location:  Publications → journals
Search results

Search: MSC category 47C05 ( Operators in algebras )

 Expand all        Collapse all Results 1 - 1 of 1

1. CMB 2002 (vol 45 pp. 309)

Xia, Jingbo
 Joint Mean Oscillation and Local Ideals in the Toeplitz Algebra II: Local Commutivity and Essential Commutant A well-known theorem of Sarason [11] asserts that if $[T_f,T_h]$ is compact 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 $f \in H^\infty + C(T)$ can be inferred from the compactness of a much smaller collection of commutators $[T_f,T_h]$. Using this strengthened result and a theorem of 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 does not satisfy the double commutant relation [12], [1]. We will also show that no {\it separable} subalgebra $\calS$ of $\calT$ is capable of conferring the membership $f \in H^\infty + C(T)$ through the compactness of the commutators $\{[T_f,S] : S \in \calS\}$. Categories:46H10, 47B35, 47C05
 top of page | contact us | privacy | site map |