Universal operators between separable Banach spaces Wiesaw Kubi s - - PowerPoint PPT Presentation

Universal operators between separable Banach spaces

Wiesław Kubi´ s

Institute of Mathematics, Academy of Sciences of the Czech Republic http://www.math.cas.cz/kubis/

Integration, Vector Measures and Related Topics Be ¸dlewo, June 15 – 21, 2014

W.Kubi´ s (http://www.math.cas.cz/kubis/) Universal operators 16 June 2014 1 / 15

Co-authors:

1

F . Cabello Sánchez (University of Extremadura, Spain)

2

• J. Garbuli´

nska-We ¸grzyn (Jan Kochanowski University in Kielce, Poland)

W.Kubi´ s (http://www.math.cas.cz/kubis/) Universal operators 16 June 2014 2 / 15

Motivation

Question:

Fix a separable space F. Does there exist a non-expansive linear

• perator P : E → F “containing" all non-expansive linear operators

from separable spaces into F?

W.Kubi´ s (http://www.math.cas.cz/kubis/) Universal operators 16 June 2014 3 / 15

Motivation

Question:

Fix a separable space F. Does there exist a non-expansive linear

• perator P : E → F “containing" all non-expansive linear operators

from separable spaces into F? Some history: Caradus (1969): Universal operators in the Hilbert space Lindenstrauss and Pełczy´ nski (1968): Isomorphically universal

• perator for non-compact operators

W.Kubi´ s (http://www.math.cas.cz/kubis/) Universal operators 16 June 2014 3 / 15

Definition

Let E, F be separable spaces. An operator P : E → F is left-universal if for every operator T : X → F with X separable and with T P there exists an isometric embedding e: X → E such that T = P ◦ e.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Universal operators 16 June 2014 4 / 15

Definition

Let E, F be separable spaces. An operator P : E → F is left-universal if for every operator T : X → F with X separable and with T P there exists an isometric embedding e: X → E such that T = P ◦ e. An operator P : E → F is universal if for every operator T : X → Y with X, Y separable and with T P there exist isometric embeddings i : X → E, j : Y → F such that P ◦ i = j ◦ T.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Universal operators 16 June 2014 4 / 15

Definition

Let E, F be separable spaces. An operator P : E → F is left-universal if for every operator T : X → F with X separable and with T P there exists an isometric embedding e: X → E such that T = P ◦ e. An operator P : E → F is universal if for every operator T : X → Y with X, Y separable and with T P there exist isometric embeddings i : X → E, j : Y → F such that P ◦ i = j ◦ T. E

P

F

X

e

• T
• E

P

F

X

i

• T

Y

j

• W.Kubi´

s (http://www.math.cas.cz/kubis/) Universal operators 16 June 2014 4 / 15

Quasi-Banach spaces

Definition

Fix p ∈ (0, 1]. A p-normed space is a vector space X endowed with a p-norm · , that is, x 0 and x = 0 ⇐ ⇒ x = 0, λx = |λ| · x, x + yp xp + yp. A quasi-Banach space is a p-Banach space for some p ∈ (0, 1].

W.Kubi´ s (http://www.math.cas.cz/kubis/) Universal operators 16 June 2014 5 / 15

Quasi-Banach spaces

Definition

Fix p ∈ (0, 1]. A p-normed space is a vector space X endowed with a p-norm · , that is, x 0 and x = 0 ⇐ ⇒ x = 0, λx = |λ| · x, x + yp xp + yp. A quasi-Banach space is a p-Banach space for some p ∈ (0, 1]. Canonical example: ℓp

W.Kubi´ s (http://www.math.cas.cz/kubis/) Universal operators 16 June 2014 5 / 15

Main result I

Theorem

Fix p ∈ (0, 1] and fix a separable p-Banach space F. There exists a left-universal non-expansive linear operator PF : E → F

W.Kubi´ s (http://www.math.cas.cz/kubis/) Universal operators 16 June 2014 6 / 15

Main result I

Theorem

Fix p ∈ (0, 1] and fix a separable p-Banach space F. There exists a left-universal non-expansive linear operator PF : E → F with the following property: (G) Given ε > 0, finite-dimensional spaces A ⊆ B, an isometric embedding e: A → E, and a non-expansive operator T : B → F such that T ↾ A = PF ◦ e, there exists an ε-isometric embedding f : B → E such that PF ◦ f − T ε.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Universal operators 16 June 2014 6 / 15

Main result I

Theorem

Fix p ∈ (0, 1] and fix a separable p-Banach space F. There exists a left-universal non-expansive linear operator PF : E → F with the following property: (G) Given ε > 0, finite-dimensional spaces A ⊆ B, an isometric embedding e: A → E, and a non-expansive operator T : B → F such that T ↾ A = PF ◦ e, there exists an ε-isometric embedding f : B → E such that PF ◦ f − T ε. Furthermore, this property describes PF uniquely, up to isometry.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Universal operators 16 June 2014 6 / 15

Main result I

Theorem

Fix p ∈ (0, 1] and fix a separable p-Banach space F. There exists a left-universal non-expansive linear operator PF : E → F with the following property: (G) Given ε > 0, finite-dimensional spaces A ⊆ B, an isometric embedding e: A → E, and a non-expansive operator T : B → F such that T ↾ A = PF ◦ e, there exists an ε-isometric embedding f : B → E such that PF ◦ f − T ε. Furthermore, this property describes PF uniquely, up to isometry.

Corollary

The operator PF is a projection.

Proof.

Apply left-universality to idF.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Universal operators 16 June 2014 6 / 15

Fix p ∈ (0, 1]. Let 0 denote the trivial p-Banach space.

Definition

The domain of the left-universal operator P0 is called the p-Gurari˘ ı space, denoted by Gp.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Universal operators 16 June 2014 7 / 15

Fix p ∈ (0, 1]. Let 0 denote the trivial p-Banach space.

Definition

The domain of the left-universal operator P0 is called the p-Gurari˘ ı space, denoted by Gp.

Fact

The p-Gurari˘ ı space is the unique separable p-Banach space G satisfying the following condition: (G0) For every ε > 0, for every finite-dimensional spaces A ⊆ B, for every isometric embedding e: A → G there exists an ε-isometric embedding f : B → G such that f ↾ A = e. For p = 1, the space Gp was constructed by Gurari˘ ı in 1966. Uniqueness was proved by Lusky in 1976.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Universal operators 16 June 2014 7 / 15

Fact

The space Gp is isometrically universal in the class of all separable p-Banach spaces.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Universal operators 16 June 2014 8 / 15

Fact

The space Gp is isometrically universal in the class of all separable p-Banach spaces.

Fact

If p < 1 then the dual of Gp is trivial.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Universal operators 16 June 2014 8 / 15

Fact

The space Gp is isometrically universal in the class of all separable p-Banach spaces.

Fact

If p < 1 then the dual of Gp is trivial.

Theorem

For every separable p-Banach space F, the kernel of PF is linearly isometric to Gp.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Universal operators 16 June 2014 8 / 15

Almost injective spaces

Definition

A p-Banach space F is locally almost 1-injective if for every finite-dimensional spaces A ⊆ B, for every bounded linear operator S : A → F, for every ε > 0 there exists a linear operator T : B → F such that T ↾ A = S and T (1 + ε)S.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Universal operators 16 June 2014 9 / 15

Almost injective spaces

Definition

A p-Banach space F is locally almost 1-injective if for every finite-dimensional spaces A ⊆ B, for every bounded linear operator S : A → F, for every ε > 0 there exists a linear operator T : B → F such that T ↾ A = S and T (1 + ε)S.

Fact

A separable Banach space is locally almost 1-injective ⇐ ⇒ it is a Lindenstrauss space, i.e. an isometric L1 predual.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Universal operators 16 June 2014 9 / 15

Lemma

Every 1-complemented subspace of Gp is locally almost 1-injective.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Universal operators 16 June 2014 10 / 15

Lemma

Every 1-complemented subspace of Gp is locally almost 1-injective.

Theorem

For a separable p-Banach space F, the following properties are equivalent:

1

F is locally almost 1-injective.

2

dom(PF) is linearly isometric to Gp.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Universal operators 16 June 2014 10 / 15

Lemma

Every 1-complemented subspace of Gp is locally almost 1-injective.

Theorem

For a separable p-Banach space F, the following properties are equivalent:

1

F is locally almost 1-injective.

2

dom(PF) is linearly isometric to Gp.

Corollary

Gp ≈ Gp ⊕ Gp.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Universal operators 16 June 2014 10 / 15

Lemma

Every 1-complemented subspace of Gp is locally almost 1-injective.

Theorem

For a separable p-Banach space F, the following properties are equivalent:

1

F is locally almost 1-injective.

2

dom(PF) is linearly isometric to Gp.

Corollary

Gp ≈ Gp ⊕ Gp.

Corollary (Wojtaszczyk 1972, Lusky 1977)

Let X be a separable Lindenstrauss space. Then there exists a linear isometric embedding X ⊆ G and a norm one projection P : G → X such that ker P is linearly isometric to G.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Universal operators 16 June 2014 10 / 15

Main result II

Theorem

Fix p ∈ (0, 1]. There exists a universal operator U: Gp → Gp,

W.Kubi´ s (http://www.math.cas.cz/kubis/) Universal operators 16 June 2014 11 / 15

Main result II

Theorem

Fix p ∈ (0, 1]. There exists a universal operator U: Gp → Gp, uniquely determined by the following property: (G∗) Given finite-dimensional spaces A0 ⊆ A1, B0 ⊆ B1, given a non-expansive operator T : A1 → B1, given ε > 0, given isometric embeddings e0 : A0 → Gp, f0 : B0 → Gp such that U ◦ e0 = f0 ◦ (T ↾ A0), there exist ε-isometric embeddings e1 : A1 → Gp, f1 : B1 → Gp such that U ◦ e1 − f1 ◦ T ε.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Universal operators 16 June 2014 11 / 15

Theorem

U = PG.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Universal operators 16 June 2014 12 / 15

• W.Kubi´

s (http://www.math.cas.cz/kubis/) Universal operators 16 June 2014 13 / 15

One of the key tools: Amalgamation

Lemma

Let f : Z → X, g : Z → Y be isometric embeddings of p-Banach

• spaces. Let V = (X ⊕ Y)/∆, where

∆ = {(f(z), −g(z)): z ∈ Z} and endow V with the p-norm (x, y) + ∆p = inf

z∈Z

• x + f(z)p + y − g(z)p

. Then the canonical operators f ′ : X → V, g′ : Y → V defined by f ′(x) = (x, 0) + ∆, g′(y) = (0, y) + ∆ are isometric embeddings satisfying f ′ ◦ f = g′ ◦ g.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Universal operators 16 June 2014 14 / 15

References

• F. CABELLO SÁNCHEZ, J. GARBULI ´

NSKA-WE ¸ GRZYN, W. KUBI ´ S,

Quasi-Banach spaces of almost universal disposition, Journal of Functional Analysis 267 (2014) 744–771 http://arxiv.org/abs/1309.7649

• J. GARBULI ´

NSKA-WE ¸ GRZYN, W. KUBI ´ S, A universal operator on

the Gurari˘ ı space, to appear in the Journal of Operator Theory http://arxiv.org/abs/1310.2380

W.Kubi´ s (http://www.math.cas.cz/kubis/) Universal operators 16 June 2014 15 / 15