# Product systems of graphs and the Toeplitz algebras of higher-rank graphs | NOVA. The University of Newcastle's Digital Repository

@inproceedings{Raeburn2003ProductSO, title={Product systems of graphs and the Toeplitz algebras of higher-rank graphs | NOVA. The University of Newcastle's Digital Repository}, author={Iain Raeburn and Aidan Sims}, year={2003} }

There has recently been much interest in the C � -algebras of directed graphs. Here we consider product systems E of directed graphs over semigroups and associated C � -algebras C � (E) and T C � (E) which generalise the higher-rank graph algebras of Kumjian-Pask and their Toeplitz analogues. We study these algebras by constructing from E a product system X(E) of Hilbert bimodules, and applying re- cent results of Fowler about the Toeplitz algebras of such systems. Fowler's hypotheses turn out…

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## References

SHOWING 1-10 OF 26 REFERENCES

### Higher Rank Graph C-Algebras

- Mathematics
- 2000

Building on recent work of Robertson and Steger, we associate a C{algebra to a combinatorial object which may be thought of as a higher rank graph. This C{algebra is shown to be isomorphic to that of…

### HIGHER-RANK GRAPHS AND THEIR $C^*$-ALGEBRAS

- MathematicsProceedings of the Edinburgh Mathematical Society
- 2003

Abstract We consider the higher-rank graphs introduced by Kumjian and Pask as models for higher-rank Cuntz–Krieger algebras. We describe a variant of the Cuntz–Krieger relations which applies to…

### COMPACTLY-ALIGNED DISCRETE PRODUCT SYSTEMS, AND GENERALIZATIONS OF ${\mathcal O}_\infty$

- Mathematics
- 1999

The universal C*-algebras of discrete product systems generalize the Toeplitz–Cuntz algebras and the Toeplitz algebras of discrete semigroups. We consider a semigroup P which is quasi-lattice ordered…

### CUNTZ-KRIEGER ALGEBRAS OF DIRECTED GRAPHS

- Mathematics
- 1998

We associate to each row-nite directed graph E a universal Cuntz-Krieger C-algebra C(E), and study how the distribution of loops in E aects the structure of C(E) .W e prove that C(E) is AF if and…

### A class ofC*-algebras and topological Markov chains

- Mathematics
- 1980

In this paper we present a class of C*-algebras and point out its close relationship to topological Markov chains, whose theory is part of symbolic dynamics. The C*-algebra construction starts from a…

### Graphs, Groupoids, and Cuntz–Krieger Algebras

- Mathematics
- 1997

We associate to each locally finite directed graphGtwo locally compact groupoidsGandG(★). The unit space ofGis the space of one–sided infinite paths inG, andG(★) is the reduction ofGto the space of…

### Semigroup Crossed Products and the Toeplitz Algebras of Nonabelian Groups

- Mathematics
- 1996

Abstract We consider the quasi-lattice ordered groups ( G , P ) recently introduced by Nica. We realise their universal Toeplitz algebra as a crossed product B P ⋊ P by a semigroup of endomorphisms,…

### Viewing AF-algebras as graph algebras

- Mathematics
- 1998

Every AF-algebra A arises as the C*-algebra of a locally finite pointed directed graph in the sense of Kumjian, Pask, Raeburn, and Renault. For AF-algebras, the diagonal subalgebra defined by…

### On higher rank graph C ∗ -algebras

- Mathematics
- 2000

Given a row-finite k-graph Λ with no sources we investigate the K-theory of the higher rank graph C *-algebra, C * (Λ). When k = 2 we are able to give explicit formulae to calculate the K-groups of C…

### The Toeplitz algebra of a Hilbert bimodule

- Mathematics
- 1998

Suppose a C*-algebra A acts by adjointable operators on a Hilbert A-module X. Pimsner constructed a C*-algebra O_X which includes, for particular choices of X, crossed products of A by Z, the Cuntz…