Paul Oblique Idempotent Or Cubic Zero Mapping. Operator Algebra Completely

The problems of characterizing maps on operator algebras preserving certain properties, subsets, functions, relations and else invariants are so called preserver problems on operator algebras. Preserver problems are new research projects on operator algebras and functional analysis, and the study of them not only enriches the theory of operator algebras and functional analysis ,but also has its practical value in quantum mechanics and else subjects. The results obtained from these preserver problems show that such maps are algebraic homomorphisms or algebraic anti-homomorphisms and then reveal the properties of algebra or geometry of operator algebras or matrix algebras.Recently, complete preserver problems on operator algebras have been studied. Many scholars studied the problem of characterizing maps completely preserving some properties of elements in operator algebras, and got a series of results. In this paper, we continue this study and discuss the maps having skew idempotency or cube-zero operators as invariants on standard operator algebras. By using the classical methods of preserver problems, the problems are converted to characterizing maps preserving idempotents, square-zero operators or rank-one operators. And then we get the characterization and classification of maps completely preserving skew-idempotents and cube-zero operators. The following are our main results.1: A characterization is given for additive maps completely preserving skew-idempotent operators on standard operator algebras on infinite dimensional real or complex Banach spaces. Our results show that such maps must be isomorphisms or conjugate-isomorphisms.2: The maps completely preserving cube-zero operators between standard operator algebras on Banach spaces and Hilbert spaces are studied respectively. It is proved that these maps are isomorphisms or (in the complex case) conjugate-isomorphisms.3: We study the maps preserving the union of {0 } and the spectrum of products of operators on standard operator algebras, and obtain the concrete forms of the maps.