Disjointness preserving operators onC *-algebras

Mappings preserving zero products

“…In case the underlying field has characteristic not 2, this condition is equivalent to J(x 2 ) = (Jx) 2 for all x in the domain. The following result was obtained essentially by Wolff [32]. For a different approach, see also the thesis of J. Schweizer, Interplay between noncommutative topology and operators on C * -algebras, Eberhard-Karls-Universität, Tübingen, 1997.…”

“…By a careful study of the original proof of Wolff in [32], we obtain a * -free version of Theorem 4.3. A great part of our arguments presented below is modeled on [32].…”

“…For a different approach, see also the thesis of J. Schweizer, Interplay between noncommutative topology and operators on C * -algebras, Eberhard-Karls-Universität, Tübingen, 1997. Note that as its original form that appeared in [32], the Jordan homomorphism J in the above theorem can be extended in a canonical way to a Jordan * -homomorphism from A into B * * . In case θ (1) is invertible, the range of J is contained in B.…”

“…However, the C * -algebra case is far more complicated. In [32], Wolff shows that if θ : A → B is a bounded linear map between unital C * -algebras preserving involution and zero products of self-adjoint elements in A then θ = θ(1)J for a Jordan * -homomorphism J from A into B * * . We obtain a * -free version of this.…”

On Maps Preserving Zero Jordan Products

“…In case à is the ordinary multiplication, a similar problem, namely that of describing maps preserving zero products, was studied in [1,15,18,30,31]. It turns out that the question of preservers on zero products is more difficult than that on zero Lie products.…”