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

Reintroduction of North Island robins to Paengaroa Scenic Reserve : factors limiting survival, nest success, and population viability in a mainland restoration area : a thesis presented in partial fulfilment of the requirements for the degree of Master of Science in Ecology at Massey University

Forty North Island robins (Petroica australis longipes) were reintroduced to Paengaroa Scenic Reserve in March 1999. I monitored the survival and breeding success of this population for two years post-release. This study aims to assess survival, nest success, and population viability of robins in Paengaroa in an attempt to discover whether habitat in the reserve is likely to support a population of robins. Survival from the time of release to the start of the first breeding season was lower at Paengaroa than at two other release sites, Boundary Stream Scenic Reserve and Tiritiri Matangi Island. This may be due to higher predator levels at Paengaroa or dispersal out of the reserve. Methods of estimating nest success were compared, and Stanley's (2000) method was found to have advantages over the traditional and Mayfield methods. Daily survival rates of nests at Paengaroa depended on both the stage in the nesting cycle and stage of the breeding season, with the survival rate lowest for early nests at the incubation stage. Nest success for the first two breeding seasons after translocation was compared to that for the first two seasons after release at Tiritiri Matangi and Boundary Stream. Paengaroa had a similar nest success rate to Tiritiri Matangi (25% and 26% respectively), and both of these sites had lower nest success than Boundary Stream (47%). Survival at Paengaroa was most affected by whether a bird was recently-translocated. a juvenile, or an adult. Recently-translocated birds and juveniles suffered similarly low survival rates, suggesting that this high mortality may be due to problems faced when finding and establishing a territory. The survival of juveniles from January to September was estimated to be 29%. The annual adult survival rate was also low (59%). Fecundity and survival estimates were used in a stochastic simulation model to predict the viability of the Paengaroa population. Under current conditions, the population was predicted to have a 17% probability of surviving 10 years. However, variation of parameters to lower and upper 95% confidence limits gave survival probabilities of 0% to 100% over 10 years. When data from the first year after translocation were excluded, the population was predicted to have a 100% probability of surviving 100 years. These results demonstrate the large uncertainly associated with small sample sizes and short-term studies. To assess whether habitat quality is likely to account for the poor overall viability predicted at Paengaroa, the habitat quality at Paengaroa was compared to that at Waimarino forest, where robins still persist. Food supply and predator levels were used to assess habitat quality, as these are obvious factors that may limit viability. Data on food and predator levels provided no indication of why robins may be non-viable at Paengaroa. The power of statistical tests was low due to small sample size, but results suggest Paengaroa has more food as well as fewer rats and stoats than Waimarino. There is a need for further research to improve our understanding of why robins are present and common in some mainland areas but have disappeared from others without any obvious difference in habitat quality. Continued research is also required to reduce the uncertainty regarding population viability at Paengaroa and to determine whether improved management is needed

Islam's Foundational Equality

In Feminist Edges of the Qur’an, Aysha Hidayatullah argues that certain Qur’anic verses are incorrigibly male-privileging and are themselves privileged. Hence, egalitarian readings of the Qur’an are unsupported and unsupportable. If, as egalitarians propose, such verses are unjust, then either the Qur’an is not God’s word or God is unjust. By contrast, I argue that no evidence suggests any such verses are incorrigibly male- privileging. Further I indicate egalitarian rereadings for relevant contenders and note that, in any case, no Qur’anic evidence warrants the primacy of such verses. Finally, since controverting egalitarian readings of such verses are available, the logical form of Hidayatullah’s argument merely shows that if they are read to exhibit injustice, those readings cannot be God’s word. Since believers hold that the Qur’an is God’s word, there is no option but to reread them

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