Σ-finite Von Neumann 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.

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.

Choquet Order and Jordan Morphisms of Operator Algebras

We show that ordinal isomorphisms of orthogonal measures on state spaces of operator algebras equipped with the Choquet order are generated by Jordan isomorphisms of associated von Neumann algebras. This yields a new Jordan invariant of σ-finite von Neumann algebras in terms of decompositions of states.

Jordan Invariants of Von Neumann Algebras Given by Abelian Subalgebras and Choquet Order on State Spaces

New complete invariants for Jordan parts of von Neumann algebras are presented. We shall prove that the poset of all finite dimensional abelian von Neumann subalgebras ordered by set theoretic inclusion is a complete Jordan invariant for von Neumann algebras. On the other hand, we exhibit an example showing that not any order isomorphism on this structure is derived from a Jordan isomorphism. We apply our results to the Choquet order of orthogonal measures on state spaces of von Neumann algebras. Among others we show that the poset of decompositions of a fixed faithful normal state on a von Neumann algebra endowed with the Choquet order is a complete Jordan invariant for σ-finite von Neumann algebras.

On weak* convergent sequences in duals of symmetric spaces of τ-measurable operators

It is shown that the pre-dual of a σ-finite von Neumann algebra has property (k) in the sense of Figiel, Johnson and Pelczynski [12]. This resolves in the affirmative an open question raised in [12]. It is shown further that a weakly sequentially complete symmetric space E of τ-measurable operators affiliated with a semifinite σ-finite von Neumann algebra has property (k).

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.

Asymptotic structure of free product von Neumann algebras

AbstractLet (M, ϕ) = (M1, ϕ1) * (M2, ϕ2) be the free product of any σ-finite von Neumann algebras endowed with any faithful normal states. We show that whenever Q ⊂ M is a von Neumann subalgebra with separable predual such that both Q and Q ∩ M1 are the ranges of faithful normal conditional expectations and such that both the intersection Q ∩ M1 and the central sequence algebra Q′ ∩ Mω are diffuse (e.g. Q is amenable), then Q must sit inside M1. This result generalizes the previous results of the first named author in [Ho14] and moreover completely settles the questions of maximal amenability and maximal property Gamma of the inclusion M1 ⊂ M in arbitrary free product von Neumann algebras.

Orthogonal Measures on State Spaces and Context Structure of Quantum Theory

An interplay between recent topos theoretic approach and standard convex theoretic approach to quantum theory is discovered. Combining new results on isomorphisms of posets of all abelian subalgebras of von Neumann algebras with classical Tomita’s theorem from state space Choquet theory, we show that order isomorphisms between the sets of orthogonal measures (resp. finitely supported orthogonal measures) on state spaces endowed with the Choquet order are given by Jordan ∗-isomorphims between corresponding operator algebras. It provides new complete Jordan invariants for σ-finite von Neumann algebras in terms of decompositions of states and shows that one can recover physical system from associated structure of convex decompositions (discrete or continuous) of a fixed state.

The characterization of a class of quantum Markov semigroups and the associated operator-valued Dirichlet forms based on Hilbert C*-module l 2(A)

In this paper, we introduce the concept of operator-valued quadratic form based on Hilbert W*-module \(l_2 \bar \otimes A\), and give a one to one correspondence between the set of positive self-adjoint regular module operators on \(l_2 \bar \otimes A\) and the set of regular quadratic forms, where A is a finite and σ-finite von Neumann algebra. Furthermore, we obtain that a strict continuous symmetric regular module operator semigroup \(\{ T_t \} _{t \in \mathbb{R}^ + } \subset L(l_2 \bar \otimes A)\) is Markovian if and only if the associated A-valued quadratic form is a Dirichlet form, where L(\(l_2 \bar \otimes A\)) is the von Neumann algebra of all adjointable module maps on \(l_2 \bar \otimes A\).

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.

An Entropy Based Treatment of Gaussian Communication Process for General Quantum Systems

The quantum entropy introduced by von Neumann around 1932 describes the amount of information of the quantum state itself. It was extended by Ohya for C*-systems before Conne-Narnhoffer-Thirring (CNT) entropy. The quantum relative entropy was first defined by Umegaki for σ-finite von Neumann algebras and it was subsequently extended by Araki and Uhlmann for general von Neumann algebras and *-algebras, respectively. By introducing a new notion, the so-called compound state, in 1983 Ohya succeeded to construct the mutual entropy in a complete quantum mechanical system (i.e., input state, output state and channel are all quantum mechanical) describing the amount of information correctly transmitted through the quantum channel. In this paper, we briefly review Ohya's S-mixing entropy and the quantum mutual entropy for general quantum systems. Based on a concept of structure equivalent, we apply the general framework of quantum communication to the Gaussian communication processes.

Second dual projection characterizations of three classes of [formula omitted]-closed, convex, bounded sets in [formula omitted]: Non-commutative generalizations

We provide characterizations of convex, compact for the topology of local convergence in measure subsets of non-commutative L1-spaces previously considered for classical L1-spaces. More precisely, if M is a semifinite and σ-finite von Neumann algebra equipped with a distinguished semifinite faithful normal trace τ, P:M∗→L1(M,τ) is the non-commutative Yosida–Hewitt projection, and C is a norm bounded subset of L1(M,τ) that is convex and closed for the topology of local convergence in measure then we isolate the precise conditions on C for which P:C¯w∗→C is compactness preserving, sequentially continuous, or continuous when C¯w∗ is equipped with the weak* topology and C with the topology of local convergence in measure.

The characterization of a class of quantum Markov semigroups and the associated operator-valued Dirichlet forms based on Hilbert W*-module

In this paper, we introduce the concept of operator-valued quadratic form based on Hilbert W*-module $l_2 \bar \otimes A$ , and give a one to one correspondence between the set of positive self-adjoint regular module operators on $l_2 \bar \otimes A$ and the set of regular quadratic forms, where A is a finite and σ-finite von Neumann algebra. Furthermore, we obtain that a strict continuous symmetric regular module operator semigroup $\{ T_t \} _{t \in \mathbb{R}^ + } \subset L(l_2 \bar \otimes A)$ is Markovian if and only if the associated A-valued quadratic form is a Dirichlet form, where L( $l_2 \bar \otimes A$ ) is the von Neumann algebra of all adjointable module maps on $l_2 \bar \otimes A$ .

Tensor Products of Noncommutative Lp-Spaces

We consider the notion of tensor product of noncommutative Lp spaces associated with finite von Neumann algebras and define the notion of tensor product of Haagerup noncommutative Lp spaces associated with σ-finite von Neumann algebras.

Topologies on central extensions of von Neumann algebras

Abstract Given a von Neumann algebra M, we consider the central extension E(M) of M. We introduce the topology t c(M) on E(M) generated by a center-valued norm and prove that it coincides with the topology of local convergence in measure on E(M) if and only if M does not have direct summands of type II. We also show that t c(M) restricted to the set E(M)h of self-adjoint elements of E(M) coincides with the order topology on E(M)h if and only if M is a σ-finite type Ifin von Neumann algebra.

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.

Embeddings of non-commutative L p -spaces into preduals of finite von Neumann algebras

Let $$\mathcal{R}$$ be a (not necessarily semi-finite) σ-finite von Neumann algebra. We prove that there exists a finite von Neumann algebra $$\mathcal{N}$$ so that for every 1 < p < 2, the Haagerup L p -space associated with $$\mathcal{R}$$ embeds isomorphically into $$\mathcal{N}_ * $$ . We also provide a proof of the following non-commutative generalization of a classical result of Rosenthal: if $$\mathcal{M}$$ is a semi-finite von Neumann algebra then every reflexive subspace of $$\mathcal{M}_ * $$ embeds isomorphically into L r ( $$\mathcal{M}$$ ) for some r > 1.

Classification and decomposition of Quantum Markov Semigroups

We show that a QMS on a σ-finite von Neumann algebra can be decomposed as the sum of several ``sub''-semigroups corresponding to transient and recurrent projections. We discuss two applications to physical models.