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Notes on automorphisms of ultrapowers of II₁ factors

David Sherman (2009)

Studia Mathematica

In functional analysis, approximative properties of an object become precise in its ultrapower. We discuss this idea and its consequences for automorphisms of II₁ factors. Here are some sample results: (1) an automorphism is approximately inner if and only if its ultrapower is ℵ₀-locally inner; (2) the ultrapower of an outer automorphism is always outer; (3) for unital *-homomorphisms from a separable nuclear C*-algebra into an ultrapower of a II₁ factor, equality of the induced traces implies unitary...

On ultrapowers of Banach spaces of type ℒ ∞

Antonio Avilés, Félix Cabello Sánchez, Jesús M. F. Castillo, Manuel González, Yolanda Moreno (2013)

Fundamenta Mathematicae

We prove that no ultraproduct of Banach spaces via a countably incomplete ultrafilter can contain c₀ complemented. This shows that a "result" widely used in the theory of ultraproducts is wrong. We then amend a number of results whose proofs have been infected by that statement. In particular we provide proofs for the following statements: (i) All M-spaces, in particular all C(K)-spaces, have ultrapowers isomorphic to ultrapowers of c₀, as also do all their complemented subspaces isomorphic to their...

Supertauberian operators and perturbations.

M. González, A. Martínez-Abejón (1993)

Extracta Mathematicae

Upper semi-Fredholm operators and tauberian operators in Banach spaces admit the following perturbative characterizations [6], [2]: An operator T: X --> Y is upper semi-Fredholm (tauberian) if and only if for every compact operator K: X --> Y the kernel N(T+K) is finite dimensional (reflexive). In [7] Tacon introduces an intermediate class between upper semi-Fredholm operators and tauberian operators, the supertauberian operators, and he studies this class using non-standard analysis....

The density condition and the strong dual density condition by operator.

Wolf-Dieter Heinrichs (1997)

Collectanea Mathematica

The aim of the present article is to introduce and investigate topological properties by operator. We obtain good stability properties for the density condition and the strong dual density condition by taking injective tensor products. Further we analyze the connection to (DF)-properties by operator.

The density condition in projective tensor products.

Wolf-Dieter Heinrichs (1999)

Revista Matemática Complutense

In this paper we modify a construction due to J. Taskinen to get a Fréchet space F which satisfies the density condition such that the complete injective tensor product l2 x~eF'b does not satisfy the strong dual density condition of Bierstedt and Bonet. In this way a question that remained open in Heinrichs (1997) is solved.

The range of a contractive projection in Lp(H).

Yves Raynaud (2004)

Revista Matemática Complutense

We show that the range of a contractive projection on a Lebesgue-Bochner space of Hilbert valued functions Lp(H) is isometric to a lp-direct sum of Hilbert-valued Lp-spaces. We explicit the structure of contractive projections. As a consequence for every 1 < p < ∞ the class Cp of lp-direct sums of Hilbert-valued Lp-spaces is axiomatizable (in the class of all Banach spaces).

Types on stable Banach spaces

José Iovino (1998)

Fundamenta Mathematicae

 We prove a geometric characterization of Banach space stability. We show that a Banach space X is stable if and only if the following condition holds. Whenever X ^ is an ultrapower of X and B is a ball in X ^ , the intersection B ∩ X can be uniformly approximated by finite unions and intersections of balls in X; furthermore, the radius of these balls can be taken arbitrarily close to the radius of B, and the norm of their centers arbitrarily close to the norm of the center of B.  The preceding condition...

Ultrapowers of Köthe function spaces.

Yves Raynaud (1997)

Collectanea Mathematica

The ultrapowers, relative to a fixed ultrafilter, of all the Köthe function spaces with non trivial concavity over the same measure space can be represented as Köthe function spaces over the same (enlarged) measure space. The existence of a uniform homeomorphism between the unit spheres of two such Köthe function spaces is reproved.

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