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

Published online by Cambridge University Press:  20 November 2018

Jingbo Xia
Jingbo Xia*
Affiliation:
Department of Mathematics, State University of New York at Buffalo, Buffalo, NY 14260, USA, e-mail: jxia@acsu.buffalo.edu
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Abstract

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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 $\left\{ \left[ {{T}_{f,}}\,S \right]\,:\,S\,\in \,\mathcal{S} \right\}$.

Keywords

46H1047B3547C05
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 45 , Issue 2 , 01 June 2002 , pp. 309 - 318
Copyright
Copyright © Canadian Mathematical Society 2002

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