Isometries of Cross Products of Sequence Spaces

Let X0, X1, . . . , Xk with k ∈ IN ∪ {∞} be sequence spaces (finite or infinite dimensional) over C or IR with absolute norms Ni for i = 0, . . . , k, (i.e., with 1unconditional bases) such that dimX0 = k. Define an absolute norm on the cross product space (also known as the X0 1-unconditional sum) X1 × · · · ×Xk by N(x1, . . . , xk) = N0(N1(x1), . . . , Nk(xk)) for all (x1, . . . , xk) ∈ X1 × · · · ×Xk. We show that every sequence space with an absolute norm has an intrinsic cross product structure of this form. The result is used to prove a characterization of isometries of complex cross product spaces that covers all the existing results. We demonstrate by examples and the theory of finite reflection groups that it is impossible to extend the complex result to the real case. Nevertheless, some new isometry theorems are obtained for real cross product spaces.