Characterizing Ring Isomorphisms Of Operator Algebras

In the present paper,we study the problem of characterizing ring isomorphisms of opera-tor algebras.Let A and B be C*-algebras.Assume that A is of real rank zero and unital with uait,and k>0 is a real number.It is shown that if Φ:4→B is an additive map preserving |·|k for all normal elements；that is,Φ(|A|k)=|Φ(A)|k for all normal elements A∈A,Φ(I) is a projection,and there exists a positive number c such that Φ(iI)Φ(iI)*≤cΦ(I)Φ(I)*, then Φ is the sum of a linear Jordan*-homomorphism and a conjugate-linear Jordan*-homomorphism.If, moreover,the map Φ commutes with |·|k on A,then Φ is the sum of a linear *-homomorphism and a conjugate-linear*-homomorphism.In the case when k≠1, the assumption Φ(I)being a projection can be deleted.For positive integers m≥k≥2 and a sequence (i1,…,im)with(i1,i2,…,im)=(1,2,…,k)and at lea,st one p such that the term ip appearing exactly once,the associated generaized product of T1,T2,…,Tk is defined as T1*T2*…*Tk=Ti1Ti2···Tim.Let X,Y be complex Banach spaces,dim X≥3,and let A(?)B(X),B(?)B(Y)be standard operator algebras.In this paper,it is also shown that,if ip∈{i1,im} or if m—1>2 is prime when ip(?){i1,im},then every bijective generalized mul-tiplicative map Φ:A→B(i.e.Φ satisfies Φ(A1*A2*…*Ak)=Φ(A1)*Φ(A2)*…*Φ(Ak)) is a scalar multiple of a ring isomorphism.