• Corpus ID: 498014

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}
}

• I. Raeburn, A. Sims
• Published 27 May 2003
• Mathematics

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 to be very interesting graph-theoretically, and indicate new relations which will have to be added to the usual Cuntz-Krieger relations to obtain a satisfactory theory of Cuntz-Krieger algebras for product systems of graphs; our algebras C � (E) and T C � (E) are universal for families of partial isometries satisfying these relations. Our main result is a uniqueness theorem for T C � (E) which has particularly in- teresting implications for the C � -algebras of non-row-finite higher-rank graphs. This theorem is apparently beyond the reach of Fowler's theory, and our proof requires a detailed analysis of the expectation onto the diagonal in T C � (E).