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 ${ B[X]\otimes B[Y]}$.
Categories: math.FA