A newC *-algebra and its essential commutant

1999 ◽  
Vol 44 (3) ◽  
pp. 204-207 ◽  
Author(s):  
Jinshan Liu ◽  
Yunxin Zhang ◽  
Kunyu Guo
Keyword(s):  
Essential Commutant

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

2002 ◽  
Vol 45 (2) ◽  
pp. 309-318
Author(s):  
Jingbo Xia
Keyword(s):  
Local Analysis ◽  
Toeplitz Algebra ◽  
Calkin Algebra ◽  
Algebra Ii ◽  
Mean Oscillation ◽  
Essential Commutant ◽  
Double Commutant ◽  
Local Ideals

AbstractA well-known theorem of Sarason [11] asserts that if [Tf, Th] 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 [Tf, Th]. 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 {[Tf, S] : S ∈ Ѕ}.

scholarly journals On the essential commutant of analytic Toeplitz operators associated with spherical isometries

2011 ◽  
Vol 261 (5) ◽  
pp. 1361-1383 ◽  
Author(s):  
Michael Didas ◽  
Jörg Eschmeier ◽  
Kevin Everard
Keyword(s):  
Toeplitz Operators ◽  
Essential Commutant

scholarly journals Essential Commutants of Semicrossed Products

2015 ◽  
Vol 58 (1) ◽  
pp. 91-104 ◽  
Author(s):  
Kei Hasegawa
Keyword(s):  
Hilbert Space ◽  
Toeplitz Operators ◽  
Von Neumann Algebra ◽  
Compact Operators ◽  
Von Neumann ◽  
Finite Dimensional ◽  
Famous Result ◽  
Essential Commutant ◽  
Countable Abelian Group ◽  
Finite Dimensional Hilbert Space

AbstractLet α: G ↷ M be a spatial action of a countable abelian group on a “spatial” von Neumann algebra M and let S be its unital subsemigroup with G = S-1S. We explicitly compute the essential commutant and the essential fixed-points, modulo the Schatten p-class or the compact operators, of the w*-semicrossed product of M by S when M' contains no non-zero compact operators. We also prove a weaker result when M is a von Neumann algebra on a finite dimensional Hilbert space and (G, S) = (ℤ, ℤ+), which extends a famous result due to Davidson (1977) for the classical analytic Toeplitz operators.

Singular Integral Operators and Essential Commutativity on the Sphere

2010 ◽  
Vol 62 (4) ◽  
pp. 889-913 ◽  
Author(s):  
Jingbo Xia
Keyword(s):  
Hardy Space ◽  
Singular Integral ◽  
Unit Sphere ◽  
Toeplitz Operators ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
Essential Commutant

AbstractLet 𝒯 be the C*-algebra generated by the Toeplitz operators {T𝜑 : 𝜑 Є L∞(S, dσ)} on the Hardy space H2(S) of the unit sphere in Cn. It is well known that 𝒯 is contained in the essential commutant of {T𝜑 : 𝜑 Є VMO∩L∞(S, dσ)}. We show that the essential commutant of {T𝜑 : 𝜑 Є VMO∩L∞(S, dσ)} is strictly larger than 𝒯.

Essential commutant of analytic Toeplitz operators

1997 ◽  
Vol 42 (7) ◽  
pp. 548-552 ◽  
Author(s):  
Xuanhao Ding ◽  
Shunhua Sun
Keyword(s):  
Toeplitz Operators ◽  
Essential Commutant
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