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Contemporary Mathematics Volume 480, 2009

Summary: Contemporary Mathematics
Volume 480, 2009
Leavitt Path Algebras and Direct Limits
K. R. Goodearl
Abstract. An introduction to Leavitt path algebras of arbitrary directed
graphs is presented, and direct limit techniques are developed, with which
many results that had previously been proved for countable graphs can be
extended to uncountable ones. Such results include characterizations of sim-
plicity, characterizations of the exchange property, and cancellation conditions
for the K-theoretic monoid of equivalence classes of idempotent matrices.
The algebras of the title descend from algebras constructed by W. G. Leavitt
[21] to exhibit rings in which free modules of specific different finite ranks are
isomorphic. Later, and independently, J. Cuntz introduced an analogous class of
C*-algebras [15]. Generalizations of these led to a large class of C*-algebras built
from directed graphs, and the construction was carried to the algebraic category
by G. Abrams and G. Aranda Pino [1]. For both historical and technical reasons,
the graphs used in constructing graph C*-algebras and Leavitt path algebras have
been assumed to be countable, although the construction does not require this.
Our motivation for this paper was to initiate the study of Leavitt path algebras of


Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara


Collections: Mathematics