Joint Mean Oscillation and Local Ideals in the Toeplitz Algebra II: Local Commutivity and Essential Commutant

Canadian mathematical bulletin

关键词: polynomial invariants of finite groups;    permutation representation;    transfer;   

DOI  :  10.4153/CMB-2002-034-9

学科分类：数学（综合）

来源: University of Toronto Press * Journals Division

【 摘 要 】

A well-known theorem of Sarason [11] asserts that if $[T_f,T_h]$ iscompact for every $h in H^infty$, then $f in H^infty + C(T)$.Using local analysis in the full Toeplitz algebra $calT = calT(L^infty)$, we show that the membership $f in H^infty + C(T)$ canbe inferred from the compactness of a much smaller collection ofcommutators $[T_f,T_h]$. Using this strengthened result and a theoremof Davidson [2], we construct a proper $C^ast$-subalgebra $calT(calL)$ of $calT$ which has the same essential commutant as that of$calT$. Thus the image of $calT (calL)$ in the Calkin algebra doesnot satisfy the double commutant relation [12], [1]. We will alsoshow that no {it separable} subalgebra $calS$ of $calT$ is capableof conferring the membership $f in H^infty + C(T)$ through thecompactness of the commutators ${[T_f,S] : S in calS}$.

【 授权许可】

Unknown