Abstract
Let A be a C * -algebra of real rank zero and B be a C * -algebra with unit I.I t is shown that if φ : A -→ B is an additive mapping which satisfies |φ(A)φ(B) |≤ φ(|AB| )f or every A, B ∈ A+ and φ(A )= I for some A ∈ As withA �≤ 1, then the restriction of mapping φ to As is a Jordan homomorphism, where As denotes the set of all self-adjoint elements. We will also show that if φ is surjective preserving the product and an absolute value, then φ is a C-linear or C-antilinear *-homomorphism on A. MSC: Primary 47B49; Secondary 46L05; 47L30
Highlights
Introduction and preliminariesIn recent years, the subject of linear preserver problems is the focus of attention of many mathematicians, and much research has been going on in this area
The subject of linear preserver problems is the focus of attention of many mathematicians, and much research has been going on in this area
We say that a mapping φ : A –→ B is preserving absolute values of a product if |φ(A)φ(B)| = φ(|AB|) (resp. |φ(A)φ(B)| ≤ φ(|AB|)) for every A, B ∈ A, where |A| = A*A
Summary
The subject of linear preserver problems is the focus of attention of many mathematicians, and much research has been going on in this area. Let A and B be two C*-algebras with unit I. We say that a mapping φ : A –→ B is preserving Sub-preserving) absolute values of a product if |φ(A)φ(B)| = φ(|AB|) By a *-homomorphism we just mean a map φ : A –→ B which preserves the ring structure and for which φ(A*) = φ(A)* for every A ∈ A. *-homomorphism if it is R-linear, φ(A*) = φ(A)* and φ(A) = φ(A ) for all A ∈ A. We say a map φ : A → B is unital if φ(I) = I.
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