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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 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]