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In 1988, Haagerup and Størmer conjectured that every pointwise inner automorphism of a type ${\rm III_1}$ factor is a composition of an inner and a modular automorphism. We study this conjecture and prove that every type ${\rm III_1}$ factor with trivial bicentralizer indeed satisfies this condition. In particular, this shows that Haagerup and Størmer's conjecture holds in full generality if Connes' bicentralizer problem has an affirmative answer. Our proof is based on Popa's intertwining theory and Marrakchi's recent work on relative bicentralizers. [ABSTRACT FROM AUTHOR]

We prove that a large class of nonamenable almost periodic type III 1 factors M, including all McDuff factors that tensorially absorb R ∞ and all free Araki–Woods factors, satisfy Haagerup–Størmer's conjecture (1988): any pointwise inner automorphism of M is the composition of an inner and a modular automorphism. [ABSTRACT FROM AUTHOR]

Let A be a unital C ∗ -algebra with unit 1 A and let a ∈ A be a positive and invertible element. Suppose that S (A) is the set of all states on A and let S a (A) = f f (a) : f ∈ S (A) , f (a) ≠ 0. The norm ‖ x ‖ a for every x ∈ A is defined by ‖ x ‖ a = sup φ ∈ S a (A) φ (x ∗ a x). In this paper, we aim to investigate the notion of Birkhoff–James orthogonality with respect to the norm ‖ · ‖ a in A , namely a-Birkhoff–James orthogonality. The characterization of a-Birkhoff–James orthogonality in A by means of the elements of generalized state space S a (A) is provided. As an application, a characterization for the best approximation to elements of A in a subspace B with respect to ‖ · ‖ a is obtained. Moreover, a formula for the distance of an element of A to the subspace B = C 1 A is given. We also study the strong version of a-Birkhoff–James orthogonality in A. The classes of C ∗ -algebras in which these two types orthogonality relationships coincide are described. In particular, we prove that the condition of the equivalence between the strong a-Birkhoff–James orthogonality and A -valued inner product orthogonality in A implies that the center of A is trivial. Finally, we show that if the (strong) a-Birkhoff–James orthogonality is right-additive (left-additive) in A , then the center of A is trivial. [ABSTRACT FROM AUTHOR]

In the setting of non-type II 1 representations, we propose a definition of deformed Fredholm module [ D T | D T | - 1 , · ] T for a modular spectral triple T , where D T is the deformed Dirac operator. D T is assumed to be invertible for the sake of simplicity, and its domain is an “essential” operator system E T . According to such a definition, we obtain [ D T | D T | - 1 , · ] T = | D T | - 1 d T (·) + d T (·) | D T | - 1 , where d T is the deformed derivation associated to D T . Since the “quantum differential” 1 / | D T | appears in a symmetric position, such a definition of Fredholm module differs from the usual one even in the undeformed case, that is in the tracial case. Therefore, it seems to be more suitable for the investigation of noncommutative manifolds in which the nontrivial modular structure might play a crucial role. We show that all models in Fidaleo and Suriano (J Funct Anal 275:1484–1531, 2018) of non-type II 1 representations of noncommutative 2-tori indeed provide modular spectral triples, and in addition deformed Fredholm modules according to the definition proposed in the present paper. Since the detailed knowledge of the spectrum of the Dirac operator plays a fundamental role in spectral geometry, we provide a characterisation of eigenvalues and eigenvectors of the deformed Dirac operator D T in terms of the periodic solutions of a particular class of eigenvalue Hill equations. [ABSTRACT FROM AUTHOR]

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\|. We show that if dim $(\mathcal {H})$ ⩾ 2, then MT is a factor when T admits an eigenvector of some special form. [ABSTRACT FROM AUTHOR]

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 (J. Funct. Anal. 171 (2000) 139–154). The proof uses an improvement of Ellis's lemma for compact convex semigroups. [ABSTRACT FROM AUTHOR]

We show that a factor M is full if and only if the C ∗ -algebra generated by its left and right regular representations contains the compact operators. We prove that the bicentralizer flow of a type III 1 factor is always ergodic. As a consequence, for any type III 1 factor M and any λ ∈ ] 0 , 1 ] , there exists an irreducible AFD type III λ subfactor with expectation in M. Moreover, any type III 1 factor M which satisfies M ≅ M ⊗ ¯ R λ for some λ ∈ ] 0 , 1 [ has trivial bicentralizer. Finally, we give a counter-example to the characterization of approximately inner automorphisms conjectured by Connes and we prove a weaker version of this conjecture. In particular, we obtain a new proof of Kawahigashi–Sutherland–Takesaki's result that every automorphism of the AFD type III 1 factor is approximately inner. [ABSTRACT FROM AUTHOR]

Let M be an arbitrary factor and $\sigma : \Gamma \curvearrowright M$ an action of a discrete group. In this paper, we study the fullness of the crossed product $M \rtimes _\sigma \Gamma $. When Γ is amenable, we obtain a complete characterization: the crossed product factor $M \rtimes _\sigma \Gamma $ is full if and only if M is full and the quotient map $\overline {\sigma } : \Gamma \rightarrow {\rm out}(M)$ has finite kernel and discrete image. This answers the question of Jones from [11]. When M is full and Γ is arbitrary, we give a sufficient condition for $M \rtimes _\sigma \Gamma $ to be full which generalizes both Jones' criterion and Choda's criterion. In particular, we show that if M is any full factor (possibly of type III) and Γ is a non-inner amenable group, then the crossed product $M \rtimes _\sigma \Gamma $ is full. [ABSTRACT FROM AUTHOR]

We investigate the structure of the relative bicentralizer algebra B (N ⊂ M , φ) for inclusions of von Neumann algebras with normal expectation where N is a type III 1 subfactor and φ ∈ N ∗ is a faithful state. We first construct a canonical flow β φ : R + ∗ ↷ B (N ⊂ M , φ) on the relative bicentralizer algebra and we show that the W ∗ -dynamical system (B (N ⊂ M , φ) , β φ) is independent of the choice of φ up to a canonical isomorphism. In the case when N = M , we deduce new results on the structure of the automorphism group of B (M , φ) and we relate the period of the flow β φ to the tensorial absorption of Powers factors. For general irreducible inclusions N ⊂ M , we relate the ergodicity of the flow β φ to the existence of irreducible AFD subfactors in M that sit with normal expectation in N. When the inclusion N ⊂ M is discrete, we prove a relative bicentralizer theorem and we use it to solve Kadison's problem when N is amenable. [ABSTRACT FROM AUTHOR]

For the noncommutative 2-torus, we define and study Fourier transforms arising from representations of states with central supports in the bidual, exhibiting a possibly nontrivial modular structure (i.e. type III representations). We then prove the associated noncommutative analogous of Riemann–Lebesgue Lemma and Hausdorff–Young Theorem. In addition, the L p -convergence result of the Cesaro means (i.e. the Fejer theorem), and the Abel means reproducing the Poisson kernel are also established, providing inversion formulae for the Fourier transforms in L p spaces, p ∈ [ 1 , 2 ] . Finally, in L 2 (M) we show how such Fourier transforms "diagonalise" appropriately some particular cases of modular Dirac operators, the latter being part of a one-parameter family of modular spectral triples naturally associated to the previously mentioned non type II 1 representations. [ABSTRACT FROM AUTHOR]

It is well known that for any irrational rotation number α , the noncommutative torus A α must have representations π such that the generated von Neumann algebra π ( A α ) ″ is of type III. Therefore, it could be of interest to exhibit and investigate such kind of representations, together with the associated spectral triples whose twist of the Dirac operator and the corresponding derivation arises from the Tomita modular operator. In the present paper, we show that this program can be carried out, at least when α is a Liouville number satisfying a faster approximation property by rationals. In this case, we exhibit several type I I ∞ and II I λ , λ ∈ [ 0 , 1 ] , factor representations and modular spectral triples. The method developed in the present paper can be generalised to CCR algebras based on a locally compact abelian group equipped with a symplectic form. [ABSTRACT FROM AUTHOR]

We construct the first II 1 factors having exactly two group measure space decompositions up to unitary conjugacy. Also, for every positive integer n , we construct a II 1 factor M that has exactly n group measure space decompositions up to conjugacy by an automorphism. [ABSTRACT FROM AUTHOR]

We show that whenever m ≥ 1 and M 1 , … , M m are nonamenable factors in a large class of von Neumann algebras that we call C ( AO ) and which contains all free Araki–Woods factors, the tensor product factor M 1 ⊗ ‾ ⋯ ⊗ ‾ M m retains the integer m and each factor M i up to stable isomorphism, after permutation of the indices. Our approach unifies the Unique Prime Factorization (UPF) results from [33,25] and moreover provides new UPF results in the case when M 1 , … , M m are free Araki–Woods factors. In order to obtain the aforementioned UPF results, we show that Connes's bicentralizer problem has a positive solution for all type I I I 1 factors in the class C ( AO ) . [ABSTRACT FROM AUTHOR]

Crossed products with noncommutative Bernoulli actions were introduced by Connes as the first examples of full factors of type I I I . This article provides a complete classification of the factors ( P , ϕ ) F n ⋊ F n , where F n is the free group and P is an amenable factor with an almost periodic state ϕ . We show that these factors are completely classified by the rank n of the free group F n and Connes's Sd-invariant. We prove similar results for free product groups, as well as for classes of generalized Bernoulli actions. [ABSTRACT FROM AUTHOR]

We explore a utilization of Boolean matrix factorization for data preprocessing in classification of Boolean data. In our previous work, we demonstrated that preprocessing that consists in replacing the original Boolean attributes by factors, i.e. new Boolean attributes obtained from the original ones by Boolean matrix factorization, can improve classification quality. The aim of this paper is to explore the question of how the various Boolean factorization methods that were proposed in the literature impact the quality of classification. In particular, we compare five factorization methods, present experimental results, and outline issues for future research. [ABSTRACT FROM AUTHOR]

We provide a unified and self-contained treatment of several of the recent uniqueness theorems for the group measure space decomposition of a II factor. We single out a large class of groups Γ, characterized by a one-cohomology property, and prove that for every free ergodic probability measure preserving action of Γ the associated II factor has a unique group measure space Cartan subalgebra up to unitary conjugacy. Our methods follow closely a recent article of Chifan-Peterson, but we replace the usage of Peterson's unbounded derivations by Thomas Sinclair's dilation into a malleable deformation by a one-parameter group of automorphisms. [ABSTRACT FROM AUTHOR]

We prove that if Γ ↷ (X, μ) is a free ergodic rigid (in the sense of Popa in Ann Math 163:809-889, 2006) probability measure preserving action of a group Γ with positive first ℓ2-Btti number, then the II1 factor L ∞ (X) ⋊ Γ has a unique group measure space Cartan subalgebra, up to unitary conjugacy. We deduce that many HT factors, including the II1 factors associated with the usual actions Γ ↷ T2 and Γ ↷ SL2 (R)/SL2 (Z), where Γ is a non-amenable subgroup of SL2(Z), have a unique group measure space decomposition. [ABSTRACT FROM AUTHOR]