Normal Conditional Expectation Research Articles

A strong subadditivity‐like inequality for quantum entropy in semifinite von Neumann algebras

AbstractLet be a semifinite von Neumann algebra with a normal faithful semifinite trace , and let , , be its subalgebras such that and that restricted to any of these subalgebras is semifinite. Denote by , , and the normal conditional expectations from onto , and , respectively, such that is invariant with respect to any of them. The quadruple , , , is said to be a commuting square if In this note, we show that the property of being a commuting square is characterized by a sort of the SSA‐like (strong subadditivity of entropy) inequality for an arbitrary normal state on , where denotes the Segal entropy of the state . The situation when we have equality in the inequality above is also investigated, and various equivalent conditions are obtained, in an especially appealing form for finite von Neumann algebras. For such algebras, one more condition is obtained under the assumption of independence of the algebras and .

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.

A planar algebraic description of conditional expectations

Let [Formula: see text] be a unital inclusion of arbitrary von Neumann algebras. We give a 2-[Formula: see text]-categorical/planar algebraic description of normal faithful conditional expectations [Formula: see text] with finite index and their duals [Formula: see text] by means of the solutions of the conjugate equations for the inclusion morphism [Formula: see text] and its conjugate morphism [Formula: see text]. In particular, the theory of index for conditional expectations admits a 2-[Formula: see text]-categorical formulation in full generality. Moreover, we show that a pair [Formula: see text] as above can be described by a Q-system, and vice versa. These results are due to Longo in the subfactor/simple tensor unit case [R. Longo, Index of subfactors and statistics of quantum fields. II. Correspondences, braid group statistics and Jones polynomial, Comm. Math. Phys. 130(1990) 285–309; A duality for Hopf algebras and for subfactors. I, Comm. Math. Phys. 159 (1994) 133–150].

Unitary conjugacy for type III subfactors and W$^*$-superrigidity

Let A,B\subset M be inclusions of \sigma -finite von Neumann algebras such that A and B are images of faithful normal conditional expectations. In this article, we investigate Popa's intertwining condition A\preceq_MB using modular actions on A , B , and M . In the main theorem, we prove that if A\preceq_MB , then an intertwining element for A\preceq_MB also intertwines some modular flows of A and B . As a result, we deduce a new characterization of A\preceq_MB in terms of the continuous cores of A , B , and M . Using this new characterization, we prove the first W ^* -superrigidity type result for group actions on amenable factors. As another application, we characterize stable strong solidity for free product factors in terms of their free product components.

Connes' bicentralizer problem for q‐deformed Araki–Woods algebras

Let $(H_{\mathbf{R}}, U_t)$ be any strongly continuous orthogonal representation of $\mathbf{R}$ on a real (separable) Hilbert space $H_{\mathbf{R}}$. For any $q\in (-1,1)$, we denote by $\Gamma_q(H_{\mathbf{R}},U_t)^{\prime\prime}$ the $q$-deformed Araki-Woods algebra introduced by Shlyakhtenko and Hiai. In this paper, we prove that $\Gamma_q(H_{\mathbf{R}},U_t)^{\prime\prime}$ has trivial bicentralizer if it is a type $\rm III_1$ factor. In particular, we obtain that $\Gamma_q(H_{\mathbf{R}},U_t)^{\prime\prime}$ always admits a maximal abelian subalgebra that is the range of a faithful normal conditional expectation. Moreover, using Sniady's work, we derive that $\Gamma_q(H_{\mathbf{R}},U_t)^{\prime\prime}$ is a full factor provided that the weakly mixing part of $(H_{\mathbf{R}}, U_t)$ is nonzero.

On the weak relative Dixmier property

We show that every inclusion of von Neumann algebras with a faithful normal conditional expectation has the weak relative Dixmier property. This answers a question of Popa \cite{Po99}. The proof uses an improvement of Ellis' lemma for compact convex semigroup.

On Period, Cycles and Fixed Points of a Quantum Channel

We consider a quantum channel acting on an infinite dimensional von Neumann algebra of operators on a separable Hilbert space. When there exists an invariant normal faithful state, the cyclic properties of such channels are investigated passing through the decoherence free algebra and the fixed points domain. Both these spaces are proved to be images of a normal conditional expectation so that their consequent atomic structure are analyzed in order to give a better description of the action of the channel and, for instance, of its Kraus form and invariant densities.

Polynomial Interpolation of Normal Conditional Expectation

In this paper we are interested in approximating the conditional expectation of a given random variable X with respect to the standard normal distribution N(0, 1). Actually we have shown that the conditional expectation E(X|Z) could be interpolated by an N degree polynomial function of Z, φN(Z) where N is the number of observations recorded for the conditional expectation E(X|Z = z). A pointwise error estimation has been proved under reasonable condition on the random variable X.

Cartan subalgebras of tensor products of free quantum group factors with arbitrary factors

Let G be a free (unitary or orthogonal) quantum group. We prove that for any nonamenable subfactor N⊂L∞(G) which is an image of a faithful normal conditional expectation, and for any σ-finite factor B, the tensor product N⊗¯B has no Cartan subalgebras. This generalizes our previous work that provides the same result when B is finite. In the proof, we establish Ozawa–Popa and Popa–Vaes’s weakly compact action on the continuous core of L∞(G)⊗¯B as the one relative to B, by using an operator-valued weight to B and the central weak amenability of G .

Jensen's inequality in finite subdiagonal algebras

Let $\mathscr{M}$ be a finite von Neumann algebra with a faithful normal tracial state $\tau$ and $\mathfrak{A}$ be a finite subdiagonal subalgebra of $\mathscr{M}$ with respect to a $\tau$-preserving faithful normal conditional expectation $\Phi$ on $\mathscr{M}$. Let $\Delta$ denote the Fuglede-Kadison determinant corresponding to $\tau$. For $X \in \mathscr{M}$, define $|X| := (X^*X)^{\frac{1}{2}}$. In 2005, Labuschagne proved the so-called Jensen's inequality for finite subdiagonal algebras i.e. $\Delta(\Phi(A)) \le \Delta(A)$ for an operator $A \in \mathfrak{A}$, thus resolving a long-standing open problem posed by Arveson in 1967. In this article, we prove the following more general result: $\tau(f( |\Phi(A)|)) \le \tau(f( |A|))$ for $A \in \mathfrak{A}$ and any increasing continuous function $f : [0, \infty) \to \mathbb{R}$ such that $f \circ \exp$ is convex on $\mathbb{R}$. Under the additional hypotheses that $A$ is invertible in $\mathscr{M}$ and $f \circ \exp$ is strictly convex, we have $\tau(f( |\Phi(A)|)) = \tau(f( |A|))$ $\Longleftrightarrow \Phi(A) = A$. As an application, we show that for $A \in \mathfrak{A}$ the point spectrum of $A$ is contained in the point spectrum of $\Phi(A)$, though such a conclusion does not hold in general for their spectra.

Strong Solidity of free Araki-Woods factors

We show that Shlyakhtenko's free Araki-Woods factors are strongly solid, meaning that for any diffuse amenable von Neumann subalgebra that is the range of a normal conditional expectation, the normalizer remains amenable. This provides the first class of nonamenable strongly solid type III factors.

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.

Quantum symmetric states on free product $C^*$-algebras

We introduce symmetric states and quantum symmetric states on universal unital free product C*-algebras an arbitrary unital C*-algebra A with itself infinitely many times, as a generalization of the notions of exchangeable and quantum exchangeable random variables. We prove existence of conditional expectations onto tail algebras in various settings and we define a natural C*-subalgebra of the tail algebra, called the tail C*-algebra. Extending and building on the proof of the noncommutative de Finetti theorem of Koestler and Speicher, we prove a de Finetti type theorem that characterizes quantum symmetric states in terms of amalgamated free products over the tail C*-algebra, and we provide a convenient description of the set of all quantum symmetric states on the free product C*-algebra in terms of C*-algebras generated by homomorphic images of A and the tail C*-algebra. This description allows a characterization of the extreme quantum symmetric states. Similar results are proved for the subset of tracial quantum symmetric states, though in terms of von Neumann algebras and normal conditional expectations. The central quantum symmetric states are those for which the tail algebra is in the center of the von Neumann algebra, and we show that the central quantum symmetric states form a Choquet simplex whose extreme points are the free product states, while the tracial central quantum symmetric states form a Choquet simplex whose extreme points are the free product traces.

Paving over arbitrary MASAs in von Neumann algebras

We consider a paving property for a maximal abelian *-subalgebra (MASA) $A$ in a von Neumann algebra $M$, that we call so-paving, involving approximation in the so-topology, rather than in norm (as in classical Kadison-Singer paving). If $A$ is the range of a normal conditional expectation, then so-paving is equivalent to norm paving in the ultrapower inclusion $A^\omega\subset M^\omega$. We conjecture that any MASA in any von Neumann algebra satisfies so-paving. We use [MSS13] to check this for all MASAs in $\mathcal B(\ell^2 \mathbb N)$, all Cartan subalgebras in amenable von Neumann algebras and in group measure space II$_1$ factors arising from profinite actions. By [P13], the conjecture also holds true for singular MASAs in II$_1$ factors, and we obtain here an improved paving size $C\varepsilon^{-2}$, which we show to be sharp.

Asymptotic structure of free Araki–Woods factors

The purpose of this paper is to investigate the structure of Shlyakhtenko’s free Araki–Woods factors using the framework of ultraproduct von Neumann algebras. We first prove that all the free Araki–Woods factors $$\Gamma (H_{\mathbb R}, U_t)^{\prime \prime }$$ are $$\omega $$ -solid in the following sense: for every von Neumann subalgebra $$Q \subset \Gamma (H_{\mathbb R}, U_t)^{\prime \prime }$$ that is the range of a faithful normal conditional expectation and such that the relative commutant $$Q' \cap M^\omega $$ is diffuse, we have that $$Q$$ is amenable. Next, we prove that the continuous cores of the free Araki–Woods factors $$\Gamma (H_{\mathbb R}, U_t)^{\prime \prime }$$ associated with mixing orthogonal representations $$U : \mathbb R \rightarrow \mathcal O(H_{\mathbb R})$$ are $$\omega $$ -solid type $$\mathrm{II_\infty }$$ factors. Finally, when the orthogonal representation $$U : \mathbb R \rightarrow \mathcal O(H_{\mathbb R})$$ is weakly mixing, we prove a dichotomy result for all the von Neumann subalgebras $$Q \subset \Gamma (H_{\mathbb R}, U_t)^{\prime \prime }$$ that are globally invariant under the modular automorphism group $$(\sigma _t^{\varphi _U})$$ of the free quasi-free state $$\varphi _U$$ .

The Schur-Horn theorem for operators with finite spectrum

The carpenter problem in the context of II1 factors, formulated by Kadison asks: Let A ⊂ M be a masa in a type II1 factor and let E be the normal conditional expectation from M onto A. Then, is it true that for every positive contraction A in A, there is a projection P in M such that E(P ) = A? In this note, we show that this is true if A has finite spectrum. We will then use this result to prove an exact Schur-Horn theorem for (positive)operators with finite spectrum and an approximate Schur-Horn theorem for general (positive)operators.

On Bi-exactness of Discrete Quantum Groups

We define Ozawa's notion of bi-exactness to discrete quantum groups, and then prove some structural properties of associated von Neumann algebras. In particular, we prove that any non amenable subfactor of free quantum group von Neumann algebras, which is an image of a faithful normal conditional expectation, has no Cartan subalgebras.

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.

Approximation properties and absence of Cartan subalgebra for free Araki–Woods factors

We show that all the free Araki–Woods factors Γ ( H R , U t ) ″ have the complete metric approximation property. Using Ozawa–Popaʼs techniques, we then prove that every nonamenable subfactor N ⊂ Γ ( H R , U t ) ″ which is the range of a normal conditional expectation has no Cartan subalgebra. We finally deduce that the type III 1 factors constructed by Connes in the ʼ70s can never be isomorphic to any free Araki–Woods factor, which answers a question of Shlyakhtenko and Vaes.