Finite Von Neumann Algebra Research Articles

On cleanness of von Neumann algebras

A unital ring is called clean (resp. strongly clean) if every element can be written as the sum of an invertible element and an idempotent (resp. an invertible element and an idempotent that commutes). T.Y. Lam proposed a question: which von Neumann algebras are clean as rings? In this paper, we characterize strongly clean von Neumann algebras and prove that all finite von Neumann algebras and all separable infinite factors are clean.

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.

Alberti–Uhlmann Problem on Hardy–Littlewood–Pólya Majorization

We fully describe the doubly stochastic orbit of a self-adjoint element in the noncommutative \(L_1\)-space affiliated with a semifinite von Neumann algebra, which answers a problem posed by Alberti and Uhlmann (Stochasticity and partial order: doubly stochastic maps and unitary mixing. VEB Deutscher Verlag der Wissenschaften, Berlin, 1982) in the 1980s, extending several results in the literature. It follows further from our methods that, for any \(\sigma \)-finite von Neumann algebra \({{\mathcal {M}}}\) equipped with a semifinite infinite faithful normal trace \(\tau \), there exists a self-adjoint operator \(y\in L_1({{\mathcal {M}}},\tau )\) such that the doubly stochastic orbit of y does not coincide with the orbit of y in the sense of Hardy–Littlewood–Pólya, which confirms a conjecture by Hiai (J Math Anal Appl 127:18–48, 1987). However, we show that Hiai’s conjecture fails for non-\(\sigma \)-finite von Neumann algebras. The main result of the present paper also answers the (noncommutative) infinite counterparts of problems due to Luxemburg (Proc Symp Anal Queen’s univ 83–144, 1967) and Ryff (Pac J Math 13:1379–1386, 1963) in the 1960s.

Free Rosenthal Inequalities on Vector-Valued Symmetric Operator Spaces

Abstract Let $\mathcal{M}$ be a finite von Neumann algebra and let $E$ be an interpolation space between $L_1(0,1)$ and $L_{\infty }(0,1)$. For every $q\in [1,\infty ]$, we introduce a general notion of a vector-valued symmetric operator space $E(\mathcal{M};\ell _q)$ via $K$-functional of the couple $(L_1(\mathcal M;\ell _q),L_{\infty }(\mathcal M;\ell _q))$. We then establish free Rosenthal inequalities on $E(\mathcal{M};\ell _q)$, which provide an equivalent characterization of the quantity $\|(x_k)_{k\geq 0}\|_{E(\mathcal{M};\ell _q)},$ where $(x_k)_{k\geq 0}\subset E(\mathcal{M})$ is a sequence of freely independent random variables.

Lattice of intermediate subalgebras

Analogous to subfactor theory, employing Watatani's notions of index and C ∗ -basic construction of certain inclusions of C ∗ -algebras, (a) we develop a Fourier theory (consisting of Fourier transforms, rotation maps and shift operators) on the relative commutants of any inclusion of simple unital C ∗ -algebras with finite Watatani index, and (b) we introduce the notions of interior and exterior angles between intermediate C ∗ -subalgebras of any inclusion of unital C ∗ -algebras admitting a finite index conditional expectation. Then, on the lines of Bakshi et al. (Trans. Amer. Math. Soc. 371 (2019) 5973–5991), we apply these concepts to obtain a bound for the cardinality of the lattice of intermediate C ∗ -subalgebras of any irreducible inclusion as in (a), and improve Longo's bound for the cardinality of intermediate subfactors of an irreducible inclusion of type I I I factors with finite index. Moreover, we also show that for a fairly large class of inclusions of finite von Neumann algebras, the lattice of intermediate von Neumann subalgebras is always finite.

Logarithmic submajorizations inequalities for operators in a finite von Neumann algebra

The aim of this paper is to study the logarithmic submajorizations inequalities for operators in a finite von Neumann algebra. Firstly, some logarithmic submajorizations inequalities due to Garg and Aulja are extended to the case of operators in a finite von Neumann algebra. As an application, we get some new Fuglede-Kadison determinant inequalities of operators in that circumstance. Secondly, we improve and generalize to the setting of finite von Neumann algebras, a generalized Hölder type generalized singular numbers inequality.

Some aspects of quantum sufficiency for finite and full von Neumann algebras

Some features of the notion of sufficiency in quantum statistics are investigated. Three kinds of this notion are considered: plain sufficiency (called simply: sufficiency), strong sufficiency and Umegaki’s sufficiency. It is shown that for a finite von Neumann algebra with a faithful family of normal states the minimal sufficient von Neumann subalgebra is sufficient in Umegaki’s sense. Moreover, a proper version of the factorization theorem of Jenčová and Petz is obtained. The structure of the minimal sufficient subalgebra is described in the case of pure states on the full algebra of all bounded linear operators on a Hilbert space.

On Murray-von Neumann algebras—I: topological, order-theoretic and analytical aspects

For a countably decomposable finite von Neumann algebra \({\mathscr {R}}\), we show that any choice of a faithful normal tracial state on \({\mathscr {R}}\) engenders the same measure topology on \({\mathscr {R}}\) in the sense of Nelson (J Funct Anal 15:103–116, 1974). Consequently it is justified to speak of ‘the’ measure topology of \({\mathscr {R}}\). Having made this observation, we extend the notion of measure topology to general finite von Neumann algebras and denominate it the \({\mathfrak {m}}\)-topology. We note that the procedure of \({\mathfrak {m}}\)-completion yields Murray-von Neumann algebras in a functorial manner and provides them with an intrinsic description as unital ordered complex topological \(*\)-algebras. This enables the study of abstract Murray-von Neumann algebras avoiding reference to a Hilbert space. Furthermore, it makes apparent the appropriate notion of Murray-von Neumann subalgebras, and the intrinsic nature of the spectrum and point spectrum of elements, independent of their ambient Murray-von Neumann algebra. In this context, we show the well-definedness of the Borel function calculus for normal elements and use it along with approximation techniques in the \({\mathfrak {m}}\)-topology to transfer many standard operator inequalities involving bounded self-adjoint operators to the setting of (unbounded) self-adjoint operators in Murray-von Neumann algebras. On the algebraic side, Murray-von Neumann algebras have been described as the Ore localization of finite von Neumann algebras with respect to their corresponding multiplicative subset of non-zero-divisors. Our discussion reveals that, in addition, there are fundamental topological, order-theoretic and analytical facets to their description.

Angles between Haagerup–Schultz projections and spectrality of operators

We investigate angles between Haagerup–Schultz projections of operators belonging to finite von Neumann algebras, in connection with a property analogous to Dunford's notion of spectrality of operators. In particular, we show that an operator is similar to the sum of a normal and an s.o.t.-quasinilpotent operator that commute if and only if the angles between its Haagerup–Schultz projections are uniformly bounded away from zero (and we call this the uniformly non-zero angles property). Moreover, we show that spectrality is equivalent to this uniformly non-zero angles property plus decomposability. Finally, using this characterization, we construct an easy example of an operator which is decomposable but not spectral, and we show that Voiculescu's circular operator is not spectral (nor is any of the circular free Poisson operators).

Some logarithmic submajorisations and determinant inequalities for operators with numerical ranges in a sector

In this paper, we utilize the method of submajorization and the properties of operators to generalize the submajorisation and determinant inequalities of sector matrices to operators in a finite von Neumann algebra.

Maximality and finiteness of type 1 subdiagonal algebras

Let A \mathfrak A be a type 1 subdiagonal algebra in a σ \sigma -finite von Neumann algebra M \mathcal M with respect to a faithful normal conditional expectation Φ \Phi . We give necessary and sufficient conditions for which A \mathfrak A is maximal among the σ \sigma -weakly closed subalgebras of M \mathcal M . In addition, we show that a type 1 subdiagonal algebra in a finite von Neumann algebra is automatically finite which gives a positive answer of Arveson’s finiteness problem in 1967 in type 1 case.

Non-global Nonlinear Lie Triple Derivable Maps on Finite von Neumann Algebras

Let $${\mathcal {M}}$$ be a finite von Neumann algebra with no central summands of type $$I_{1}$$ . Assume that $$\delta :{\mathcal {M}}\rightarrow {\mathcal {M}}$$ is a nonlinear map satisfying $$\delta ([[A,B],C])=[[\delta (A),B],C]+[[A,\delta (B)],C]+[[A,B],\delta (C)]$$ for any $$A,B,C\in {\mathcal {M}}$$ with $$ABC=0$$ . Then, we prove that there exists an additive derivation $$d:{\mathcal {M}}\rightarrow {\mathcal {M}}$$ , such that $$\delta (A)=d(A)+\tau (A)$$ for any $$A\in {\mathcal {M}}$$ , where $$\tau :{\mathcal {M}}\rightarrow {\mathcal {Z}}_{{\mathcal {M}}}$$ is a nonlinear map, such that $$\tau ([[A,B],C])=0$$ for any $$A,B,C\in {\mathcal {M}}$$ with $$ABC=0$$ .

Harnack type inequalities for operators in logarithmic submajorisation

The aim of this paper is to study the Harnack type logarithmic submajorisation and Fuglede-Kadison determinant inequalities for operators in a finite von Neumann algebra. In particular, the Harnack type determinant inequalities due to Lin-Zhang [15] and Yang-Zhang [27] are extended to the case of operators in a finite von Neumann algebra.

Gradient forms and strong solidity of free quantum groups

Consider the free orthogonal quantum groups O_N^+(F) and free unitary quantum groups U_N^+(F) with N ge 3. In the case F = text {id}_N it was proved both by Isono and Fima-Vergnioux that the associated finite von Neumann algebra L_infty (O_N^+) is strongly solid. Moreover, Isono obtains strong solidity also for L_infty (U_N^+) . In this paper we prove for general F in GL_N(mathbb {C}) that the von Neumann algebras L_infty (O_N^+(F)) and L_infty (U_N^+(F)) are strongly solid. A crucial part in our proof is the study of coarse properties of gradient bimodules associated with Dirichlet forms on these algebras and constructions of derivations due to Cipriani–Sauvageot.

A noncommutative generalisation of a problem of Steinhaus

We extend the Révész and Komlós theorems to arbitrary finite von Neumann algebras, and in doing so solve an open problem of Randrianantoanina, removing the need for hyperfiniteness. The main result is the noncommutative Komlós theorem, which states that every norm-bounded sequence of operators in L1(M), for any finite von Neumann algebra M, admits a subsequence, such that for any further subsequence, the Cesàro averages converge bilaterally almost uniformly. This is a natural extension of Komlós' original result to the noncommutative setting.The necessary techniques which facilitate the proof also allow us to extend the Révész theorem to the noncommutative setting, which gives a similar subsequential law for series over bounded sequences in L2(M).

On the von Neumann algebras associated to Yang–Baxter operators

Bożejko and Speicher associated a finite von Neumann algebra MT to a self-adjoint operator T on a complex Hilbert space of the form $\mathcal {H}\otimes \mathcal {H}$ which satisfies the Yang–Baxter relation and $ \left\| T \right\| < 1$. We show that if dim$(\mathcal {H})$ ⩾ 2, then MT is a factor when T admits an eigenvector of some special form.

Characterization of subdiagonal algebras on noncommutative Calderón–Lozanovskiĭ spaces

Let $$\mathcal {M}$$ be a finite von Neumann algebra acting on a Hilbert space $$\mathcal {H}$$ with a faithful, normal, finite trace $$\tau$$ and let $$\mathcal {A}$$ be a tracial subalgebra of $$\mathcal {M}$$ . Let E be an $$\alpha$$ -convex symmetric quasi-Banach space on I for some $$0<\alpha <\infty$$ and let $$\varphi$$ be an Orlicz function. It is observed that $$\mathcal {A}$$ has $$E_\varphi$$ -factorization if and only if $$\mathcal {A}$$ is a subdiagonal algebra. Moreover, we present some characterizations of subdiagonal algebras.

On subspaces spanned by freely independent random variables in noncommutative Lp-spaces

This paper deals with new phenomena appearing in the structure of Banach spaces arising in noncommutative integration theory. Let E be a fully symmetric Banach function space on [0, 1] and M be a finite von Neumann algebra. Let [xk] be the closed subspace spanned by a sequence (xk) of freely independent mean zero random variables from E(ℳ). The subspace [xk] is complemented in E(ℳ) if and only if the closed subspace spanned by the pairwise orthogonal sequence (xk ⊗ ek) is complemented in a certain symmetric operator space $$Z_E^2\left({{\cal M}\overline \otimes {\ell_\infty}} \right)$$ . We obtain noncommutative (free) analogues of classical results of Dor and Starbird as well as those of Kadec and Pelczynski. We show that [xk] is complemented in L1(ℳ)provided (xk) is equivalent in L1(ℳ) to the standard basis of l2, while this never happens in the classical case. We prove that a sequence of freely independent copies of a mean zero random variable x in Lp(ℳ), 1 ≤ p ≤ 2, is equivalent to the standard basis in some Orlicz sequence space ℳΦ and give a precise description of the connection between the Orlicz function Φ and the distribution of the given random variable x. Finally, we prove that [xk] spanned by a sequence of freely independent copies of a mean zero random variable is complemented in E(ℳ) if and only if (xk) is equivalent in E(ℳ) to the standard basis of l2.

Relative entropy for von Neumann subalgebras

We revisit the connection between index and relative entropy for an inclusion of finite von Neumann algebras. We observe that the Pimsner–Popa index connects to sandwiched [Formula: see text]-Rényi relative entropy for all [Formula: see text], including Umegaki’s relative entropy at [Formula: see text]. Based on that, we introduce a new notation of relative entropy to a subalgebra which generalizes subfactors index. This relative entropy has application in estimating decoherence time of quantum Markov semigroups.

The Second Nonlinear Mixed Lie Triple Derivations on Finite Von Neumann Algebras

Let $${\mathcal {M}}$$ be a finite von Neumann algebra with no central summands of type $${I}_{1}$$ . Suppose that $$L: {\mathcal {M}}\rightarrow {\mathcal {M}}$$ is the second nonlinear mixed Lie triple derivation. Then L is an additive $$*$$ -derivation. We also show that each local and 2-local second Lie triple derivation on finite von Neumann algebras with no central summands of type $${I}_{1}$$ is a $$*$$ -derivation. Besides, each local and 2-local second Lie triple derivation on factor von Neumann algebras $${\mathcal {M}}$$ with dim $${\mathcal {M}}>1$$ is also a $$*$$ -derivation.