200 research outputs found

### 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

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