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Articles 1 - 8 of 8
Full-Text Articles in Physical Sciences and Mathematics
Homology For Higher-Rank Graphs And Twisted C*-Algebras, Alex Kumjian, David Pask, Aidan Sims
Homology For Higher-Rank Graphs And Twisted C*-Algebras, Alex Kumjian, David Pask, Aidan Sims
Faculty of Informatics - Papers (Archive)
We introduce a homology theory for k-graphs and explore its fundamental properties. We establish connections with algebraic topology by showing that the homology of a k-graph coincides with the homology of its topological realisation as described by Kaliszewski et al. We exhibit combinatorial versions of a number of standard topological constructions, and show that they are compatible, from a homological point of view, with their topological counterparts. We show how to twist the C*-algebra of a k-graph by a T-valued 2-cocycle and demonstrate that examples include all noncommutative tori. In the appendices, we construct a cubical set …
Generalised Morphisms Of K-Graphs: K-Morphs, Alex Kumjian, David Pask, Aidan Sims
Generalised Morphisms Of K-Graphs: K-Morphs, Alex Kumjian, David Pask, Aidan Sims
Faculty of Informatics - Papers (Archive)
In a number of recent papers, (k+l)-graphs have been constructed from k-graphs by inserting new edges in the last l dimensions. These constructions have been motivated by C*-algebraic considerations, so they have not been treated systematically at the level of higher-rank graphs themselves. Here we introduce k-morphs, which provide a systematic unifying framework for these various constructions. We think of k-morphs as the analogue, at the level of k-graphs, of C*-correspondences between C*-algebras. To make this analogy explicit, we introduce a category whose objects are k-graphs and whose …
Periodic 2-Graphs Arising From Subshifts, David Pask, Iain Raeburn, Natasha A. Weaver
Periodic 2-Graphs Arising From Subshifts, David Pask, Iain Raeburn, Natasha A. Weaver
Faculty of Informatics - Papers (Archive)
Higher-rank graphs were introduced by Kumjian and Pask to provide models for higher-rank Cuntz– Krieger algebras. In a previous paper, we constructed 2 graphs whose path spaces are rank two subshifts of finite type, and showed that this construction yields aperiodic 2 graphs whose C algebras are simple and are not ordinary graph algebras. Here we show that the construction also gives a family of periodic 2 graphs which we call domino graphs. We investigate the combinatorial structure of domino graphs, finding interesting points of contact with the existing combinatorial literature, and prove a structure theorem for the C algebras …
Aperiodicity And Cofinality For Finitely Aligned Higher-Rank Graphs, Peter Lewin, Aidan Sims
Aperiodicity And Cofinality For Finitely Aligned Higher-Rank Graphs, Peter Lewin, Aidan Sims
Faculty of Informatics - Papers (Archive)
We introduce new formulations of aperiodicity and cofinality for finitely aligned higher-rank graphs $\Lambda$, and prove that $C^*(\Lambda)$ is simple if and only if $\Lambda$ is aperiodic and cofinal. The main advantage of our versions of aperiodicity and cofinality over existing ones is that ours are stated in terms of finite paths. To prove our main result, we first characterise each of aperiodicity and cofinality of $\Lambda$ in terms of the ideal structure of $C^*(\Lambda)$. In an appendix we show how our new cofinality condition simplifies in a number of special cases which have been treated previously in the literature; …
A Direct Approach To Co-Universal Algebras Associated To Directed Graphs, Aidan Sims, S B. Webster
A Direct Approach To Co-Universal Algebras Associated To Directed Graphs, Aidan Sims, S B. Webster
Faculty of Informatics - Papers (Archive)
We prove directly that if $E$ is a directed graph in which every cycle has an entrance, then there exists a $C^*$-algebra which is co-universal for Toeplitz-Cuntz-Krieger $E$-families. In particular, our proof does not invoke ideal-structure theory for graph algebras, nor does it involve use of the gauge action or its fixed point algebra.
A Family Of 2-Graphs Arising From Two-Dimensional Subshifts, David Pask, Iain Raeburn, Natasha A. Weaver
A Family Of 2-Graphs Arising From Two-Dimensional Subshifts, David Pask, Iain Raeburn, Natasha A. Weaver
Faculty of Informatics - Papers (Archive)
Higher-rank graphs (or k-graphs) were introduced by Kumjian and Pask to provide combinatorial models for the higher-rank Cuntz-Krieger C*-algebras of Robertson and Steiger. Here we consider a family of finite 2-graphs whose path spaces are dynamical systems of algebraic origin, as studied by Schmidt and others.
Simplicity Of C*-Algebras Associated To Row-Finite Locally Convex Higher-Rank Graphs, David Robertson, Aidan Sims
Simplicity Of C*-Algebras Associated To Row-Finite Locally Convex Higher-Rank Graphs, David Robertson, Aidan Sims
Faculty of Informatics - Papers (Archive)
In a previous work, the authors showed that the C*-algebra C*(\Lambda) of a row-finite higher-rank graph \Lambda with no sources is simple if and only if \Lambda is both cofinal and aperiodic. In this paper, we generalise this result to row-finite higher-rank graphs which are locally convex (but may contain sources). Our main tool is Farthing's "removing sources" construction which embeds a row-finite locally convex higher-rank graph in a row-finite higher-rank graph with no sources in such a way that the associated C*-algebras are Morita equivalent.
Fundamental Groupoids Of K-Graphs, David A. Pask, Iain F. Raeburn, John C. Quigg
Fundamental Groupoids Of K-Graphs, David A. Pask, Iain F. Raeburn, John C. Quigg
Faculty of Informatics - Papers (Archive)
k-graphs are higher-rank analogues of directed graphs which were first developed to provide combinatorial models for operator algebras of Cuntz– Krieger type. Here we develop a theory of the fundamental groupoid of a kgraph, and relate it to the fundamental groupoid of an associated graph called the 1-skeleton. We also explore the failure, in general, of k-graphs to faithfully embed into their fundamental groupoids.