A double commutant relation in the Calkin algebra on the Bergman space. (English) Zbl 1394.46044
Summary: Let \(\mathcal{T}\) be the Toeplitz algebra on the Bergman space \(L_a^2(\mathbb{B}, d v)\) of the unit ball in \(\mathbb{C}^n\). We show that the image of \(\mathcal{T}\) in the Calkin algebra satisfies the double commutant relation: \(\pi(\mathcal{T}) = \{\pi(\mathcal{T}) \}^{\prime\prime}\). This is a surprising result, for it is the opposite of what happens in the Hardy-space case [J.-B. Xia, Trans. Am. Math. Soc. 360, No. 2, 1089–1102 (2008; Zbl 1144.42002); Can. J. Math. 62, No. 4, 889–913 (2010; Zbl 1205.32006)].
MSC:
| 46L05 | General theory of \(C^*\)-algebras |
| 47L80 | Algebras of specific types of operators (Toeplitz, integral, pseudodifferential, etc.) |
| 32A36 | Bergman spaces of functions in several complex variables |
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