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

2002; Cambridge University Press; Volume: 45; Issue: 2 Linguagem: Inglês

10.4153/cmb-2002-034-9

1496-4287

Jingbo Xia,

Advanced Banach Space Theory

Abstract A well-known theorem of Sarason [11] asserts that if [ T f , T h ] is compact for every h ∈ H ∞ , then f ∈ H ∞ + C(T) . Using local analysis in the full Toeplitz algebra τ = τ ( L ∞ ), we show that the membership f ∈ H ∞ + 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 * -subalgebra τ ( L )) of τ which has the same essential commutant as that of τ . Thus the image of τ ( ℒ ) in the Calkin algebra does not satisfy the double commutant relation [12], [1]. We will also show that no separable subalgebra Ѕ of τ is capable of conferring the membership f ∈ H ∞ + C(T) through the compactness of the commutators {[ T f , S ] : S ∈ Ѕ }.