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The dimension of any module over an algebra of affiliated operators ${\mathcal U}$ of a finite von Neumann algebra ${\mathcal A}$ is defined using a trace on ${\mathcal A}.$ All zero-dimensional ${\mathcal U}$-modules constitute the torsion…

A discussion of the number of degrees of freedom, and their dynamical properties, in higher derivative gravitational theories is presented. The complete non-linear sigma model for these degrees of freedom is exhibited using the method of…

We show that a free product of a II_1-factor and a finite von Neumann algebra with amalgamation over a finite dimensional subalgebra is always a II_1-factor, and provide an algorithm for describing it in terms of free products (with…

An essentially free group action of $\Gamma$ on $(X,\mu)$ is called W*-superrigid if the crossed product von Neumann algebra $L^\infty(X) \rtimes \Gamma$ completely remembers the group $\Gamma$ and its action on $(X,\mu)$. We prove…

We define a free probability analogue of the Wasserstein metric, which extends the classical one. In dimension one, we prove that the square of the Wasserstein distance to the semi-circle distribution is majorized by a modified free entropy…

We investigate a construction which associates a finite von Neumann algebra $M(\Gamma,\mu)$ to a finite weighted graph $(\Gamma,\mu)$. Pleasantly, but not surprisingly, the von Neumann algebra associated to to a `flower with $n$ petals' is…

Let $R$ be a commutative Noetherian local ring. We prove a variety of new formulae for modules of finite quasi-projective or finite quasi-injective dimension. These include the Derived Depth Formula, itself an extension of Auslander famous…

We consider two extensions of free probability that have been studied in the research literature, and are based on the notions of c-freeness and respectively of infinitesimal freeness for noncommutative random variables. In a 2012 paper,…

We introduce the notion of proper proximality for finite von Neumann algebras, which naturally extends the notion of proper proximality for groups. Apart from the group von Neumann algebras of properly proximal groups, we provide a number…

We study $(\sigma,\tau)$-derivations of a group ring $RG$ of a finite group $G$ over an integral domain $R$ with $1$. As an application we extend a well known result on derivation of an integral group ring $\Bbb{Z}G$ to…

By proving that certain free stochastic differential equations have stationary solutions, we give a lower estimate on the microstates free entropy dimension of certain $n$-tuples $X_{1},...,X_{n}$: we show that Abstract. By proving that…

Let $G$ be a noncompact semisimple algebraic group with trivial center, $S < G$ a maximal split torus, $H < G$ the centralizer of $S$ in $G$ and $\Gamma < G$ an irreducible lattice. Consider the group measure space von Neumann algebra…

We consider converses to the density theorem for irreducible, projective, unitary group representations restricted to lattices using the dimension theory of Hilbert modules over twisted group von Neumann algebras. We show that under the…

We prove that certain free products of factors of type ${\rm I}$ and other von Neumann algebras with respect to nontracial, almost periodic states are almost periodic free Araki-Woods factors. In particular, they have the free absorption…

To any trace preserving action $\sigma: G \curvearrowright A$ of a countable discrete group on a finite von Neumann algebra $A$ and any orthogonal representation $\pi:G \to \mathcal O(\ell^2_{\mathbb{R}}(G))$, we associate the generalized…

Let $G$ be a group, $R$ an integral domain, and $V_G$ the subspace of the group algebra $R[G]$ consisting of all the elements of $R[G]$ whose coefficient of the identity element $1_G$ of $G$ is equal to zero. Motivated by the Mathieu…

Given a type I von Neumann algebra $M$ with a faithful normal semi-finite trace $\tau,$ let $S_0(M, \tau)$ be the algebra of all $\tau$-compact operators affiliated with $M.$ We give a complete description of all derivations on the algebra…

Let $K[X_d]=K[x_1,\ldots,x_d]$ be the polynomial algebra in $d$ variables over a field $K$ of characteristic 0. The classical theorem of Weitzenb\"ock from 1932 states that for linear locally nilpotent derivations $\delta$ (known as…

It is shown that the differential geometry of space-time, can be expressed in terms of the algebra of operators on a bundle of Hilbert spaces. The price for this is that the algebra of smooth functions on space-time has to be made…

Let $K$ be an algebraically closed non-Archimedean field. Leonard Lipshitz has introduced a manageable notion of subanalytic sets of the unit polydisc. This class contains the class of affinoid sets and is stable under projection. We…