Commutativity up to a factor is one of the invariants of isomophisms among operator algebras.The study of the question that whether the converse proposition holds true is helpful to understand the relations between commutativity up to a factor and the algebraic as well as the geometric properties of operator algebras.By using some techniques in operator theory

without the assumption that maps are unit-preserving

it is shown that every subjective additive map among the whole infinite dimensional Hilbert operator algebras that preserves commutativity up to a factor is a scalar multiple of an isomorphism

or(in some situations) a conjugate isomorphism or a conjugate anti-isomorphism.