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

Xia, Jingbo

Canadian mathematical bulletin, Tome 45 (2002) no. 2, pp. 309-318

Résumé

A well-known theorem of Sarason [11] asserts that if $\left[ {{T}_{f}},\,{{T}_{h}} \right]$ is compact for every $h\,\in \,{{H}^{\infty }}$ , then $f\,\in \,{{H}^{\infty }}\,+\,C\left( T \right)$ . Using local analysis in the full Toeplitz algebra $T\,=\,T\left( {{L}^{\infty }} \right)$ , we show that the membership $f\,\in \,{{H}^{\infty }}\,+\,C\left( T \right)$ can be inferred from the compactness of a much smaller collection of commutators $\left[ {{T}_{f}},\,{{T}_{h}} \right]$ . Using this strengthened result and a theorem of Davidson [2], we construct a proper ${{C}^{*}}$ -subalgebra $T\left( \mathcal{L} \right)$ of $T$ which has the same essential commutant as that of $T$ . Thus the image of $T\left( \mathcal{L} \right)$ in the Calkin algebra does not satisfy the double commutant relation [12], [1]. We will also show that no separable subalgebra $\mathcal{S}$ of $T$ is capable of conferring the membership $f\,\in \,{{H}^{\infty }}\,+\,C\left( T \right)$ through the compactness of the commutators $\left\{ \left[ {{T}_{f,}}\,S \right]\,:\,S\,\in \,\mathcal{S} \right\}$ .

Détail

DOI : 10.4153/CMB-2002-034-9

Mots-clés : 46H10, 47B35, 47C05