Restriction of Uniform Crossnorms

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 $\Vert \cdot {\Vert }_{\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 $B[X]\otimes B[Y]$.