The graded Grothendieck group and the classification of Leavitt path algebras

This paper is an attempt to show that, parallel to Elliott's classification of AF $C^*$-algebras by means of $K$-theory, the graded $K_0$-group classifies Leavitt path algebras completely. In this direction, we prove this claim at two extremes, namely, for the class of acyclic graphs (graphs with no cycles) and comet and polycephaly graphs (graphs which each head is connected to a cycle or a collection of loops).Comment: 30 pages more polishe

SK1 of Azumaya algebras over hensel pairs

Let A be an Azumaya algebra of constant rank n^2 over a Hensel pair (R,I) where R is a semilocal ring with n invertible in R. Then the reduced Whitehead group SK(A) coincides with its reduction SK(A/IA).Comment: 7 page

Relative unitary commutator calculus and applications

This note revisits localisation and patching method in the setting of generalised unitary groups. Introducing certain subgroups of relative elementary unitary groups, we develop relative versions of the conjugation calculus and the commutator calculus in unitary groups, which are both more general, and substantially easier than the ones available in the literature. For the general linear group such relative commutator calculus has been recently developed by the first and the third authors. As an application we prove the mixed commutator formula, for two form ideals of a form ring. This answers two problems posed in a paper by Alexei Stepanov and the second author.Comment: polishe