ADDITIVE PRESERVERS OF TENSOR PRODUCT OF RANK ONE HERMITIAN MATRICES

Let K be a field of characteristic not two or three with an involution and F be its fixed field. Let H-m be the F-vector space of all m-square Hermitian matrices over K. Let rho(m) denote the set of all rank-one matrices in H-m. In the tensor product space circle times(k)(i-1) H-mi, let circle times(k)(i=1) rho(mi) denote the set of all decomposable elements circle times(k)(i=1) A(i) such that A(i) is an element of rho(mi), i = 1, ... ,k. In this paper, additive maps T from H-m circle times H-n to H-s circle times H-t such that T(rho(m) circle times rho(n)) subset of (rho(s) circle times rho(t)) boolean OR {0} are characterized. From this, a characterization of linear maps is found between tensor products of two real vector spaces of complex Hermitian matrices that send separable pure states to separable pure states. Also classified in this paper are almost surjective additive maps L from circle times(k)(i=1) H-mi to circle times(l)(i=1) H-ni such that L (circle times(k)(i=1) rho(mi)) subset of circle times(l)(i=1) rho(ni) where 2 <= k <= l. When K is algebraically closed and K = F, it is shown that every linear map on circle times(k)(i=1) H-mi that preserves circle times(k)(i=1) rho(mi) pm is induced by k bijective linear rank-one preservers on H-mi, i = 1, ... ,k.