A Density Theorem on the Operator Algebra in Pontrjagin Space1

Kaplansky density theorem is extremely useful in dealing with ∗-algebras of operators. This theorem says that if is a C∗-algebra of B(H) with a strong closure , then the unit ball 1 of is strongly dense in the unit ball 1 of .Many deep results for the operator algebra are deduced with this technical tool.The main purpose of this paper is to give a relevant density result in Pontrjagin space. Suppose that is an operator algebra in Pontrjagin space.Denote 1 = A A ∈ σ A∗A ⊂ z ≤ 1 For any set of operators , denote the strong closure of by s . We prove the following theorem. Theorem. Let be a norm closed self-adjoint algebra of operators in Pontrjagin space, = s . Then 1 s ⊃ 1.