Abstract. We present a uniqueness theorem for k-graph C*-algebras that requires neither an aperiodicity nor a gauge invariance assumption. Specifically, we prove that for the injectivity of a representation of a k-graph C*-algebra, it is sufficient that the representation be injective on a distinguished abelian C*-subalgebra. A crucial part of the proof is the application of an abstract uniqueness theorem, which says that such a uniqueness property follows from the existence of a jointly faithful collection of states on the ambient C*-algebra, each of which is the unique extension of a state on the distinguished abelian C*-subalgebra.

Abstract. The reduced C * -algebra of the interior of the isotropy in any Hausdorf etale groupoid G embeds as a C * -subalgebra M of the reduced C * -algebra of G. We prove that the set of pure states of M with unique extension is dense, and deduce that any representation of the reduced C * -algebra of G that is injective on M is faithful. We prove that there is a conditional expectation from the reduced C * -algebra of G onto M if and only if the interior of the isotropy in G is closed. Using this, we prove that when the interior of the isotropy is abelian and closed, M is a Cartan subalgebra. We prove that for a large class of groupoids G with abelian isotropy-including all Deaconu-Renault groupoids associated to discrete abelian groups-M is a maximal abelian subalgebra. In the specific case of k-graph groupoids, we deduce that M is always maximal abelian, but show by example that it is not always Cartan.

The set of diagrams consisting of an annulus with a finite family of curves connecting some points on the boundary to each other defines a category in which a contractible closed curve counts for a certain complex number δ. For δ = 2 cos(π/n) this category admits a C * -structure and we determine all Hilbert space representations of this category for these values, at least in the case where the number of internal boundary points is even. This result has applications to subfactors and planar algebras.

We introduce a certain natural abelian C*‐subalgebra of a graph C*‐algebra, which has functorial properties that can be used for characterizing injectivity of representations of the ambient C*‐algebra. In particular, a short proof of the Cuntz–Krieger Uniqueness Theorem, for graphs that may have loops without entries, is given.

A Hilbert module over a planar algebra P is essentially a Hilbert module over a canonically defined algebra spanned by the annular tangles in P. It follows that any planar algebra Q containing P is a module over P, and in particular, any subfactor planar algebra is a module over the Temperley-Lieb planar algebra with the same modulus. We describe a positivity result that allows us to describe irreducible Temperley-Lieb planar algebra modules, and apply the result to decompose the planar algebras determined by the Coxeter graphs A n (n 3), D n (n 4), E 6 , E 7 , and E 8 .