A note on universal operators between separable Banach spaces

We compare two types of universal operators constructed relatively recently by Cabello Sánchez, and the authors. The first operator Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Omega }$$\end{document} acts on the Gurariĭ space, while the second one PS\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf {P}}_{{{\mathbb {S}}}}$$\end{document} has values in a fixed separable Banach space S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb {S}}}$$\end{document}. We show that if S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb {S}}}$$\end{document} is the Gurariĭ space, then both operators are isometric. We also prove that, for a fixed space S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb {S}}}$$\end{document}, the operator PS\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf {P}}_{{{\mathbb {S}}}}$$\end{document} is isometrically unique. Finally, we show that Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Omega }$$\end{document} is generic in the sense of a natural infinite game.

universal if its restrictions to closed subspaces are, up to linear isometries, all linear operators of norm not exceeding U . To be more precise, a bounded linear operator U : V → W acting between separable Banach spaces is universal if for every linear operator T : X → Y with X , Y separable and T ≤ U , there exist linear isometric embeddings i : 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 → W is left-universal (for operators into W ) if for every linear operator T : X → W with X separable and T ≤ U there exists a linear isometric embedding i : X → V for which the diagram 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. Proof (1) Fix a separable Banach space X . Taking the zero operator T : X → 0, we see that V contains an isometric copy of X . Taking the identity 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 id W , we obtain the required isometric embedding e : W → V .
(3) Assume U is universal, fix λ > 0 and fix T : If U is left-universal, the argument is the same, the only difference is that j = id W . 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 ⊕ W with the maximum norm and let π : V ⊕ W → W be the canonical projection. Given a non-expansive operator T : X → W with X separable, choose an isometric embedding e : X → V and define j : Then j is an isometric embedding and π • j = T , showing that π is left-universal. Of course, if additionally W is isometrically universal, then π is a universal operator.
Perhaps the most well known universal Banach space is 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 C( . This leads to (at least potentially) many other universal operators, namely: is commutative. As W is isometrically universal, we may additionally assume that V = W , replacing by i • and e by i • e , where i is a fixed isometric embedding of V into W . It is evident that now is a universal operator, because of the universality of π.
As a consequence, there exists a universal operator on C([0, 1]). We do not know whether there exists a left-universal operator on C([0, 1]). The situation changes when replacing [0, 1] with the Cantor set 2 N . The space C(2 N ) is linearly isomorphic (but not isometric) to C([0, 1]) and it is isometrically universal, too. Furthermore, C(2 N ) ⊕ C(2 N ) with the maximum norm is linearly isometric to C(2 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 C(2 N ).
Another, not so well known, universal Banach space is the Gurariȋ space. This is the unique, up to a linear isometry, separable Banach space G satisfying the following condition: For every ε > 0, for every finite-dimensional spaces X 0 ⊆ X , for every linear isometric embedding f 0 : X 0 → G there exists a linear ε-isometric embedding f : X → G such that f X 0 = f 0 .
By an ε-isometric embedding (briefly: ε-embedding) we mean a linear operator f satisfying for every x in the domain of f . The space G was constructed by Gurariȋ [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 : G → G with the following property: Furthermore, is a universal operator and property (G) specifies it uniquely, up to a linear isometry.
According to [5], we shall call condition (G) the Gurariȋ property. What makes this operator of particular interest is perhaps its almost homogeneity: We now describe the left-universal operators constructed in [2]. Fix a separable Banach space S. Theorem 1.6 ([2, Section 6]) There exists a non-expansive linear operator P S : V S → S with V S a separable Banach space, satisfying the following condition: ( ‡) For every finite-dimensional spaces X 0 ⊆ X , for every non-expansive linear operator T : X → S, for every linear isometric embedding e : X 0 → V S such that P S •e = T X 0 , for every ε > 0 there exists an ε-embedding f :

Furthermore, P S is left-universal for operators into S.
We shall say that an operator P has the left-Gurariȋ property if it satisfies ( ‡) in place of P S . Of course, unlike the Gurariȋ property, the left-Gurariȋ property involves a parameter S, namely, the common range of the operators. The left-universality of P S implies that it is right-invertible, i.e., there is an isometric embedding J : S → V S such that P S • J = id S . The operator J • P S is a non-expansive projection of V S onto an isometric copy of S.
Actually, the operator P S was constructed in [2] in case where S had some additional property, needed only for determining the domain of P S . Moreover, [2] deals with p-Banach spaces, where p ∈ (0, 1], however p = 1 gives exactly the result stated above. Operators P S have the following property which can be called almost left-homogeneity. In this note we present a proof that condition ( ‡) determines P S uniquely, up to linear isometries. The arguments will also provide a proof of Theorem 1.7. Furthermore, we show that = P G and that is a generic operator in the space of all non-expansive operators on the Gurariȋ space into itself, in the sense of a natural variant of the Banach-Mazur game.

Properties of Ä and P S
Let us recall the following easy fact concerning finite-dimensional normed spaces (cf. [ Proof Fix M > 0 satisfying the following condition: Such M clearly exists, because of compactness of the unit ball of E. Now, given ε > 0, let We conclude that f ≤ ε. The following result, in the case S = G, can be found in [1]. Proof Obviously, (b) is stronger than ( ‡). Fix ε > 0 and fix a vector basis A of X such that A 0 = X 0 ∩ A is a basis of X 0 . We may assume that a = 1 for every a ∈ A. Fix δ > 0 and apply the left-Gurariȋ property for δ instead of ε. We obtain a δ-embedding f : X → V such that f X 0 − e ≤ δ and P • f − T ≤ δ. Define f : X → V by the conditions f (a) = e(a) for a ∈ A 0 and f (a) = f (a) for a ∈ A \ A 0 . Note that f (a) − f (a) ≤ δ for every a ∈ A. Thus, if δ is small enough, then by Lemma 2.1, we can obtain that f is an ε-embedding. Furthermore, P • f − P • f ≤ ε (recall that δ depends on ε and the norm of X only), therefore The arguments above show that for every ε > 0 there exists an ε-embedding f : X → V extending e and satisfying P • f − T ≤ ε.
Let us apply this property for δ instead of ε, where δ is taken from Lemma 2.1. We obtain a δ-embedding f : X → V extending e and satisfying P • f − T ≤ δ. Now recall that P is left-universal (Theorem 1.6), therefore there is an isometric embedding J : S → V such that P • J is the identity (take the identity of S in the definition of left-universality). Given a ∈ A \ A 0 , the vector and the same obviously holds for a ∈ A 0 . Thus P • f = T .
The proof of the next result is just a suitable adaptation of the arguments above, therefore we skip it.

Proposition 2.3 Let : V → W be a linear operator. The following conditions are equivalent.
(a) has the Gurariȋ property (G). (b) Given ε > 0, given a non-expansive operator T : X → Y between finite-dimensional spaces, given X 0 ⊆ X, Y 0 ⊆ Y and isometric embeddings i 0 : The last result of this section is the key step towards identifying with P G .

Theorem 2.4 The operator has the left-Gurariȋ property (i.e., it satisfies condition ( ‡) of Theorem 1.6 with S = G). In particular, it is left-universal.
Proof Fix a non-expansive linear operator T : X → G with X finite-dimensional, and fix an isometric embedding e : X 0 → G, where X 0 is a linear subspace of X and T X 0 = • e. Let Y 0 = Y = T [X ] ⊆ G and consider T as an operator from X to Y . Applying the Gurariȋ property with i = e and j the inclusion Y 0 ⊆ G, we obtain an ε-embedding e : X → G which is ε-close to e and satisfies • e − T ≤ ε. This is precisely condition ( ‡) from Theorem 1.6.
In order to conclude that = P G , it remains to show that ( ‡) determines the operator uniquely. This is done in the next section.

Uniqueness of P S
Before proving that the left-Gurariȋ property determines the operator uniquely, we quote the following crucial lemma from [3]. Lemma 3.1 Let ε > 0 and let f : X → Y be an ε-embedding, where X , Y are Banach spaces. Let π : X → S, : Y → S be non-expansive linear operators such that • f −π ≤ ε. Then there exists a norm on Z = X ⊕Y such that the canonical embeddings i : Note that the operator t satisfies t • i = π and t • j = . Actually, the norm mentioned in the lemma above does not depend on the operators π, . It is defined by the following formula: where · X , · 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
S be a separable Banach space and let π : E → S, π : E → S be nonexpansive linear operators, both with the left-Gurariȋ property. If E, E are separable Banach spaces, then there exists a linear isometry i : E → E such that π = π • i. In particular, π and π are linearly isometric to P S .
Proof It suffices to prove the following Claim 3.3 Let E 0 ⊆ E be a finite-dimensional space, 0 < ε < 1, let i 0 : E 0 → E be an ε-embedding such that π • i 0 = π E 0 . Then for every v ∈ E, v ∈ E , for every η > 0 there exists an η-embedding i 1 : E 1 → E such that E 1 is finite-dimensional, E 0 ⊆ E 1 ⊆ E and the following conditions are satisfied: Using Claim 3.3 together with the separability of E and E , we can construct a sequence i n : E n → E of linear operators such that i n is a 2 −n -embedding, n∈ω E n is dense in E, for every n ∈ ω. It is evident that {i n } n∈ω converges pointwise to a linear isometry whose completion i is the required bijection from E onto E satisfying π • i = π. Thus, it remains to prove Claim 3.3. This will be carried out by making two applications of Lemma 3.1. Fix 0 < δ < 1; more precise estimations for δ will be given later. Let E 0 ⊆ E be a finitedimensional space containing v and such that i 0 [E 0 ] ⊆ E 0 . Applying Lemma 3.1, we obtain linear isometric embeddings e 1 : E 0 → W 0 , f 1 : E 0 → W 0 and a non-expansive operator t 0 : W 0 → S such that t 0 • e 1 = π E 0 , t 0 • f 1 = π E 0 , and e 1 − f 1 • i 0 ≤ ε. Knowing that π has the left-Gurariȋ property, by Theorem 2.2 applied to the isometric embedding e 1 , we obtain a δ-embedding g 1 : W 0 → E such that g 1 • e 1 is the identity on E 0 and π • g 1 = t 0 . Now note that g 1 • f 1 is a δ-embedding of E 0 into a finite-dimensional subspace E 1 of E. Without loss of generality, we may assume that v ∈ E 1 . Applying Lemma 3.1 again to g 1 • f 1 , we obtain linear isometric embeddings e 2 : E 1 → W 1 , f 2 : E 0 → W 1 and a nonexpansive linear operator t 1 : W 1 → S such that t 1 • e 2 = π E 1 , t 1 • f 2 = π E 0 , and Knowing that π has the left-Gurariȋ property and using Theorem 2.2 for the isometric embedding f 2 , we obtain a δ-embedding g 2 : W 1 → E such that g 2 • f 2 is the identity on E 0 and π • 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 ε-commutative, and the triangle E 0 E 1 W 1 is δ-commutative.
It remains to check that i 1 := g 2 • e 2 is the required δ-embedding. First, recall that v ∈ E 1 , v ∈ E 0 and v = g 2 ( f 2 (v )). Thus, using the fact that g 2 ≤ 1 + δ, we get Here we have used the fact that π • g 2 = t 1 and t 1 • e 2 = π E 1 . Furthermore, given x ∈ E 0 , we have because g 2 ≤ 1 + δ. On the other hand, Here we have used the following facts: e 2 is an isometric embedding, g 1 is a δ-embedding, i 0 is an ε-embedding, x . Summarizing, if (1 + δ)δ v < η and 7δ < η then conditions (1), (2) are satisfied. This completes the proof.
Note that if S is the trivial space, the proof above reduces to the well known uniqueness of the Gurariȋ space, shown by this way in [8]. Furthermore, the arguments above can be applied to π = π = P S and i 0 = h, thus proving Theorem 1.7. Theorems 2.4 and 3.2 yield the following result, announced before.
In particular, V G = G. It has been shown in [2] that V S = G as long as S is a (separable) Lindenstrauss space, namely, an isometric L 1 predual or (equivalently) a locally almost 1injective 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 Gurariȋ space. The non-trivial direction was proved by Wojtaszczyk [10]. Thus, since P S is a projection, if V S is linearly isometric to G then S is necessarily a Lindenstrauss space.

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 → F 0 , where E 0 , F 0 are finite-dimensional normed spaces. Adam responds by a non-expansive linear operator T 1 : E 1 → F 1 , such that E 1 ⊇ E 0 , F 1 ⊇ 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 → 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 → F n } n∈ω . Let T ∞ : E ∞ → F ∞ denote the completion of its union, namely, E ∞ is the completion of {E n } n∈ω , F ∞ is the completion of {F n } n∈ω and T ∞ E n = T n for every n ∈ ω. So far, we cannot say who wins the game.
Let us say that a (necessarily non-expansive) linear operator U : X → Y is generic if Adam has a strategy making the operator T ∞ isometric to U . Recall that operators U , V are isometric if there are bijective linear isometries i, j such that U • j = i • V .

Theorem 4.1 The operator is generic.
Proof Let us fix a non-expansive linear operator U : G → G between separable Banach spaces satisfying (G). Adam's strategy can be described as follows.
Fix a countable set {v n : a n → b n } n∈N linearly dense in U : G → G. Let T 0 : E 0 → F 0 be the first move of Eve. Adam finds isometric embeddings i 0 : E 0 → G, j 0 : F 0 → G and finite-dimensional spaces E 0 ⊂ E 1 , F 0 ⊂ F 1 together with isometric embeddings i 1 : E 1 → G , j 1 : F 1 → G and non-expansive linear operators T 1 : Suppose now that n = 2k > 0 and T n : E n → F n was the last move of Eve. We assume that linear isometric embeddings i n−1 : E n−1 → G, j n−1 : F n−1 → G have already been fixed. Using (G) from Theorem 1.4 we choose linear isometric embeddings i n : E n → G, j n : F n → G such that i n E n−1 is 2 −k -close to i n−1 , j n F n−1 is 2 −k -close to j n−1 and U • i n is 2 −k -close to j n • T n .
Let {T n : E n → F n } n∈N be the chain of non-expansive operators between finitedimensional normed spaces resulting from a fixed play, when Adam was using his strategy. In particular, Adam has recorded sequences {T n : E n → F n } n∈N , {i n : E n → G} n∈N , { j n : F n → G} n∈N of linear isometric embeddings such that i 2n+1 E 2n−1 is 2 −n -close to i 2n−1 and j 2n+1 F 2n−1 is 2 −n -close to j 2n−1 for each n ∈ N.
Let T ∞ : E ∞ → F ∞ denote the completion of those unions, namely, E ∞ is the completion of {E n } n∈ω , F ∞ is the completion of {F n } n∈ω and T ∞ E n = T n for every n ∈ ω. The assumptions that i 2n+1 [E 2n+1 ] contains all the vectors a 0 , . . . , a n and j 2n+1 [F 2n+1 ] contains all the vectors b 0 , . . . , b n ensures that both i ∞ [E ∞ ], j ∞ [F ∞ ] are dense in G, where i ∞ : E ∞ → G, j ∞ : F ∞ → G are pointwise limits of {i n } n∈N and { j n } n∈N , respectively. More precisely, i ∞ E k is the pointwise limit of {i n E k } n≥k and j ∞ F k is the pointwise limit of { j n F k } n≥k for every k ∈ n ∈ N. In particular, both i ∞ and j ∞ are surjective linear isometries.
Finally, U • i ∞ = j ∞ • T ∞ , because U • i 2k is 2 −k -close to j 2k • T 2k for every k ∈ N. This completes the proof.
Question 1 Is generic in the space of all non-expansive operators on the Gurariȋ space? In this case, being "generic" means that the set {i • • j : i, j bijective linear isometries of G} is residual in the space of all non-expansive operators on 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 S. If S is separable then similar arguments as in the proof of Theorem 4.1 show that the second player has a strategy leading to P S . Thus, a variant of Question 1 makes sense: Is it true that isometric copies of P S form a residual set in a suitable space of operators?
After concluding that = P G , it seems that the "parametrized" construction of universal projections is better in the sense that it "captures" both the Gurariȋ space G (when the range is the trivial space {0}) and the universal operator (when the range equals G), but also other examples, including projections from the Gurariȋ space onto any separable Lindenstrauss space (see [10] and [2]).