The University of Southampton
University of Southampton Institutional Repository

Local properties of accessible injective operator ideals

Local properties of accessible injective operator ideals
Local properties of accessible injective operator ideals
In addition to Pisier’s counterexample of a non-accessible maximal Banach ideal, we will give a large class of maximal Banach ideals which are accessible. The first step is implied by the observation that a “good behaviour” of trace duality, which is canonically induced by conjugate operator ideals can be extended to adjoint Banach ideals, if and only if these adjoint ideals satisfy an accessibility condition (theorem 3.1). This observation leads in a natural way to a characterization of accessible injective Banach ideals, where we also recognize the appearance of the ideal of absolutely summing operators (prop. 4.1). By the famous Grothendieck inequality, every operator from L_1 to a Hilbert space is absolutely summing, and therefore our search for such ideals will be directed towards Hilbert space factorization—via an operator version of Grothendieck’s inequality (lemma 4.2). As a consequence, we obtain a class of injective ideals, which are “quasi-accessible”, and with the help of tensor stability, we improve the corresponding norm inequalities, to get accessibility (theorem 4.1 and 4.2). In the last chapter of this paper we give applications, which are implied by a non-trivial link of the above mentioned considerations to normed products of operator ideals.
accessibility, banach spaces, conjugate operator ideals, hilbert space factorization, grothendieck’s inequality, tensor norms, tensor stability
0011-4642
119-133
Oertel, Frank
5026be9a-a787-477f-bf94-23c72bd08ef5
Oertel, Frank
5026be9a-a787-477f-bf94-23c72bd08ef5

Oertel, Frank (1998) Local properties of accessible injective operator ideals. Czechoslovak Mathematical Journal, 48 (1), 119-133.

Record type: Article

Abstract

In addition to Pisier’s counterexample of a non-accessible maximal Banach ideal, we will give a large class of maximal Banach ideals which are accessible. The first step is implied by the observation that a “good behaviour” of trace duality, which is canonically induced by conjugate operator ideals can be extended to adjoint Banach ideals, if and only if these adjoint ideals satisfy an accessibility condition (theorem 3.1). This observation leads in a natural way to a characterization of accessible injective Banach ideals, where we also recognize the appearance of the ideal of absolutely summing operators (prop. 4.1). By the famous Grothendieck inequality, every operator from L_1 to a Hilbert space is absolutely summing, and therefore our search for such ideals will be directed towards Hilbert space factorization—via an operator version of Grothendieck’s inequality (lemma 4.2). As a consequence, we obtain a class of injective ideals, which are “quasi-accessible”, and with the help of tensor stability, we improve the corresponding norm inequalities, to get accessibility (theorem 4.1 and 4.2). In the last chapter of this paper we give applications, which are implied by a non-trivial link of the above mentioned considerations to normed products of operator ideals.

Full text not available from this repository.

More information

Published date: 1998
Keywords: accessibility, banach spaces, conjugate operator ideals, hilbert space factorization, grothendieck’s inequality, tensor norms, tensor stability
Organisations: Mathematical Sciences

Identifiers

Local EPrints ID: 347307
URI: https://eprints.soton.ac.uk/id/eprint/347307
ISSN: 0011-4642
PURE UUID: 4aff0499-fa6c-4968-91d7-99359bbe8192

Catalogue record

Date deposited: 18 Jan 2013 11:05
Last modified: 18 Jul 2017 04:59

Export record

Download statistics

Downloads from ePrints over the past year. Other digital versions may also be available to download e.g. from the publisher's website.

View more statistics

Atom RSS 1.0 RSS 2.0

Contact ePrints Soton: eprints@soton.ac.uk

ePrints Soton supports OAI 2.0 with a base URL of https://eprints.soton.ac.uk/cgi/oai2

This repository has been built using EPrints software, developed at the University of Southampton, but available to everyone to use.

We use cookies to ensure that we give you the best experience on our website. If you continue without changing your settings, we will assume that you are happy to receive cookies on the University of Southampton website.

×