Coset enumeration, based on the methods described by Todd and Coxeter, is one of the basic tools for investigating nitely presented groups. The process is not well understood, and various pathological presentations of, for example, the trivial group have been suggested as challenge problems. Here we consider one such family of presentations proposed by B.H. Neumann. We show that the problems ar...

A Murray-von Neumann algebra is the algebra of operators affiliated with a finite von Neumann algebra. In this article, we first present a brief introduction to the theory of derivations of operator algebras from both the physical and mathematical points of view. We then describe our recent work on derivations of Murray-von Neumann algebras. We show that the "extended derivations" of a Murray-v...

We study power series over the group ring CF of a free group F . We prove that the von Neumann trace maps rational power series over CF to algebraic power series. Using the Riemann-Stieltjes formula, we deduce the rationality and positivity of Novikov-Shubin invariants of matrices over CF .

Murray-von Neumann algebras are algebras of operators affiliated with finite von Neumann algebras. In this article, we study commutativity and affiliation of self-adjoint operators (possibly unbounded). We show that a maximal abelian self-adjoint subalgebra A of the Murray-von Neumann algebra A(f)(R) associated with a finite von Neumann algebra R is the Murray-von Neumann algebra A(f)(A(0)), wh...

For a Hausdorff zero-dimensional topological space X and totally ordered field F with interval topology, let $$C_c(X,F)$$ be the ring of all F-valued continuous functions on countable range. It is proved that if either an uncountable or subfield $${\mathbb {R}}$$ , then structure $$\beta _0X$$ Banaschewski Compactification X. The ideals $$\{O^{p,F}_c:p\in \beta _0X\}$$ in are introduced as modi...

From symplectic reflection algebras [12], some algebras are naturally introduced. We show that these algebras are non-homogeneous N Koszul algebras. The Koszul property was generalized to homogeneous algebras of degree N > 2 in [6]. In the present paper, the extension of the Koszul property to non-homogeneous algebras is realized through a PBW theorem. This PBW theorem is the generalization to ...

Theorem: An AW*-algebra is the ring generated by its projections if and only if it has no abelian summand. Corollary: Every equivalence in an A W*-algebra may be implemented by a partial isometry in the ring generated by the projections of the algebra. The corollary is extended to certain finite Baer »-rings. An early proposition in the theory of von Neumann algebras is that every such algebra ...