Pontryagin Space J-symmetric Operator Algebra Of A Number Of Issues

The thesis mainly concerns J-symmetric operator algebra on Pontryagin space. We discussed the generators of commutative J-von Neumann algebras on separable Pontryagin space,the relevant Kaplansky density theorem for J-symmetric operator algebra on Pontryagin space.The J-symmetric ideal,J-approximate unit, irreducibility of JC^*- algebras on∏₁ space,J-symmetry for J-reductive algebra on Pontryagin space are discussed.Moreover,we discussed the disconnection of spectral space of commutative J-von Neumann algebra on separable∏_k space,derivations of J-von Neumann algebra on∏₁ space,the double commutant of commutative J-von Neumann algebra on separable∏₂ space.In chapter 1,we give some preliminaries of indefinite inner product space and the main results of this thesis.It is well known that if(?) is a commutative von Neumann algebra on separable Hilbert space,it can be generated by a selfadjoint operator.However,the relevant result is false for commutative J-von Neumann algebra on separable∏_k space.In chapter 2,based on the results of V.Strauss,Sh.Z.Yan,Y.S.Tong,we concern the generators of commutative J-von Neumann algebra on separable∏_k space.We show that a commutative J-von Neumann algebra(?) in Pontryagin space can be decomposed into the sum of a commutative nilpotent J-von Neumann algebra and a commutative J-von Neumann algebra that has finite number of generators.As the application of the results,we give a necessary and sufficient conditon of J-unitary equivalence of commutative J-von Neumann algebra on separable∏₁ space.Kaplansky density theorem is extremely useful in dealing with ^*-algebra of operators.There are two natural directions for generalizing the Kaplansky density theorem to∏_k space.Y.S.Tong got perfect results in one direction.In another direction,S.Sh.Masharipova generalized the Kaplansky density theorem to∏₁ space. In chapter 3,we discussed the relevant Kaplansky density theorem for a type of J-symmetric operator algebra(?) on∏_k space.We prove that there is a bounded subset(?)_c(?),such that(?)_c is strongly dense in ball of the strong closer of(?). Moreover,the result of S.Sh.Masharipova is the special case of the results in this thesis.It is well known that every closed ideal of C^*-algebra is symmetric.Moreover, every closed ideal of C^*-algebra admits approximate unit.Kadison and Richard.V proved the startling fact that if a C^*- algebra of operators acting on a Hilbert space has no nontrivial invariant closed linear subspaces,then it has no nontrivial invariant subspace.However,these results are not true for JC^*-algebra.In chapter 4,we discussed J-symmetric ideal,J-approximate unit,irreducibility of JC^*- algebras on∏₁ space.In chapter 5,the J-reductive algebra in∏_k space is discussed.We show that the non-degenerate J-reductive algebra containing a maximal commutative J-von Neumann algebra must be a J-von Neumann algebra,the non-degenerate J-reductive algebra consisting of J-normal operators must be a J-von Neumann algebra.Moreover, we prove that the degenerate J-reductive algebra(?) on∏₁ space,containing a maximal commutative J-von neumann algebra which does not belong to class M¹, must be a J-von Neumann algebra.Every derivation of a von Neumann algebra is inner.The spectral of commutative von Neumann algebra on separable Hilbert space is extremely disconnected. A von Neumann algebra equals its double commutant.However,these results can not be extended to the J-von Neumann algebra on Pontryagin space.In chapter 6, based on the results of others,we discuss the disconnection of commmutative J-von Neumann algebra on separable∏_k space,the inner derivations of J-von Neumann algebra on∏₁ space and the double commutant of commmutative J-von Neumann algebra on separable∏₂ space.