Restriction of Uniform Crossnorms

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 $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{\alpha }$$\mathrm{.}$ 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 $B[Y]\mathrm{.}$ It is proved that such a restriction is a reasonable crossnorm on $B[X]\otimes B[Y]$.