Pontrjagin空间上一般算子代数弱闭和一致闭的等价条件

本文研究Pontrjagin空间上一般算子代数弱闭和一致闭的等价条件,得到定理:设C0(U),C1(U,L,R,D,V),C2a(U),C2b(U,R),C3a(U),C3b(U,R)分别是Ⅱk空间上第0,Ⅰ,Ⅱa,Ⅱb,Ⅲa和Ⅲb类的算子代数,则(1)C0(U),C2a(U)或C3a(U)为一致闭(弱闭)的等价条件是U是Hibert空间G上的C*-代数(W*-代数;(2)C1(U,L,R,D,V)为一致闭(弱闭)的等价条件是U是Hibert空间H上的C*-代数(W*-代数),并且R是闭子空间,V是闭算子,L对称闭的;(3)C2b(U,R)或C3b(U,R)为一致闭(弱闭)的等价条件是U是Hibert空间H上的C*-代数(W*-代数),并且R是闭子空间．

Equivalence Conditions of Weakly and Uniformly Closeduess on the General Operator Algebra in Pontrjagin Space

The main purpose of this paper is to study that problem, it is equivalent conditions of weakly and uniformly closed on the generate operator algebra in Pontrja-gin space. We prove the following theorem: Suppose that C0(U),C1(U,L,R,D,V),C2a(U),C2b(U,R),C3a(U),C3b(U,R) are, respectively, classes 0,Ⅰ,Ⅱa,Ⅱb,Ⅲa andⅢb of general symmetric operator algebras on spaceⅡk. Then (1)C0(U), C2a(U) or C3a(U) is uniformly (weakly) closed if and only if u is C*-algebra (W*-algebra) on the Hilbert space H; (2) C1(U,L,R,D,V) is uniformly (weakly) closed if and only if U is C*-algebra (W*-algebra) on the Hilbert space H, and R is closed subspace, V is a closed operator, L is symmetrically closed; (3) C2b(U,R) or C3B(U,R) is uniformly (weakly) closed if and only if U is C*-algebra (W*-algebra) on the Hilbert space H, and R is a closed subspace.