关于幂等算子代数的注记

给出幂等算子代数的一个刻画.定义了希尔波特空间H的幂等算子代数.设Ω是B（H）上的一个子代数,且满足Ω^1=Ω,Ω^n=Ω^（n-1）Ω＋Ω^（n-2）Ω^2＋…＋ΩΩ^（n-1）,n=1,2,…,当Ω^2=Ω时,Ω是幂等的.经过研究,得出了幂等算子代数的一些重要性质.从而,进一步得到一个算子代数是幂等算子代数的充分条件.如果Ω不含单位元,对Ω中的每一个元A,都存在一个非零复数λA,使得R（A）真包含于N（A-λA）,那么,Ω就是幂等算子代数.

A note on idempotent operator algebras

A characterization for an idempotent operator algebra is established.Idempotent operator algebras acting on a Hilbert space H are defined. Let Ω be a subalgebra of B(H). The powers of Ω are defined as follows:Ω1=Ω,Ωn=Ω (n-1)Ω +Ω (n-2)Ω 2+...+ΩΩ (n-1),n=1,2,.... Then Ω is idempotent when Ω2=Ω. Some important properties of idempotent operator algebras are presented.Some sufficient conditions for a operator algebra to be an idempotent operator algebra are discussed. If Ω is not unital and for each element A in Ω, that exists a nonzero number λA such that R(A)N(A-λA), then Ω is an idempotent operator algebra.