Vol 52, No 3 (2018)

The article continues a series of papers on the absolute of finitely generated groups. The absolute of a group with a fixed system of generators is defined as the set of ergodic Markov measures for which the system of cotransition probabilities is the same as for the simple (right) random walk generated by the uniform distribution on the generators. The absolute is a new boundary of a group, generated by random walks on the group.

We divide the absolute into two parts, Laplacian and degenerate, and describe the connection between the absolute, homogeneous Markov processes, and the Laplace operator; prove that the Laplacian part is preserved under taking certain central extensions of groups; reduce the computation of the Laplacian part of the absolute of a nilpotent group to that of its abelianization; consider a number of fundamental examples (free groups, commutative groups, the discrete Heisenberg group).

Given C^*-algebras A and B, we generalize the notion of a quasi-homomorphism from A to B in the sense of Cuntz by considering quasi-homomorphisms from some C^*-algebra C to B such that C surjects onto A and the two maps forming the quasi-homomorphism agree on the kernel of this surjection. Under an additional assumption, the group of homotopy classes of such generalized quasi-homomorphisms coincides with KK(A, B). This makes the definition of the Kasparov bifunctor slightly more symmetric and provides more flexibility in constructing elements of KK-groups. These generalized quasi-homomorphisms can be viewed as pairs of maps directly from A (instead of various C’s), but these maps need not be *-homomorphisms.

In this paper we find new pairs of self-adjoint commuting differential operators of rank 2 with rational coefficients and prove that any curve of genus 2 written as a hyperelliptic curve is the spectral curve of a pair of commuting differential operators with rational coefficients. We also study the case where curves of genus 3 are the spectral curves of pairs of commuting differential operators of rank 2 with rational coefficients.

Let g be a linear combination with quasipolynomial coefficients of shifts of the Jacobi theta function and its derivatives in the argument. All entire functions f: ℂ → ℂ satisfying f(x+y)g(x−y) = α₁(x)β₁(y)+· · ·+α_r(x)β_r(y) for some r ∈ ℕ and α_j, β_j: ℂ → ℂ are described.

This paper studies the tracial stability of C^*-algebras, which is a general property of stability of relations in a Hilbert–Schmidt-type norm defined by a trace on a C^*-algebra. Precise definitions are formulated in terms of tracial ultraproducts. For nuclear C^*-algebras, a characterization of matricial tracial stability in terms of approximation of tracial states by traces of finite-dimensional representations is obtained. For the nonnuclear case, new obstructions and counterexamples are constructed in terms of free entropy theory.

The aim of this paper is to give an answer to the question, posed by Alsina, Sikorska, and Tomás, as to whether each norm derivative preserving mapping is necessarily linear.