Local properties of accessible injective operator ideals

Title:

Local properties of accessible injective operator ideals

Creator:

Oertel, Frank

Identifier:

https://cdk.lib.cas.cz/client/handle/uuid:38f6bf67-032f-4e45-90c1-3d9093cb4ddd
uuid:38f6bf67-032f-4e45-90c1-3d9093cb4ddd

Subject:

accessibility, Banach spaces, conjugate operator ideals, Hilbert space factorization, Grothendieck’s inequality, tensor norms, and tensor stability

Type:

model:article and TEXT

Format:

bez média and svazek

Description:

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.

Language:

English

Rights:

http://creativecommons.org/publicdomain/mark/1.0/
policy:public

Coverage:

119-133

Source:

Czechoslovak Mathematical Journal | 1998 Volume:48 | Number:1

Harvested from:

CDK

Metadata only:

false