Symmetries of tensor products (Q789686)

Let U and V be normed vector spaces over the field \(F={\mathbb{R}}\) or \({\mathbb{C}}\). Endow the tensor product \(W=U\otimes V\) with the canonical norm (which is defined, e.g., by the condition that \(\{y\in W:\| y\| \leq 1\}=\quad convex\quad hull\quad of\quad \{p\otimes q:p\in U,\quad q\in V\quad and\quad \| p\| =\| q\| =1\}.\) Then any linear automorphism c of W of the form \(c=a\otimes b\), where a and b are isometries of U and V respectively, is an isometry of W. Moreover if U and V are isometrically isomorphic, then any automorphism c of the form \[ (*)\quad c(x\otimes y)=(ay)\otimes(bx), \] where a:\(V\to U\) and b:\(U\to V\) are isometries, is also an isometry of W. It is proved that every isometry of W has either the form \(c=a\otimes b\), where a and b are isometries of U and V respectively, or the form (*) in the case the dual norms on \(U^*\) and \(V^*\) are smooth (i.e. for each point x of the unit sphere the function \(t\mapsto \| x+tv\|\) is differentiable at \(t=0\) for all \(v\in V)\). A sufficient condition is also given that every c in the identity component of the group of isometries of W have the form \(c=a\otimes b\), where a and b are isometries of U and V respectively.