5,344 research outputs found
Leavitt path algebras: the first decade
The algebraic structures known as {\it Leavitt path algebras} were initially
developed in 2004 by Ara, Moreno and Pardo, and almost simultaneously (using a
different approach) by the author and Aranda Pino.
During the intervening decade, these algebras have attracted significant
interest and attention, not only from ring theorists, but from analysts working
in C-algebras, group theorists, and symbolic dynamicists as well. The goal
of this article is threefold: to introduce the notion of Leavitt path algebras
to the general mathematical community; to present some of the important results
in the subject; and to describe some of the field's currently unresolved
questions.Comment: 53 pages. To appear, Bulletin of Mathematical Sciences. (page
numbering in arXiv version will differ from page numbering in BMS published
version; numbering of Theorems, etc ... will be the same in both versions
Convex subshifts, separated Bratteli diagrams, and ideal structure of tame separated graph algebras
We introduce a new class of partial actions of free groups on totally
disconnected compact Hausdorff spaces, which we call convex subshifts. These
serve as an abstract framework for the partial actions associated with finite
separated graphs in much the same way as classical subshifts generalize the
edge shift of a finite graph. We define the notion of a finite type convex
subshift and show that any such subshift is Kakutani equivalent to the partial
action associated with a finite bipartite separated graph. We then study the
ideal structure of both the full and the reduced tame graph C*-algebras,
and , of a separated graph , and
of the abelianized Leavitt path algebra as well. These
algebras are the (reduced) crossed products with respect to the above-mentioned
partial actions, and we prove that there is a lattice isomorphism between the
lattice of induced ideals and the lattice of hereditary -saturated
subsets of a certain infinite separated graph built
from , called the separated Bratteli diagram of . We finally use
these tools to study simplicity and primeness of the tame separated graph
algebras.Comment: 60 page
Stable rank of graph algebras. Type I graph algebras and their limits
For an arbitrary countable directed graph E we show that the only possible
values of the stable rank of the associated Cuntz-Krieger algebra C*(E) are 1,
2 or \infty. Explicit criteria for each of these three cases are given. We
characterize graph algebras of type I, and graph algebras which are inductive
limits of C*-algebras of type I. We also show that a gauge-invariant ideal of a
graph algebra is itself isomorphic to a graph algebra.Comment: 13 pages, LaTe
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