Simplicity and Chain Conditions for Ultragraph Leavitt Path Algebras via Partial Skew Group Ring Theory

Abstract

We realize Leavitt ultragraph path algebras as partial skew group rings. Using this realization we characterize artinian ultragraph path algebras and give simplicity criteria for these algebras. MSC[2010]: 16S35, 16S99, 16P20.

Irreducible and permutative representations of ultragraph Leavitt path algebras

“…Remark 4.3 The above branching system can also be seen as a partial action of the free group on the edges of the ultragraph and be used to realized L R (G) as a partial skew group ring, see [8,22,24].…”

Recent developments around partial actions

“…More recent C * and topological advances include the groupoid approach to the enveloping C * -algebras associated to partial actions of countable discrete groups on (locally) compact spaces in [156], the use of inverse semigroup expansions to treat C *crossed products by twisted partial actions via twisted global actions of the expansion in [68], the full (respectively, reduced) partial C * -crossed product descriptions of full (respectively, reduced) C * -algebras of countable E-unitary or strongly 0-E-unitary inverse semigroups as well as of tight groupoids of countable strongly 0-E-unitary inverse semigroups in [235], the study of the continuous orbit equivalence for partial dynamical systems and of the partial transformation groupoids with applications to graph C * -algebras and semigroup C * -algebras in [224], the partial group action approach to produce a Bratteli-Vershik model linked to a minimal homeomorphism between open subsets with finite disjoint complements of the Cantor set in [179], new developments on the globalization problem for partial actions on C * -algebras and Hilbert bimodules in [168], the employment of the partial crossed product theory to the investigation of the Cuntz-Li C * -algebras related to an integral domain in [61] with a further development in [281] for C * -algebras associated with an injective endomorphism of a group with finite cokernel. Moreover, partial crossed products turned out to be useful to deal with C * -algebras arising from self-similar graph actions in [162], with C * -algebras associated to any stationary ordered Bratteli diagram in [185], as well as with ultragraph C * -algebras and related infinite alphabet shifts in [187][188][189]191]. In addition, in [212], partial coactions of C * -bialgebras, in particular of C * -quantum groups, on C * -algebras were defined and studied, and a globalization result was obtained.…”

On the ideals of ultragraph Leavitt path algebras

“…The Leavitt path algebra associated with an ultragraph was defined by Imanfar, Pourabbas, and Larki in [25]. In [21], a slightly different definition appeared, and in [13] de Castro, Gonçalves and van Wyk showed that the resulting algebras are isomorphic. As in the C*-algebraic setting, the ultragraph Leavitt path algebras unify the study of Leavitt path algebras associated with graphs and the algebras associated with infinite matrices.…”