In this short note we present a common characterization of the logarithmic function and the subspace of all trace zero elements in finite von Neumann factors. Our aim is to prove the following statement. Theorem 1. Let A be a von Neumann factor and f :]0,∞[→ R be a nonconstant continuous function. Set S A f = span{ f (AB A)− (2 f (A)+ f (B)) : A,B ∈A −1 + }. Then either we have S A f = A or we ...

This is an expanded version of the talk given by first author at conference “Topology, Geometry, and Dynamics: Rokhlin – 100”. The purpose this was to explain our current results on classification rational symplectic 4-manifolds equipped with anti-symplectic involution. A detailed exposition will appear elsewhere.

We give a classification theorem for unital separable simple nuclear C∗-algebras with tracial topological rank zero which satisfy the Universal Coefficient Theorem. We prove that if A and B are two such C∗-algebras and (K0(A),K0(A)+, [1A], K1(A)) = (K0(B), K0(B)+, [1B ], K1(B)), then A = B.

We investigate the notion of tracial $\mathcal Z$-stability beyond unital C*-algebras, and we prove that this is equivalent to in class separable simple nuclear C*-algebras.

We establish the tracial stability of a certain class graph products C*-algebras. This result involves development pincushion finite graphs. then apply this in two ways. The first application yields selective version Lin's Theorem for almost commuting operators. second addresses some approximation properties right-angled Artin groups. In particular, we show that full C*-algebra any group is qua...

We prove a central limit theorem for non-commutative random variables in a von Neumann algebra with a tracial state: Any non-commutative polynomial of averages of i.i.d. samples converges to a classical limit. The proof is based on a central limit theorem for ordered joint distributions together with a commutator estimate related to the Baker-Campbell-Hausdorff expansion. The result can be cons...

This paper investigates the first part of Hilbert’s 16th problem which asks about topology of the real projective algebraic curves. Using the Rokhlin-Viro-Fiedler method of complex orientation, we obtain new restrictions on the arrangements of ovals of projective algebraic curves of odd degree d = 4k + 1, k ≥ 2, with nests of depth k.