A universal operator on the Gurarii space

Abstract

We construct a nonexpansive linear operator on the Gurarii space that "captures" all nonexpansive linear operators between separable Banach spaces. Some additional properties involving its restrictions to finite-dimensional subspaces describe this operator uniquely up to an isometry.Comment: 17 pages, extended subsection 2.2. Final versio

Fraïssé limits in functional analysis

“…An analogue of Rota's universal operator for the class of operators on arbitrary separable Banach spaces was constructed by Garbulińska-Wȩgrzyn and Kubiś in [45]. In this section we will prove a general result concerning the existence of a "universal morphism" defined on the Fraïssé limit M of a Fraïssé class C as in Theorem 2.8.…”

“…In addition to the results above, our general framework will apply to produce commutative and noncommutative analogs of universal operators in the sense of Rota, generalizing work of Garbulińska-Wȩgrzyn and Kubiś [45] The rest of the paper is organized as follows. In Section 2 we present the general framework of Fraïssé classes generated by injective objects, and provide a characterization of the corresponding limits.…”

Gurarii operators are generic

“…In [1], [2] Garbulińska-Wȩgrzyn and Kubiś constructed a universal operator Ω : G → G using the technique of Fraisee limits. More precisely, they defined the notion of a Gurariȋ operator (which is an operator counterpart of the notion of a Gurariȋ space), constructed a Gurariȋ operator (as the Fraisse limit in a suitable category) and proved that every Gurariȋ operator is universal.…”

“…An example of a Gurariȋ operator Ω : G → G was constructed in [1]. By [1,Theorem 3.5], any Gurariȋ operator G : X → Y is isometric to the Gurariȋ operator Ω : G → G in the sense that there exist bijective isometries i : X → G and j : Y → G such that Ω • i = j • G. Therefore, a Gurariȋ operator is unique up to an isometry (like the Gurariȋ space). By [1,Theorem 3.3], every Gurariȋ operator is universal.…”

“…By [1,Theorem 3.5], any Gurariȋ operator G : X → Y is isometric to the Gurariȋ operator Ω : G → G in the sense that there exist bijective isometries i : X → G and j : Y → G such that Ω • i = j • G. Therefore, a Gurariȋ operator is unique up to an isometry (like the Gurariȋ space). By [1,Theorem 3.3], every Gurariȋ operator is universal. Therefore, the set G(G) of Gurariȋ operators from a Gurariȋ space G to itself is a subset of the set U(G) of nonexpansive universal operators from G to G. In its turn, U(G) is a subset of the space B(G) of all nonexpansive operators from G to G. The space B(G) is endowed with the strong operator topology.…”