Analytic Toeplitz algebras and the Hilbert transform associated with a subdiagonal algebra

Ji, Guoxing

doi:10.1007/s11425-013-4684-z

Cited by 20 publications

(24 citation statements)

References 16 publications

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“…Notice that (v(k) λ [(g k + 1l)h 1/2 + iH(g k h 1/2 )]) converges weakly in L 2 to w k [(g k + 1l)h 1/2 + iH(gh 1/2 )]. By Lemma 4.5 we have that (a(k) * λ + d(k) λ + a(k) λ )h 1/2 → g k h 1/2 in L 2 -norm, and so also H((a(k) * λ + d(k) λ + a(k) λ )h 1/2 ) → H(g k h 1/2 ) in L 2 -norm by the continuity of H established in [27]. Since the v(k) λ 's are contractive, it easily follows that…”

Section: A Kaplansky Density Type Resultsmentioning

confidence: 86%

“…Inspired by this fact, the Hardy spaces H p (A) (1 ≤ p < ∞) are defined to be the closure in L p (M ) of the subspace h c/p Ah (1−c)/p where 0 ≤ c ≤ 1. (We remind the reader that the closures for the various values of c all agree [27]).…”

Section: Introductionmentioning

confidence: 70%

“…extend to a unique bounded operator on L p (M ) (the Hilbert transform) not dependent on the parameter θ. See [27,Theorem 3.2]. In the case p = ∞, the Hilbert transform H is only partially defined on M by means of the formula by…”

Section: A Kaplansky Density Type Resultsmentioning

confidence: 99%

“…Letg (resp.g n ) be the Hilbert transform of gh 1 2 (resp. g n h 1 2 ) as in [27,Section 3], and let f = gh Proof. If g n is as defined above, then by Lemma 3.3 we have h is a 'standard form' for M , where P and J respectively denote the naturally associated cone and the modular conjugation.…”

Section: A Kaplansky Density Type Resultsmentioning

confidence: 99%

“…With the theory of subdiagonal subalgebras of von Neumann algebras with a faithful normal tracial state reaching a level of maturity, several authors turned their attention to the more general σ-finite von Neumann algebras. Important structural results were obtained by Ji, Ohwada, Saito, Bekjan, and Xu [28,29,43,26,27,4]. Recently in [34] the second author used Haagerup's reduction theory [21] to make several significant advances in generalizing aspects of the earlier theory to the σfinite case, most notably the Beurling invariant subspace theory.…”

Section: Introductionmentioning

confidence: 99%

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Ueda’s peak set theorem for general von Neumann algebras

Blecher

Labuschagne

2018

Trans. Amer. Math. Soc.

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Abstract. We extend Ueda's peak set theorem for subdiagonal subalgebras of tracial finite von Neumann algebras, to σ-finite von Neumann algebras (that is, von Neumann algebras with a faithful state; which includes those on a separable Hilbert space, or with separable predual.) To achieve this extension completely new strategies had to be invented at certain key points, ultimately resulting in a more operator algebraic proof of the result. Ueda showed in the case of finite von Neumann algebras that his peak set theorem is the fountainhead of many other very elegant results, like the uniqueness of the predual of such subalgebras, a highly refined F & M Riesz type theorem, and a Gleason-Whitney theorem. The same is true in our more general setting, and indeed we obtain a quite strong variant of the last mentioned theorem. We also show that set theoretic issues dash hopes for extending the theorem to some other large general classes of von Neumann algebras, for example finite or semi-finite ones. Indeed certain cases of Ueda's peak set theorem, for a von Neumann algebra M , may be seen as 'set theoretic statements' about M that require the sets to not be 'too large'.

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Section: A Kaplansky Density Type Resultsmentioning

confidence: 86%

Section: Introductionmentioning

confidence: 70%

Section: A Kaplansky Density Type Resultsmentioning

confidence: 99%

Section: A Kaplansky Density Type Resultsmentioning

confidence: 99%