Discrete cores of type III free product factors

We give a general description of the discrete decompositions of type III factors arising as central summands of free product von Neumann algebras based on our previous works. This enables us to give several precise structural results on type III free product factors.Comment: Expanded versio

Some analysis on amalgamated free products of von Neumann algebras in non-tracial setting

Several techniques together with some partial answers are given to the questions of factoriality, type classification and fullness for amalgamated free product von Neumann algebras.Comment: 3rd version; no essential change; typos correcte

HNN Extensions of von Neumann Algebras

Reduced HNN extensions of von Neumann algebras (as well as $C^*$-algebras) will be introduced, and their modular theory, factoriality and ultraproducts will be discussed. In several concrete settings, detailed analysis on them will be also carried out.Comment: Slightly revised versio

Realization of rigid C∗^*-tensor categories via Tomita bimodules

Starting from a (small) rigid C$^*$-tensor category $\mathscr{C}$ with simple unit, we construct von Neumann algebras associated to each of its objects. These algebras are factors and can be either semifinite (of type II$_1$ or II$_\infty$, depending on whether the spectrum of the category is finite or infinite) or they can be of type III$_\lambda$, $\lambda\in (0,1]$. The choice of type is tuned by the choice of Tomita structure (defined in the paper) on certain bimodules we use in the construction. Moreover, if the spectrum is infinite we realize the whole tensor category directly as endomorphisms of these algebras, with finite Jones index, by exhibiting a fully faithful unitary tensor functor $F:\mathscr{C} \hookrightarrow End_0(\Phi)$ where $\Phi$ is a factor (of type II or III). The construction relies on methods from free probability (full Fock space, amalgamated free products), it does not depend on amenability assumptions, and it can be applied to categories with uncountable spectrum (hence it provides an alternative answer to a conjecture of Yamagami \cite{Y3}). Even in the case of uncountably generated categories, we can refine the previous equivalence to obtain realizations on $\sigma$-finite factors as endomorphisms (in the type III case) and as bimodules (in the type II case). In the case of trivial Tomita structure, we recover the same algebra obtained in \cite{PopaS} and \cite{AMD}, namely the (countably generated) free group factor $L(F_\infty)$ if the given category has denumerable spectrum, while we get the free group factor with uncountably many generators if the spectrum is infinite and non-denumerable.Comment: 39 page

Type III factors with unique Cartan decomposition

We prove that for any free ergodic nonsingular nonamenable action \Gamma\ \actson (X,\mu) of all \Gamma\ in a large class of groups including all hyperbolic groups, the associated group measure space factor $L^\infty(X) \rtimes \Gamma$ has L^\infty(X) as its unique Cartan subalgebra, up to unitary conjugacy. This generalizes the probability measure preserving case that was established in [PV12]. We also prove primeness and indecomposability results for such crossed products, for the corresponding orbit equivalence relations and for arbitrary amalgamated free products $M_1 *_B M_2$ over a subalgebra B of type I.Comment: v2: we only fixed a LaTeX issu