Entirely On Standard Operator Algebras Security The Involution Mapping Studies

Preserver problems on operator algebras are to study the maps leaving some proper-ties of elements in algebras invariant (for example, properties, functions, relations or elseinvariants). Preserver problems are new research projects on operator algebras, and often,the results obtained on this problem show that such maps are algebraic holomorphisms oranti-algebraic holomorphisms, and then they reveal the properties of algebra or geometryof operator algebras or matrix algebras. And the study of them not only enriches thetheory of functional analysis and operator algebras, but also has its practical value inquantum mechanics and else subjects.Recently, completely preserver problems on operator algebras have been studied.Many researchers studied the problem of characterizing maps completely preserving someproperties of elements in operator algebras, and have gotten a series of results. In thispaper, we continue this study and discuss the maps having involution or anti-involutionas invariants on operator algebras. By using the classical methods of preserver problems,which are converted to characterizing maps preserving idempotents, square-zero operatorsor rank-one operators. And then we get the characterization and classifcation of mapscompletely preserving involution or anti-involution.The following are our main results.1A characterization is given for additive maps completely preserving involution onstandard operator algebras on infnite dimensional real or complex Banach spaces. Our re-sults show that such map is either an isomorphism multiplied by a constant or a conjugate-isomorphism multiplied by a constant.2Following the result1, a characterization is given for general maps completelypreserving involution on standard operator algebras. And we get the results.3We study the maps completely preserving anti-involution between sets of rank oneoperators and the concrete results are given.