Graded Steinberg algebras and their representations

Abstract

We study the category of left unital graded modules over the Steinberg algebra of a graded ample Hausdorff groupoid. In the first part of the paper, we show that this category is isomorphic to the category of unital left modules over the Steinberg algebra of the skew-product groupoid arising from the grading. To do this, we show that the Steinberg algebra of the skew product is graded isomorphic to a natural generalisation of the the Cohen-Montgomery smash product of the Steinberg algebra of the underlying groupoid with the grading group. In the second part of the paper, we study the minimal (that is, irreducible) representations in the category of graded modules of a Steinberg algebra, and establish a connection between the annihilator ideals of these minimal representations, and effectiveness of the groupoid. Specialising our results, we produce a representation of the monoid of graded finitely generated projective modules over a Leavitt path algebra. We deduce that the lattice of order-ideals in the $K_0$-group of the Leavitt path algebra is isomorphic to the lattice of graded ideals of the algebra. We also investigate the graded monoid for Kumjian--Pask algebras of row-finite $k$-graphs with no sources. We prove that these algebras are graded von Neumann regular rings, and record some structural consequences of this.

28 pages

References in corpus (5)

• Graph inverse semigroups, groupoids and their C*-algebras
• A Groupoid Approach to Discrete Inverse Semigroup Algebras
• Gauge-invariant ideals in the C*-algebras of finitely aligned higher-rank graphs
• The co-universal C*-algebra of a row-finite graph
• A note on the isomorphism conjectures for Leavitt path algebras

Cited by in corpus (11)

• Lifting morphisms between graded Grothendieck groups of Leavitt path algebras
• Classification conjectures for Leavitt path algebras
• Graded Semigroups
• Realizing ultragraph Leavitt path algebras as Steinberg algebras
• Tensor products of Steinberg algebras
• Weighted Leavitt path algebras -- an overview
• On induced graded simple modules over graded Steinberg algebras with applications to Leavitt path algebras
• Comparability in the graph monoid
• Graded $K$-Theory, Filtered $K$-theory and the classification of graph algebras
• The talented monoid of a Leavitt path algebra
• Higher-rank graphs and the graded $K$-theory of Kumjian-Pask algebras