Maximal Subdiagonal Algebra Research Articles

Beurling type representation for certain invariant subspaces of maximal subdiagonal algebras

Let A be a maximal subdiagonal algebra in a σ-finite von Neumann algebra M with respect to a faithful normal conditional expectation Φ. We consider certain invariant subspaces of A in the Haagerup noncommutative Lp space Lp(M) (1≤p≤∞). Suppose that φ is a faithful normal state on M with φ∘Φ=φ and {σtφ:t∈R} is the modular automorphism group of M associated with φ, we give a Beurling type representation for those right invariant subspaces of A in Lp(M) which are also invariant under {σtφ:t∈R}. Furthermore, we give an algebraic characterization for a subdiagonal algebra to be type 1 and also give a type decomposition for a central projection in M associated with the center of D.

Noncommutative Calderón–Lozanovskiĭ–Hardy spaces

Let $${\mathcal {M}}$$ be a diffuse von Neumann algebra with a faithful normal semi-finite trace $$\tau $$ , and let E be a symmetric quasi-Banach space. Then for any Orlicz function $$\varphi $$ , we can define the noncommutative Calderon–Lozanovskiĭ spaces $$E_\varphi ({\mathcal {M}})$$ . These spaces share many properties with their classical counterparts. In particular, new multiplication operator spaces and complex interpolation spaces of such spaces are given under a wide range of conditions. Moreover, letting $${\mathcal {A}}$$ be a maximal subdiagonal algebra of $${\mathcal {M}}$$ , we introduce the noncommutative Calderon–Lozanovskiĭ–Hardy spaces $$H^\varphi ({\mathcal {A}})$$ and transfer the recent results of the noncommutative Hardy spaces to the noncommutative Calderon–Lozanovskiĭ–Hardy spaces.

An extension of the Beurling-Chen-Hadwin-Shen theorem for noncommutative Hardy spaces associated with finite von Neumann algebras

In 2015, Yanni Chen, Don Hadwin and Junhao Shen proved a noncommutative version of Beurling’s theorem for a continuous unitarily invariant norm α on a tracial von Neumann algebra (M, τ) such that α is one dominating with respect to τ . The role of H∞ is played by a maximal subdiagonal algebra A. In the talk, we first will show that if α is a continuous normalized unitarily invariant norm on (M, τ), then there exists a faithful normal tracial state ρ on M and a constant c > 0 such that α is a c times one norm-dominating norm on (M,ρ). Moreover, ρ(x) = τ(xg), where x ∈M , g is positive in L1(Z, τ), where Z is the center of M . Here c and ρ are not unique. However, if there is a c and ρ so that the Fuglede-Kadison determinant of g is positive, then Beurling-Chen-Hadwin-Shen theorem holds for L(α)(M, τ). The key ingredients in the proof of our result include a factorization theorem and a density theorem for for L(α)(M,ρ). Talk time: 2016-07-19 03:00 PM— 2016-07-19 03:20 PM Talk location: Cupples I Room 207

Interpolation of Haagerup noncommutative Hardy spaces

Let M be a σ-finite von Neumann algebra equipped with a normal faithful state φ, and let A be a maximal subdiagonal algebra of M. We prove a Stein–Weiss-type interpolation theorem of Haagerup noncommutative Hp-spaces associated with A.

Subdiagonal algebras with Beurling type invariant subspaces

Let A be a maximal subdiagonal algebra in a σ-finite von Neumann algebra M. If every right invariant subspace of A in the noncommutative Hardy space H2 is of Beurling type, then we say A is of type 1. We determine generators of these algebras and consider a Riesz type factorization theorem for the noncommutative H1 space. We show that the right analytic Toeplitz algebra on the noncommutative Hardy space Hp associated with a type 1 subdiagonal algebra with multiplicity 1 is hereditary reflexive.

Szegö type factorization of Haagerup noncommutative Hardy spaces

AbstractLet M be a σ-finite von Neumann algebra equipped with a normal faithful state φ, and let A be a maximal subdiagonal algebra of M. We proved a Szegö type factorization theorem for the Haagerup noncommutative Hp-spaces.

Noncommutative Hardy–Lorentz spaces associated with semifinite subdiagonal algebras

Let A be a maximal subdiagonal algebra of semifinite von Neumann algebra M. For 0<p≤∞, we define the noncommutative Hardy–Lorentz spaces Hp,ω(A), and give some properties of these spaces. We obtain that the Herglotz maps are bounded linear maps from Λωp(M) into Λωp(M), and with this result we characterize the dual spaces of Hp,ω(A) for 1<p<∞. We also present the Hartman–Wintner spectral inclusion theorem of Toeplitz operators and Pisier’s interpolation theorem for this case.

Von Neumann Algebras and Extensions of Inverse Semigroups

AbstractIn the 1970s, Feldman and Moore classified separably acting von Neumann algebras containing Cartan maximal abelian self-adjoint subalgebras (MASAs) using measured equivalence relations and 2-cocycles on such equivalence relations. In this paper we give a new classification in terms of extensions of inverse semigroups. Our approach is more algebraic in character and less point-based than that of Feldman and Moore. As an application, we give a restatement of the spectral theorem for bimodules in terms of subsets of inverse semigroups. We also show how our viewpoint leads naturally to a description of maximal subdiagonal algebras.

Noncommutative Hardy space associated with semi-finite subdiagonal algebras

Let M be a von Neumann algebra with a faithful normal semi-finite trace τ, and let A be maximal subdiagonal algebra of M. Then for 0<p<∞, we define the noncommutative Hp-space. We obtain that the conjugation and Herglotz maps are bounded linear maps from Lp(M) into Lp(M) for 1<p<∞, and continuous map from L1(M) into L1,∞(M). We also give the dual space of Hp(A) for 1≤p<∞, and extend Pisier's interpolation theorem to this case.

Some results on noncommutative Hardy-Lorentz spaces

Let \(\mathcal{A}\) be a maximal subdiagonal algebra of a finite von Neumann algebra ℳ. For \(0< p<\infty\), we define the noncommutative Hardy-Lorentz spaces and establish the Riesz and Szegö factorizations on these spaces. We also present some results of Jordan morphism on these spaces.

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

Let \(\mathcal{M}\) be a σ-finite von Neumann algebra and let \(\mathfrak{A} \subseteq \mathcal{M}\) be a maximal subdiagonal algebra with respect to a faithful normal conditional expectation Φ. Based on the Haagerup’s noncommutative Lp space \(L^p \left( \mathcal{M} \right)\) associated with \(\mathcal{M}\), we consider Toeplitz operators and the Hilbert transform associated with \(\mathfrak{A}\). We prove that the commutant of left analytic Toeplitz algebra on noncommutative Hardy space \(H^2 \left( \mathcal{M} \right)\) is just the right analytic Toeplitz algebra. Furthermore, the Hilbert transform on noncommutative \(L^p \left( \mathcal{M} \right)\) is shown to be bounded for 1 < p < ∞. As an application, we consider a noncommutative analog of the space BMO and identify the dual space of noncommutative \(H^1 \left( \mathcal{M} \right)\) as a concrete space of operators.

A Noncommutative Version of H p and Characterizations of Subdiagonal Algebras

Let \({\mathcal M}\) be a σ-finite von Neumann algebra and \({\mathfrak A}\) a maximal subdiagonal algebra of \({\mathcal M}\) with respect to a faithful normal conditional expectation \({\Phi}\) . Based on Haagerup’s noncommutative Lp space \({L^p(\mathcal M)}\) associated with \({\mathcal M}\) , we give a noncommutative version of Hp space relative to \({\mathfrak A}\) . If h0 is the image of a faithful normal state \({\varphi}\) in \({L^1(\mathcal M)}\) such that \({\varphi\circ \Phi=\varphi}\) , then it is shown that the closure of \({\{\mathfrak Ah_0^{\frac1p}\}}\) in \({L^p(\mathcal M)}\) for 1 ≤ p < ∞ is independent of the choice of the state preserving \({\Phi}\) . Moreover, several characterizations for a subalgebra of the von Neumann algebra \({\mathcal M}\) to be a maximal subdiagonal algebra are given.

Toeplitz and Hankel operators associated with subdiagonal algebras

Let M be a σ-finite von Neumann algebra and let A ⊂ M be a maximal subdiagonal algebra with respect to some faithful normal expectation E on M. Let φ be a normal faithful E-invariant state on M, let L 2 (M, φ) be the non-commutative Lebesgue space in the sense of U. Haagerup, and consider the Hardy space H 2 (A, φ) C L 2 (M, φ) associated with the pair (A, φ). For each x ∈ M, the Toeplitz operator T x ∈ B(H 2 (A, φ)) and the Hankel operator H x ∈ B ( H 2 (A, φ), H 2 (A, φ)⊥) are defined as in the classical case of the unit circle. We show that the mapping x ↦ T x is completely isometric on M and therefore σ(x) C σ(T x ) for all x ∈ M. We also show that ∥H x ∥ = dist(x, A) for every x ∈ M.

Commutants of certain analytic operator algebras

We prove that algebraic commutants of maximal subdiagonal algebras and of analytic operator algebras determined by flows in a σ \sigma -finite von Neumann algebra are self-adjoint.

A noncommutative Szegö theorem for subdiagonal subalgebras of von Neumann algebras

For almost forty years now the most frustrating open problem regarding the theory of finite maximal subdiagonal algebras has been the question regarding the universal validity of a non-commutative Szegö theorem and Jensen inequality (Arveson, 1967). These two properties are known to be equivalent. Despite extensive efforts by many authors, their validity has to date only been established in some very special cases. In the present note we solve the general problem in the affirmative by proving the universal validity of Szegö’s theorem for finite maximal subdiagonal algebras.

Hilbert transform associated with finite maximal subdiagonal algebras

AbstractLet ℳ be a von Neumann algebra with a faithful normal trace τ, and let H∞ be a finite, maximal. subdiagonal algebra of ℳ. We prove that the Hilbert transform associated with H∞ is a linear continuous map from L1 (ℳ, τ) into L1.∞ (ℳ, τ). This provides a non-commutative version of a classical theorem of Kolmogorov on weak type boundedness of the Hilbert transform. We also show that if a positive measurable operator b is such that b log+b ∈ L1 (ℳ, τ) then its conjugate b, relative to H∞ belongs to L1 (ℳ, τ). These results generalize classical facts from function algebra theory to a non-commutative setting.

Factorization in Subdiagonal Algebras

Let M be aσ-finite von Neumann algebra and let A be a maximal subdiagonal algebra of M with respect to a faithful normal conditional expectationΦ. We show that ifSis an invertible operator in M, then there exists an isometryWin M such that bothS−1WandW*Sbelong to A. We also give several characterizations of a maximal subdiagonal algebra A such that every invertible operatorSin M can be factored asUA, whereUis a unitary operator inMand bothAandA−1are in A.

Noncommutative 𝐻² spaces

Let M \mathcal {M} be a von Neumann algebra with a faithful, finite, normal tracial state τ \tau , and let A \mathcal {A} be a finite, maximal subdiagonal algebra of M \mathcal {M} . Let H 2 H^2 be the closure of A \mathcal {A} in the noncommutative Lebesgue space L 2 ( M , τ ) L^2(\mathcal {M},\tau ) . Then H 2 H^2 possesses several of the properties of the classical Hardy space on the circle, including a commutant lifting theorem, some results on Toeplitz operators, an H 1 H^1 factorization theorem, Nehari’s Theorem, and harmonic conjugates which are L 2 L^2 bounded.

Generalized interpolation in finite maximal subdiagonal algebras

Non-selfadjoint operator algebras have been studied since the paper of Kadison and Singer in 1960. In [1], Arveson introduced the notion of subdiagonal algebras as the generalization of weak *-Dirichlet algebras and studied the analyticity of operator algebras. After that, we have many papers about non-selfadjoint algebras in this direction: nest algebras, CSL algebras, reflexive algebras, analytic operator algebras, analytic crossed products and so on. Since the notion of subdiagonal algebras is the analogue of weak *-Dirichlet algebras, subdiagonal algebras have many fruitful properties from the theory of function algebras. Thus, we have several attempts in this direction: Beurling–Lax–Halmos theorem for invariant subspaces, maximality, factorization theorem and so on.

Isometric isomorphisms of Cartan bimodule algebras

This paper is dedicated to proving a single result: that isometric isomorphisms of Cartan bimodule algebras can be extended to ∗-isomorphisms of the generated von Neumann algebras. If M is a von Neumann algebra with Cartan subalgebra A , then a (Cartan) bimodule algebra is simple a σ-weakly closed subalgebra S of M which contains A ; S generates M if there is no proper von Neumann subalgebra of M containing S . The key idea is that a theorem of Muhly, Saito, and Solel concerning isomorphisms of maximal subdiagonal algebras can be extended to this much wider class of algebras if we restrict our attention to isometric isomorphisms. Although the algebras are not assumed to be hyperfinite, a finite-dimensional result of Davidson and Power lies at the leart of the proof. What makes the use of finite-dimensional techniques possible is the existence of “sufficiently many” finite subequivalence relations of an arbitrary countable measurable equivalence relation.