ such that for every Banach space X, for every linear operator $f:E \rightarrow X$ the following implication holds:

$$\begin{aligned} \max _{a \in A} \Vert f(a)\Vert \le \delta \implies \Vert f\Vert \le \varepsilon . \end{aligned}$$

Proof

Fix $M>0$ satisfying the following condition:

(*):

$\max _{a\in A}|{\lambda }_a| \le M$ whenever $x = \sum _{a \in A}{\lambda }_a a$ and $\Vert x\Vert \le 1$.

Such M clearly exists, because of compactness of the unit ball of E. Now, given $\varepsilon >0$, let $\delta = \varepsilon /(M\cdot |A|)$. Suppose $\max _{a\in A}\Vert f(a)\Vert \le \delta$. Then, given $x = \sum _{a \in A}{\lambda }_a a$ with $\Vert x\Vert \le 1$, we have

$$\begin{aligned} \Vert f(x)\Vert \le \sum _{a \in A}|{\lambda }_a| \cdot \Vert f(a)\Vert \le |A| \cdot M \cdot \delta = \varepsilon . \end{aligned}$$

We conclude that $\Vert f\Vert \le \varepsilon$. $\square$

The following result, in the case ${{\mathbb {S}}}= {{\mathbb {G}}}$, can be found in [1].

Theorem 2.2

Let $P:V \rightarrow {{\mathbb {S}}}$ be a linear operator. The following conditions are equivalent.

• (a) P has the left-Gurariĭ property ($\ddagger$).

• (b) For every finite-dimensional spaces $X_0 \subseteq X$, for every non-expansive linear operator $T:X \rightarrow {{\mathbb {S}}}$, for every linear isometric embedding $e:X_0 \rightarrow V$ such that $P \circ e = T \restriction X_0$, for every $\varepsilon >0$ there exists an $\varepsilon$-embedding $f:X \rightarrow V$ satisfying

  $$\begin{aligned} {f \restriction X_0 = e} \qquad \text {and} \qquad {P \circ f = T}. \end{aligned}$$

Proof

Obviously, (b) is stronger than ($\ddagger$).

Fix $\varepsilon >0$ and fix a vector basis A of X such that $A_0 = X_0 \cap A$ is a basis of $X_0$. We may assume that $\Vert a\Vert =1$ for every $a \in A$. Fix $\delta >0$ and apply the left-Gurariĭ property for $\delta$ instead of $\varepsilon$. We obtain a $\delta$-embedding $f:X \rightarrow V$ such that $\Vert f\restriction X_0 - e\Vert \le \delta$ and $\Vert P \circ f - T\Vert \le \delta$. Define $f':X \rightarrow V$ by the conditions $f'(a) = e(a)$ for $a \in A_0$ and $f'(a) = f(a)$ for $a \in A \setminus A_0$. Note that $\Vert f'(a)-f(a)\Vert \le \delta$ for every $a \in A$. Thus, if $\delta$ is small enough, then by Lemma 2.1, we can obtain that $f'$ is an $\varepsilon$-embedding. Furthermore, $\Vert P \circ f' - P \circ f\Vert \le \varepsilon$ (recall that $\delta$ depends on $\varepsilon$ and the norm of X only), therefore $\Vert P \circ f' - T\Vert \le \varepsilon + \delta$.

The arguments above show that for every $\varepsilon >0$ there exists an $\varepsilon$-embedding $f':X \rightarrow V$ extending e and satisfying $\Vert P \circ f' - T\Vert \le \varepsilon$.

Let us apply this property for $\delta$ instead of $\varepsilon$, where $\delta$ is taken from Lemma 2.1. We obtain a $\delta$-embedding $f:X \rightarrow V$ extending e and satisfying $\Vert P \circ f - T\Vert \le \delta$.

Now recall that P is left-universal (Theorem 1.6), therefore there is an isometric embedding $J:{{\mathbb {S}}} \rightarrow V$ such that $P \circ J$ is the identity (take the identity of ${{\mathbb {S}}}$ in the definition of left-universality). Given $a \in A \setminus A_0$, the vector

$$\begin{aligned} w_a = P(f(a)) - T(a) \end{aligned}$$

has norm $\le \delta$. Define $f':X \rightarrow V$ by the conditions $f'\restriction X_0 = e$ and

$$\begin{aligned} f'(a) = f(a) - J(w_a) \end{aligned}$$

for $a \in A \setminus A_0$. Lemma 2.1 implies that $f'$ is an $\varepsilon$-embedding, because $\Vert f'(a) - f(a)\Vert = \Vert w_a\Vert \le \delta$ for $a \in A \setminus A_0$. Finally, given $a \in A \setminus A_0$, we have

$$\begin{aligned} P f' (a) = P f (a) - w_a = T(a) \end{aligned}$$

and the same obviously holds for $a \in A_0$. Thus $P \circ f' = T$. $\square$

The proof of the next result is just a suitable adaptation of the arguments above, therefore we skip it.

Proposition 2.3

Let $\Omega :V \rightarrow W$ be a linear operator. The following conditions are equivalent.

• (a) $\Omega$ has the Gurariĭ property (G).

• (b) Given $\varepsilon >0$, given a non-expansive operator $T:X \rightarrow Y$ between finite-dimensional spaces, given $X_0 \subseteq X$, $Y_0 \subseteq Y$ and isometric embeddings $i_0:X_0 \rightarrow V$, $j_0:Y_0 \rightarrow W$ such that $\Omega \circ i_0 = j_0 \circ (T\restriction X_0)$, there exist $\varepsilon$-embeddings $i:X \rightarrow V$, $j:Y \rightarrow W$ satisfying

  $$\begin{aligned} i\restriction X_0 = i_0, \quad j\restriction Y_0 = j_0, \qquad \text {and}\qquad \Omega \circ i = j \circ T. \end{aligned}$$

The last result of this section is the key step towards identifying ${\Omega }$ with ${\mathbf {P}}_{{{\mathbb {G}}}}$.

Theorem 2.4

The operator ${\Omega }$ has the left-Gurariĭ property (i.e., it satisfies condition $(\ddagger )$ of Theorem 1.6 with ${{\mathbb {S}}}= {{\mathbb {G}}}$). In particular, it is left-universal.

Proof

Fix a non-expansive linear operator $T:X \rightarrow {{\mathbb {G}}}$ with X finite-dimensional, and fix an isometric embedding $e:X_0 \rightarrow {{\mathbb {G}}}$, where $X_0$ is a linear subspace of X and $T\restriction X_0 = {\Omega }\circ e$. Let $Y_0 = Y = T[X] \subseteq {{\mathbb {G}}}$ and consider T as an operator from X to Y. Applying the Gurariĭ property with $i = e$ and j the inclusion $Y_0 \subseteq {{\mathbb {G}}}$, we obtain an $\varepsilon$-embedding $e':X \rightarrow {{\mathbb {G}}}$ which is $\varepsilon$-close to e and satisfies $\Vert {\Omega }\circ e' - T\Vert \le \varepsilon$. This is precisely condition ($\ddagger$) from Theorem 1.6. $\square$

In order to conclude that ${\Omega }= {\mathbf {P}}_{{{\mathbb {G}}}}$, it remains to show that ($\ddagger$) determines the operator uniquely. This is done in the next section.

3 Uniqueness of ${\mathbf {P}}_{{{\mathbb {S}}}}$

Before proving that the left-Gurariĭ property determines the operator uniquely, we quote the following crucial lemma from [3].

Lemma 3.1

Let $\varepsilon >0$ and let $f:X \rightarrow Y$ be an $\varepsilon$-embedding, where X, Y are Banach spaces. Let $\pi :X \rightarrow {{\mathbb {S}}}$, $\varrho :Y \rightarrow {{\mathbb {S}}}$ be non-expansive linear operators such that $\Vert \varrho \circ f - \pi \Vert \le \varepsilon$. Then there exists a norm on $Z = X \oplus Y$ such that the canonical embeddings $i:X \rightarrow Z$, $j:Y \rightarrow Z$ are isometric, $\Vert j \circ f - i\Vert \le \varepsilon$ and the operator $t:Z \rightarrow {{\mathbb {S}}}$ defined by $t(x,y) = \pi (x) + \varrho (y)$ is non-expansive.

Note that the operator t satisfies $t \circ i = \pi$ and $t \circ j = \varrho$. Actually, the norm mentioned in the lemma above does not depend on the operators $\pi$, $\varrho$. It is defined by the following formula:

where $\Vert \cdot \Vert _X$, $\Vert \cdot \Vert _Y$ denote the norm of X and Y, respectively. An easy exercise shows that ($*$) is the required norm, proving Lemma 3.1.

Theorem 3.2

Let ${{\mathbb {S}}}$ be a separable Banach space and let $\pi :E \rightarrow {{\mathbb {S}}}$, $\pi ':E' \rightarrow {{\mathbb {S}}}$ be non-expansive linear operators, both with the left-Gurariĭ property. If E, $E'$ are separable Banach spaces, then there exists a linear isometry $i:E \rightarrow E'$ such that $\pi = \pi ' \circ i$. In particular, $\pi$ and $\pi '$ are linearly isometric to ${\mathbf {P}}_{{{\mathbb {S}}}}$.

Proof

It suffices to prove the following

Claim 3.3

Let $E_0 \subseteq E$ be a finite-dimensional space, $0<\varepsilon <1$, let $i_0:E_0 \rightarrow E'$ be an $\varepsilon$-embedding such that $\pi ' \circ i_0 = \pi \restriction E_0$. Then for every $v \in E$, $v' \in E'$, for every $\eta >0$ there exists an $\eta$-embedding $i_1:E_1 \rightarrow E'$ such that $E_1$ is finite-dimensional, $E_0 \subseteq E_1 \subseteq E$ and the following conditions are satisfied:

• (1) $v \in E_1$ and ${\text {dist}}(v',i_1[E_1]) < \eta$;

• (2) $\Vert i_0 - i_1 \restriction E_0\Vert < \varepsilon + \eta$ and $\pi ' \circ i_1 = \pi$.

Using Claim 3.3 together with the separability of E and $E'$, we can construct a sequence $i_n:E_n \rightarrow E'$ of linear operators such that $i_n$ is a $2^{-n}$-embedding, $\bigcup _{n\in \omega }E_n$ is dense in E, $\bigcup _{n\in \omega }i_n[E_n]$ is dense in $E'$ and

$$\begin{aligned} \Vert i_n - i_{n+1} \restriction E_n\Vert \le 2^{-n} + 2^{-n-1} \qquad \text {and}\qquad \pi ' \circ i_{n+1} = \pi \end{aligned}$$

for every ${n\in \omega }$. It is evident that ${\{{i}_n\}_{{n\in \omega }}}$ converges pointwise to a linear isometry whose completion i is the required bijection from E onto $E'$ satisfying $\pi ' \circ i = \pi$. Thus, it remains to prove Claim 3.3.

This will be carried out by making two applications of Lemma 3.1.

Fix $0<\delta <1$; more precise estimations for $\delta$ will be given later. Let $E_0' \subseteq E'$ be a finite-dimensional space containing $v'$ and such that $i_0[E_0] \subseteq E_0'$. Applying Lemma 3.1, we obtain linear isometric embeddings $e_1:E_0 \rightarrow W_0$, $f_1:E_0' \rightarrow W_0$ and a non-expansive operator $t_0:W_0 \rightarrow {{\mathbb {S}}}$ such that $t_0 \circ e_1 = \pi \restriction E_0$, $t_0 \circ f_1 = \pi ' \restriction E_0'$, and $\Vert e_1 - f_1 \circ i_0\Vert \le \varepsilon$. Knowing that $\pi$ has the left-Gurariĭ property, by Theorem 2.2 applied to the isometric embedding $e_1$, we obtain a $\delta$-embedding $g_1:W_0 \rightarrow E$ such that $g_1 \circ e_1$ is the identity on $E_0$ and $\pi \circ g_1 = t_0$.

Now note that $g_1 \circ f_1$ is a $\delta$-embedding of $E_0'$ into a finite-dimensional subspace $E_1$ of E. Without loss of generality, we may assume that $v \in E_1$. Applying Lemma 3.1 again to $g_1 \circ f_1$, we obtain linear isometric embeddings $e_2:E_1 \rightarrow W_1$, $f_2:E_0' \rightarrow W_1$ and a non-expansive linear operator $t_1:W_1 \rightarrow {{\mathbb {S}}}$ such that $t_1 \circ e_2 = \pi \restriction E_1$, $t_1 \circ f_2 = \pi ' \restriction E_0'$, and $\Vert e_2 \circ g_1 \circ f_1 - f_2\Vert \le \delta$. Knowing that $\pi '$ has the left-Gurariĭ property and using Theorem 2.2 for the isometric embedding $f_2$, we obtain a $\delta$-embedding $g_2:W_1 \rightarrow E'$ such that $g_2 \circ f_2$ is the identity on $E_0'$ and $\pi ' \circ g_2 = t_1$. The configuration is described in the following diagram, where the horizontal arrows are inclusions, the triangle $E_0 E_0' W_0$ is $\varepsilon$-commutative, and the triangle $E_0' E_1 W_1$ is $\delta$-commutative.

It remains to check that $i_1 := g_2 \circ e_2$ is the required $\delta$-embedding.

First, recall that $v \in E_1$, $v' \in E_0'$ and $v' = g_2(f_2(v'))$. Thus, using the fact that $\Vert g_2\Vert \le 1+\delta$, we get

$$\begin{aligned} \Vert i_1 g_1 f_1(v') - v'\Vert&= \Vert g_2 e_2 g_1 f_1(v') - g_2 f_2(v')\Vert \\&\le (1+\delta ) \Vert e_2 g_1 f_1(v') - f_2(v')\Vert \\&\le (1+\delta )\delta \Vert v'\Vert . \end{aligned}$$

Now if $(1+\delta )\delta \Vert v'\Vert <\eta$, then we conclude that ${\text {dist}}(v',i_1[E_1]) < \eta$, therefore condition (1) is satisfied.

Given $x \in E_1$, note that

$$\begin{aligned} \pi ' i_1(x) = \pi ' g_2 e_2 (x) = t_1 e_2(x) = \pi (x). \end{aligned}$$

Here we have used the fact that $\pi ' \circ g_2 = t_1$ and $t_1 \circ e_2 = \pi \restriction E_1$.

Furthermore, given $x \in E_0$, we have

$$\begin{aligned} \Vert i_1(x) - i_0(x)\Vert&= \Vert g_2 e_2(x) - i_0(x)\Vert = \Vert g_2 e_2 g_1 e_1(x) - g_2 f_2 i_0(x)\Vert \\&\le (1+\delta ) \Vert e_2 g_1 e_1(x) - f_2 i_0(x)\Vert , \end{aligned}$$

because $\Vert g_2\Vert \le 1+\delta$. On the other hand,

$$\begin{aligned} \Vert e_2 g_1 e_1(x) - f_2 i_0(x)\Vert&\le \Vert e_2 g_1 e_1(x) - e_2 g_1 f_1 i_0(x)\Vert + \Vert e_2 g_1 f_1 i_0(x) - f_2 i_0(x)\Vert \\&= \Vert g_1 e_1(x) - g_1 f_1 i_0(x)\Vert + \Vert e_2 g_1 f_1 i_0(x) - f_2 i_0(x)\Vert \\&\le (1+\delta )\Vert e_1(x) - f_1 i_0(x)\Vert + \delta \Vert i_0(x)\Vert \\&\le (1+\delta ) \varepsilon \Vert x\Vert + \delta (1+\varepsilon ) \Vert x\Vert \le (\varepsilon +3\delta ) \Vert x\Vert . \end{aligned}$$

Here we have used the following facts: $e_2$ is an isometric embedding, $g_1$ is a $\delta$-embedding, $i_0$ is an $\varepsilon$-embedding, $\Vert e_2 g_1 f_1 - f_2\Vert \le \delta$, $\Vert e_1 - f_1 i_0\Vert \le \varepsilon$ and $\varepsilon <1$.

Finally, $\Vert i_1(x) - i_0(x)\Vert \le (1+\delta ) (\varepsilon + 3\delta ) \Vert x\Vert \le (\varepsilon + 7\delta ) \Vert x\Vert$. Summarizing, if $(1+\delta )\delta \Vert v'\Vert <\eta$ and $7\delta < \eta$ then conditions (1), (2) are satisfied. This completes the proof. $\square$

Note that if ${{\mathbb {S}}}$ is the trivial space, the proof above reduces to the well known uniqueness of the Gurariĭ space, shown by this way in [8]. Furthermore, the arguments above can be applied to $\pi = \pi ' = {\mathbf {P}}_{{{\mathbb {S}}}}$ and $i_0 = h$, thus proving Theorem 1.7. Theorems 2.4 and 3.2 yield the following result, announced before.

Corollary 3.4

${\Omega }= {\mathbf {P}}_{{{\mathbb {G}}}}$.

In particular, $V_{{\mathbb {G}}}= {{\mathbb {G}}}$. It has been shown in [2] that $V_{{\mathbb {S}}}= {{\mathbb {G}}}$ as long as ${{\mathbb {S}}}$ is a (separable) Lindenstrauss space, namely, an isometric $L_1$ predual or (equivalently) a locally almost 1-injective space. Instead of going into details, let us just say that Lindenstrauss spaces are those (separable) Banach spaces that are linearly isometric to a 1-complemented subspace of the Gurariĭ space. The non-trivial direction was proved by Wojtaszczyk [10]. Thus, since ${\mathbf {P}}_{{{\mathbb {S}}}}$ is a projection, if $V_{{\mathbb {S}}}$ is linearly isometric to ${{\mathbb {G}}}$ then ${{\mathbb {S}}}$ is necessarily a Lindenstrauss space.

4 Generic operators

Inspired by the result of [7], let us consider the following infinite game for two players Eve and Adam. Namely, Eve starts by choosing a non-expansive linear operator $T_0:E_0 \rightarrow F_0$, where $E_0$, $F_0$ are finite-dimensional normed spaces. Adam responds by a non-expansive linear operator $T_1:E_1 \rightarrow F_1$, such that $E_1 \supseteq E_0$, $F_1 \supseteq F_0$ are again finite-dimensional and $T_1$ extends $T_0$. Eve responds by a further non-expansive linear extension $T_2:E_2 \rightarrow F_2$, and so on. So at each stage of the game we have a linear operator between finite-dimensional normed spaces. After infinitely many steps we obtain a chain of non-expansive operators $\{T_n:E_n \rightarrow F_n\}_{{n\in \omega }}$. Let $T_\infty :E_\infty \rightarrow F_\infty$ denote the completion of its union, namely, $E_\infty$ is the completion of $\{E_n\}_{{n\in \omega }}$, $F_\infty$ is the completion of $\{F_n\}_{{n\in \omega }}$ and $T_\infty \restriction E_n = T_n$ for every ${n\in \omega }$. So far, we cannot say who wins the game.

Let us say that a (necessarily non-expansive) linear operator $U:X \rightarrow Y$ is generic if Adam has a strategy making the operator $T_\infty$ isometric to U. Recall that operators U, V are isometric if there are bijective linear isometries i, j such that $U \circ j = i \circ V$.

Theorem 4.1

The operator ${\Omega }$ is generic.

Proof

Let us fix a non-expansive linear operator $U:{{\mathbb {G}}}\rightarrow {{\mathbb {G}}}$ between separable Banach spaces satisfying (G). Adam’s strategy can be described as follows.

Fix a countable set $\{v_n: a_n\rightarrow b_n\}_{n\in {{\mathbb {N}}}}$ linearly dense in $U:{{\mathbb {G}}}\rightarrow {{\mathbb {G}}}$. Let $T_0:E_0 \rightarrow F_0$ be the first move of Eve. Adam finds isometric embeddings $i_0:E_0\rightarrow {{\mathbb {G}}}$, $j_0:F_0\rightarrow {{\mathbb {G}}}$ and finite-dimensional spaces $E_0\subset E_1$, $F_0 \subset F_1$ together with isometric embeddings $i_1:E_1\rightarrow {{\mathbb {G}}}$ , $j_1:F_1\rightarrow {{\mathbb {G}}}$ and non-expansive linear operators $T_1:E_1 \rightarrow F_1$ such that $T_1$ extends $T_0$, $a_0\in i_1[E_1]$, $b_0\in j_1[F_1]$.

Suppose now that $n = 2k > 0$ and $T_n:E_n \rightarrow F_n$ was the last move of Eve. We assume that linear isometric embeddings $i_{n-1} : E_{n-1}\rightarrow {{\mathbb {G}}}$, $j_{n-1} : F_{n-1}\rightarrow {{\mathbb {G}}}$ have already been fixed. Using (G) from Theorem 1.4 we choose linear isometric embeddings $i_{n} : E_{n}\rightarrow {{\mathbb {G}}}$, $j_{n} : F_{n}\rightarrow {{\mathbb {G}}}$ such that $i_n\restriction E_{n-1}$ is $2^{-k}$-close to $i_{n-1}$, $j_n\restriction F_{n-1}$ is $2^{-k}$-close to $j_{n-1}$ and $U\circ i_n$ is $2^{-k}$-close to $j_n\circ T_n$.

Let $\{T_n:E_n\rightarrow F_n \}_{n\in {{\mathbb {N}}}}$ be the chain of non-expansive operators between finite-dimensional normed spaces resulting from a fixed play, when Adam was using his strategy. In particular, Adam has recorded sequences $\{T_n:E_n\rightarrow F_n\}_{n\in {{\mathbb {N}}}}$, $\{i_n:E_n\rightarrow {{\mathbb {G}}}\}_{n\in {{\mathbb {N}}}}$, $\{j_n:F_n\rightarrow {{\mathbb {G}}}\}_{n\in {{\mathbb {N}}}}$ of linear isometric embeddings such that $i_{2n+1}\restriction E_{2n-1}$ is $2^{-n}$-close to $i_{2n-1}$ and $j_{2n+1}\restriction F_{2n-1}$ is $2^{-n}$-close to $j_{2n-1}$ for each $n \in {{\mathbb {N}}}$.

Let $T_\infty :E_\infty \rightarrow F_\infty$ denote the completion of those unions, namely, $E_\infty$ is the completion of $\{E_n\}_{{n\in \omega }}$, $F_\infty$ is the completion of $\{F_n\}_{{n\in \omega }}$ and $T_\infty \restriction E_n = T_n$ for every ${n\in \omega }$. The assumptions that $i_{2n+1}[E_{2n+1}]$ contains all the vectors $a_0, \dots , a_n$ and $j_{2n+1}[F_{2n+1}]$ contains all the vectors $b_0, \dots , b_n$ ensures that both $i_\infty [E_\infty ]$, $j_\infty [F_\infty ]$ are dense in ${{\mathbb {G}}}$, where $i_\infty :E_\infty \rightarrow {{\mathbb {G}}}$, $j_\infty :F_\infty \rightarrow {{\mathbb {G}}}$ are pointwise limits of $\{i_n\}_{n\in {{\mathbb {N}}}}$ and $\{j_n\}_{n\in {{\mathbb {N}}}}$, respectively. More precisely, $i_\infty \restriction E_k$ is the pointwise limit of $\{i_n \restriction E_k\}_{n \ge k}$ and $j_\infty \restriction F_k$ is the pointwise limit of $\{j_n \restriction F_k\}_{n \ge k}$ for every $k \in n\in {{\mathbb {N}}}$. In particular, both $i_\infty$ and $j_\infty$ are surjective linear isometries.

Finally, $U \circ i_\infty = j_\infty \circ T_\infty$, because $U \circ i_{2k}$ is $2^{-k}$-close to $j_{2k} \circ T_{2k}$ for every $k \in {{\mathbb {N}}}$. This completes the proof. $\square$

Question 1

Is ${\Omega }$ generic in the space of all non-expansive operators on the Gurariĭ space? In this case, being “generic” means that the set

$$\begin{aligned} \{i \circ {\Omega }\circ j:i,j \text { bijective linear isometries of }{{\mathbb {G}}}\} \end{aligned}$$

is residual in the space of all non-expansive operators on ${{\mathbb {G}}}$. Here, it is natural to consider the pointwise convergence (i.e., strong operator) topology.

One could also consider a “parametrized” variant of the game above, where the two players build a chain of non-expansive operators from finite-dimensional normed spaces into a fixed Banach space ${{\mathbb {S}}}$. If ${{\mathbb {S}}}$ is separable then similar arguments as in the proof of Theorem 4.1 show that the second player has a strategy leading to ${\mathbf {P}}_{{{\mathbb {S}}}}$. Thus, a variant of Question 1 makes sense: Is it true that isometric copies of ${\mathbf {P}}_{{{\mathbb {S}}}}$ form a residual set in a suitable space of operators?

After concluding that ${\Omega }= {\mathbf {P}}_{{{\mathbb {G}}}}$, it seems that the “parametrized” construction of universal projections is better in the sense that it “captures” both the Gurariĭ space ${{\mathbb {G}}}$ (when the range is the trivial space $\{0\}$) and the universal operator ${\Omega }$ (when the range equals ${{\mathbb {G}}}$), but also other examples, including projections from the Gurariĭ space onto any separable Lindenstrauss space (see [10] and [2]).