Linear Maps Preserving Idempotent Operators

Operator algebra theory is produced in the1930s, it has different equations, forexample, systematical control, mathematical statistics and so on, all haveunexpected links and penetration with it. For the past forty years, some scholarsbegan to pay attention on some abstract keep characterization of the problem inB (X). Because many retentive problems on operator algebra can be summarizedthat the rank can be kept or the rank does not increase. Therefore, linearly oradditive mapping that keeps the rank one or the rank one does not increase onoperator algebra is particularly important. In this paper, to obtain conclusion, we usethe condition to proof that keeps the rank or the rank does not increase.Suppose X and Y are Banach space upon. Respectively, let B (X)and B (Y)be the algebras of all bounded linear operators on X,Y. Let F (X)denote the set of all finite-rank operators in B (X). Let a standard operator algebraon X is a norm closed subalgebra of B (X)which contains the identity and allfinite-rank operarors. Let and be standard operator algebra.In this paper, after we have introduced the context and the importence ofidempotence preserving, based on the forms of preserving idempotent operators inboth direction, we emphasize on the forms of linear idemotence preserving mapsand weakly continuous linear idempotence preserving maps, and we obtain thefollowing conclusion：(1) We impair the condition of linear idempotence preserving surjective mapsφ：A→B in both direction on finite-rank operators, giving the concrete forms oflinear idempotence preserving surjective maps on standard operator algebra, theref-ore we give the concrete forms of linear idempotence preserving maps on B (X)and M_n(C).(2) We give the concrete forms of weakly continuous linear idempotence pres- erving maps on standard operator algebra, therefore we give the concrete forms ofweakly continuous linear idempotence preserving maps on B (X).