arXiv 0907.37602009-07math.OA

自然数上仿射半群的Toeplitz代数的相变

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

Marcelo Laca, Iain Raeburn

备注：: 38 pages

摘要：

证明了有理数的保向仿射变换群${\mathbb Q \rtimes \mathbb Q^*_+}$是由它的子半群${\mathbb N \rtimes \mathbb N^\times}$拟格序的。相关的Toeplitz $C^*$ -代数${\mathcal T}({\mathbb N \rtimes \mathbb N^\times})$对于在Nica意义下协变的等距表示是泛的.我们给出了这个Toeplitz代数的生成元和关系，并利用这一点来表明，$C^*$ -代数${\mathcal Q_\mathbb N}$最近推出的Cuntz是边界商的$({\mathbb Q \rtimes \mathbb Q^*_+}, {\mathbb N \rtimes \mathbb N^\times})$的意义上的脆和拉卡。Toeplitz代数${\mathcal T}({\mathbb N \rtimes \mathbb N^\times})$带有一个自然动力学$σ$，它在商${\mathcal Q_\mathbb N}$上导出了Cuntz所考虑的动力学$σ$，我们的主要结果是计算了动力学系统$({\mathcal T}({\mathbb N \rtimes \mathbb N^\times}), {\mathbb R},σ)$对于所有逆温度$β$的KMS $_β$（平衡）态.对于$β\in [1, 2]$，存在唯一的KMS $_β$状态，并且KMS $_1$状态因子通过商映射到${\mathcal Q_\mathbb N}$上，给出了由Cuntz发现的唯一KMS状态。在$β=2$处存在相变，并且对于$β>2$，KMS $_β$状态由圆上的概率度量索引。在$β=\infty$处存在进一步的相变，其中KMS $_\infty$状态由圆上的概率测度索引，但基态由经典Toeplitz代数${\mathcal T}(\mathbb N)$上的状态索引。

英文摘要：

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 $σ$, which induces the one considered by Cuntz on the quotient ${\mathcal Q_\mathbb N}$, and our main result is the computation of the KMS$_β$ (equilibrium) states of the dynamical system $({\mathcal T}({\mathbb N \rtimes \mathbb N^\times}), {\mathbb R},σ)$ for all values of the inverse temperature $β$. For $β\in [1, 2]$ there is a unique KMS$_β$ 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 $β=2$ there is a phase transition, and for $β>2$ the KMS$_β$ states are indexed by probability measures on the circle. There is a further phase transition at $β=\infty$, 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 ${\mathcal T}(\mathbb N)$.

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