Operator Algebras On Pontrjagin Space

We consider the operator algebras on the Pontrjagin spaces, and study the classification problem of degenerate operator algebras, the derivations problem of operator algebras , and the symmetric problem of ideals in operator algebras. The paper consists of four chapters.In chapter 1 , we introduce simply the background and study actuality of problems of operator algebras on the Pontrjagin spaces, and useful main concepts and results for following chapters.In chapter 2, we give the classification concept of degenerate operator algebras on the nk spaces by symmetric ideals , and show that thus operator algebra consists of six classes. And give general forms of every class of degenerate operator algebras by the representations of this algebra and constructions of symmetric ideals . Further, give the equivalence conditions for weakly closed and uniformly closed of degenerate operator algebras on the k spaces , and equivalence conditions for commutativity of every class of operator algebras, and bicommutity theorem of degenerate operator algebras of class 0, II a and IIIa on the k spaces.In chapter 3, we prove that the ideals in the nodegenerate J.V.N-algebras and JC*-algebras on the IIk spaces are symmetric. For the degenerate JC*-algebras, give some ideals of symmetric and nosymmetric. And explain that in the JC*-algebras of class 0, I and II have not nosymmmetric ideal, however, in the algebras of class III have some nosymmmetric ideal. Finally, give two examples on the ideal symmetry .In chapter 4, we prove that the inner derivation of operator algebras is a invariant concept on the unitary equivalence transformations , and give the equivalence condition for inner derivations of nodegenerate J.V.N-algebras on the k spaces. For the degenerate J.V.N-algebras of class 0, II a and IIIa on the H1 spaces, the derivations on them are inner . And explain that the derivations on the J.V.N-algebras of class I ,IIb and IIIb are not , generally , inner by examples.