Non-rough norms and dentability in spaces of operators

TL;DR

This paper investigates the conditions under which spaces of bounded linear operators and their tensor products possess non-rough norms, revealing their dependence on the properties of the underlying Banach spaces.

Contribution

It establishes necessary and sufficient conditions for non-rough norms in operator spaces and tensor products, and explores stability of small diameter properties.

Findings

L(X,Y) has non-rough norm iff X* and Y have non-rough norm

Injective tensor product has non-rough norm iff both spaces have non-rough norm

Non-rough norms are not stable under projective tensor product

Abstract

In this work, we study non-rough norms in L(X,Y), the space of bounded linear operators between Banach spaces X and Y. We prove that L(X,Y) has non-rough norm if and only if X* and Y have non-rough norm. We show that the injective tensor product of X and Y has non-rough norm if and only if both X and Y have non-rough norm. We also give an example to show that non-rough norms are not stable under projective tensor product. We also study a related concept namely the small diameter properties in the context of L(X,Y)*. These results leads to a discussion on stability of the small diameter properties for projective and injective tensor product spaces.