Maps Preserving Generalized Projection Of Products And Triple Products On Operator Spaces

Operator space theory is a vital subject in the mathematical learning system. At present, it has been widely and quickly developed after scholars research. Mean-while, we know that projection is a kind of operator which satisfies A2= A= A* by functional analysis. Now if the operator satisfies the A2= A*, we obviously find that this operator is a generalization of the projection and we call it the generalized projection. We know that the research on preservation problem about projection which is a kind of classical operators has already tended to mature. But, how about preservation problem about the generalized projection? Therefore, based on the particularity of the generalized projection and the structure of Jordan triple product, the paper mainly studies maps preserving product and Jordan triple prod-uct of generalized projection. Firstly, we discuss the characterization of additive map preserving nonzero generalized projection of product of two operators. In the proof process, the first few steps proved that the map preserves orthogonality of projection operators in both directions. On this basics, we obtain that this map is isomorphism or anti-isomorphism by a scalar by famous Uhlhorn’s theorem. Sec-ondly, we consider the characterization of additive map bilateral preserving nonzero generalized projection of Jordan triple product of two operators and the characteri-zation of linear map unilateral preserving nonzero generalized projection of Jordan triple product, of two operators. Through the research, we find such maps are similar with the former map in structure. The study of this topic, not only enrich the operator space theory, but also play a very important significance on spectral decomposition and the preservation problem about generalized projection.