Isomorphisms between unital C∗-algebras

Abstract

It is shown that every almost linear bijection h : A → B of a unital C ∗-algebra A onto a unital C ∗-algebra B is a C ∗-algebra isomorphism when h ( 2 n u y ) = h ( 2 n u ) h ( y ) for all unitaries u ∈ A , all y ∈ A , and n = 0 , 1 , 2 , … , and that almost linear continuous bijection h : A → B of a unital C ∗-algebra A of real rank zero onto a unital C ∗-algebra B is a C ∗-algebra isomorphism when h ( 2 n u y ) = h ( 2 n u ) h ( y ) for all u ∈ { v ∈ A | v = v ∗ , ‖ v ‖ = 1 , v is invertible } , all y ∈ A , and n = 0 , 1 , 2 , … . Assume that X and Y are left normed modules over a unital C ∗-algebra A . It is shown that every surjective isometry T : X → Y , satisfying T ( 0 ) = 0 and T ( u x ) = u T ( x ) for all x ∈ X and all unitaries u ∈ A , is an A -linear isomorphism. This is applied to investigate C ∗-algebra isomorphisms between unital C ∗-algebras.