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如所周知,遗传及全遗传C~*-子代数在C~*-代数的Morita等价理论及相关课题研究中起着很重要的作用。Edwards在文献[3]中把遗传C~*-子代数概念推广到了非结合代数——JB代数中,并获得了 命题A(文献[3],定理2.3)设A是JB-代数,则A的遗传JB-子代数与A的二次理想(即内理想)一致。 最近Edwards与Rttimann在文献[4]中证明了 命题B(文献[4],推论2.2) 设A是JB-代数,B为其JB-子代数,则B是A的二次理想(内理想)的充要条件是:B~(*+)中的任意正线性泛函到A~(*+)中的保范扩张唯一。 本文从此出发,给出了JB-子代数成为遗传JB-子代数的若干充要条件。进而又给出了全遗传JB-子代数的一个刻画。
As we all know, the genetic and hereditary C ~ * - subalgebras play an important role in the Morita equivalence theory of C ~ * - algebras and related topics. Edwards [3] extended the concept of genetic C * - subalgebras to unbound algebra - JB algebras and obtained Proposition A ([3], Theorem 2.3). Let A be a JB-algebra, then A The genetic JB-subalgebras are consistent with A’s second-order ideal (ie internal ideal). Recently, Edwards and Rttimann proved Proposition B in [4] (Reference [4], Corollary 2.2). Let A be a JB-algebra and B be a JB-subalgebra. Then B is a quadratic ideal of A The necessary and sufficient conditions are: The existence of any positive linear function in B ~ (* +) to the uniqueness of the canonical expansion in A ~ (* +). Starting from this, we give some necessary and sufficient conditions for JB-subalgebra to be genetic JB-subalgebra. Then we give a characterization of the whole genetic JB-subalgebra.