Restriction of Uniform Crossnorms

TL;DR

This paper investigates how uniform crossnorms on tensor product spaces induce reasonable crossnorms on the tensor product of operator algebras, providing insights into the structure of these norms.

Contribution

It demonstrates that the restriction of the induced uniform norm on operators on tensor products to the tensor product of operator algebras is a reasonable crossnorm.

Findings

The restriction of the uniform norm is a reasonable crossnorm.

Provides a characterization of norms on tensor products of operator algebras.

Enhances understanding of tensor norms in operator space theory.

Abstract

Consider the space $B[X\otimes_\alpha Y]$ of operators on the tensors product ${X\otimes_\alpha Y}$ of normed spaces $X$ and $Y$ equipped with a uniform crossnorm ${\|\cdot\|_\alpha}$$.$ Take the induced uniform norm on $B[X\otimes_\alpha Y]$ and consider its restriction to the tensor product ${B[X]\otimes B[Y]}$ of the algebra of operators $B[X]$ and $B[Y].$ It is proved that such a restriction is a reasonable crossnorm on ${B[X]\otimes B[Y]}$.