Research On Mappings Of Invariants Such As Completely Preserving Commutation And Oblique Commutation On Standard Operator Algebras

The commutativity and the skew-commutativity between the operators are very important concepts in mathematical theory, and has an important role in the considerable measure of quantum mechanics and its spectrum analysis. So, the commutativity and the skew-commutativity between the operators have also been extensively studied in the field of mathematics. The preserver maps are studied in preserver problems. Usually, certain properties, functions, subsets, and transformation are as invariants. The results obtained from these preserver problems always show that such maps are algebra homomorphisms or algebra anti-homomorphisms, and reveal the properties of algebra or geometry of operator algebras or matrix algebras.In recent years, many scholars have studied the problem of characterizing maps completely preserving certain properties on operator algebras, and got a series beautiful and extensively results. Based on the above of the results, we continue this study and discuss the preserver maps. We choose the commutativity, Jordan zero product, commutativity up to a factor, and the skew-commutativity of operators as invariants in this paper. The problems are converted to characterizing maps preserving idempotents, rank-one operators or projection operators. And then we get the characterizations and classifications of maps completely preserving the commutativity, Jordan zero-product, commutativity up to a factor and skew Lie zero product of operators on standard operator algebras.The following are our main results:1. The maps completely preserving commutativity and Jordan zero-product on standard operator algebras on infinite dimensional (real or complex) Banach spaces are studied respectively. Our results show that such maps must be a scalar multiple of isomorphisms or a scalar multiple of conjugate isomorphisms.2. Characterizations are given for maps completely preserving commutativity up to a factor or commutativity up to different factors on standard operator algebras on infinite dimensional (real or complex) Banach spaces. Our results show that such maps must be a scalar multiple of isomorphisms or (in the complex case) a scalar multiple of conjugate isomorphisms.3. We study the maps completely preserving skew Lie zero product on standard operator algebras on infinite dimensional complex Hilbert spaces, and obtain the concrete forms of the maps.