关于二宽度CSL代数的Jacobson根

On the Jacobson Radical of the Width-Two CSL Algebra

Ｈｏｐｅｎｗａｓｓｅｒ Ａ［１］猜想ＣＳＬ代数上满足 Ｒｉｎｇｒｏｓｅ条件的算子集正是它的Ｊａｃｏｂｓｏｎ根，Ｄａｖｉｄｓｏｎ Ｋ．Ｒ．［２］证明了对于二宽度 ＣＳＬ代数，上述猜想是完全正确的．本文不仅清楚地刻画了二宽度ＣＳＬ代数Ｊａｃｏｂｓｏｎ根的结构，而且为研究ＣＳＬ代数的根提供了一种途径．设是由可分Ｈｉｌｂｅｒｔ空间上的套Ｍ和Ｎ生成的二宽度 ＣＳＬ，且 Ｗ＝ Ｍ∩Ｎ；本文得到二宽度 ＣＳＬ代数的 Ｊａｃｏｂｓｏｎ根与套Ｗ的根Ｒｗ，强根三者之间的一个重要关系同时也给出了真包含Ｒｗ的充分必要条件是Ｍ≠Ｎ且Ｍ≠Ｎ⊥．

Hopenwasser A.[1] guessed that the operator set of the CSL algebra satisfying Ringrose condition just is its Jacobson radical; Davidson K. R.[2] proved that this conjecture is true on the width-two CSL. In this paper, the structure of Jacobson radical on the width-two CSL is described clearly, and an. aproach for analyzing the CSLs radical is provided. Let be a width-two CSL generated by nests M and N in a separable Hilbert space, and W = M∩ N. We prove an important relationship , where is the Jacobson radical of alga and Rw, are the Jacobson radical and the stronger radical of nest W respectively. Also, it has been derivedt that the sufficient and necessary conditions for are M ≠ K and M≠K⊥.