On graded C*-algebras

We show that every topological grading of a C*-algebra by a discrete abelian group is implemented by an action of the compact dual group.Comment: To appear in Bull Aust Math So

Phase transition on the Toeplitz algebra of the affine semigroup over the natural numbers

We show that the group ${\mathbb Q \rtimes \mathbb Q^*_+}$ of orientation-preserving affine transformations of the rational numbers is quasi-lattice ordered by its subsemigroup ${\mathbb N \rtimes \mathbb N^\times}$. The associated Toeplitz $C^*$-algebra ${\mathcal T}({\mathbb N \rtimes \mathbb N^\times})$ is universal for isometric representations which are covariant in the sense of Nica. We give a presentation of this Toeplitz algebra in terms of generators and relations, and use this to show that the $C^*$-algebra ${\mathcal Q_\mathbb N}$ recently introduced by Cuntz is the boundary quotient of $({\mathbb Q \rtimes \mathbb Q^*_+}, {\mathbb N \rtimes \mathbb N^\times})$ in the sense of Crisp and Laca. The Toeplitz algebra ${\mathcal T}({\mathbb N \rtimes \mathbb N^\times})$ carries a natural dynamics $\sigma$, which induces the one considered by Cuntz on the quotient ${\mathcal Q_\mathbb N}$, and our main result is the computation of the KMS$_\beta$ (equilibrium) states of the dynamical system $({\mathcal T}({\mathbb N \rtimes \mathbb N^\times}), {\mathbb R},\sigma)$ for all values of the inverse temperature $\beta$. For $\beta \in [1, 2]$ there is a unique KMS$_\beta$ state, and the KMS$_1$ state factors through the quotient map onto ${\mathcal Q_\mathbb N}$, giving the unique KMS state discovered by Cuntz. At $\beta =2$ there is a phase transition, and for $\beta>2$ the KMS$_\beta$ states are indexed by probability measures on the circle. There is a further phase transition at $\beta=\infty$, where the KMS$_\infty$ states are indexed by the probability measures on the circle, but the ground states are indexed by the states on the classical Toeplitz algebra ${\mathcal T}(\mathbb N)$.Comment: 38 page

Two families of Exel-Larsen crossed products

Larsen has recently extended Exel's construction of crossed products from single endomorphisms to abelian semigroups of endomorphisms, and here we study two families of her crossed products. First, we look at the natural action of the multiplicative semigroup $\mathbb{N}^\times$ on a compact abelian group $\Gamma$, and the induced action on $C(\Gamma)$. We prove a uniqueness theorem for the crossed product, and we find a class of connected compact abelian groups $\Gamma$ for which the crossed product is purely infinite simple. Second, we consider some natural actions of the additive semigroup $\mathbb{N}^2$ on the UHF cores in 2-graph algebras, as introduced by Yang, and confirm that these actions have properties similar to those of single endomorphisms of the core in Cuntz algebras.Comment: 17 page

Twisted actions and the obstruction to extending unitary representations of subgroups

Suppose that $G$ is a locally compact group and $\pi$ is a (not necessarily irreducible) unitary representation of a closed normal subgroup $N$ of $G$ on a Hilbert space $H$. We extend results of Clifford and Mackey to determine when $\pi$ extends to a unitary representation of $G$ on the same space $H$ in terms of a cohomological obstruction