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Journal of Operator Theory

Volume 76, Issue 1, Summer 2016  pp. 107-131.

Toeplitz operators and Toeplitz algebra with symbols of vanishing oscillation

Authors:  Jingbo Xia (1) and Dechao Zheng (2)
Author institution: (1) Department of Mathematics, State University of New York at Buffalo, Buffalo, NY 14260, U.S.A.
(2) Center of Mathematics, Chongqing University,\break Chongqing, 401331, P.R. China \textit{and} Department of Mathematics, Vanderbilt University, Nashville, TN 37240, U.S.A.

Summary:  We study the $C^\ast $-algebra generated by Toeplitz operators with symbols of vanishing (mean) oscillation on the Bergman space of the unit ball. We show that the index calculation for Fredholm operators in this $C^\ast $-algebra can be easily and completely reduced to the classic case of Toeplitz operators with symbols that are continuous on the closed unit ball. Moreover, in addition to a number of other properties, we show that this $C^\ast $-algebra has uncountably many Fredholm components.

Keywords: Fredholm index, Toeplitz algebra, vanishing oscillation

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