Tracial states on the \(\theta\)-deformed plane
A tracial state \(\phi:{\mathcal A}\to\mathbb{C}\), \({\mathcal A}\) is a unital \(*\)-algebra (in this paper, \(^-\) is used instead of \(*\)) such that \[ \phi(\overline aA)\geq 0,\;\phi(\mathbf{1}_A)= 1,\;\phi(xy)= \phi(yx), \] where \(\mathbf{1}_A\) is the unit element of \({\mathcal A}\). The set of tracial states forms a convex set which is called the tracial state space. In this paper, tracial states on the \(\theta\)-deformed \(2m\)-plane \(C^{\text{alg}}(\mathbb{R}^{2m}_\theta)\) are constructed and the extremal points of the tracial state space of \(C^{\text{alg}}(\mathbb{R}^{2m}_\theta)\) are obtained. Here, \(C^{\text{alg}}(\mathbb{R}^{2m}_\theta)\) is the unital \(*\)-algebra generated by \(m\) elements \(z^i\), \(i= 1,\ldots,m\), with the commutation relation \[ z^i z^j= \lambda^{ij} z^j z^i,\;\overline z^i\overline z^j= \lambda^{ij}\overline z^j\overline z^i,\;z^i\overline z^j= \lambda^{ji}\overline z^j z^i\;(1\leq i,j\leq m), \] where \(\lambda^{ij}= e^{2\pi i\theta_{ij}}\) and \(\theta= (\theta_{ij})\) is an anti-symmetric real-valued matrix of degree \(m\). To construct tracial states, a function \(\phi: C^{\text{alg}}(\mathbb{R}^{2m}_\theta)\to\mathbb C\) is introduced by \(\phi(z^i z^j)= \phi(z^j\overline z^i)= \lambda^{ij}\), \(\phi(z^i\overline z^j)= \phi(\overline z^j\overline z^i)= \lambda^{ji}\), \(\phi(z^j z^i)= \phi(\overline z^j z^i)= \phi(\overline z^i\overline z^j)= \phi(\overline z^i z^j)= 1\), \(\phi(z^i)= \phi(\overline z^i)= 0\), \(\phi(\mathbf{1})= 1\). As for higher-order monomials, putting \(z^1= w_1\), \(\overline z^1= w_2,\ldots, z^m= w_{2m-1}\), \(\overline z^m= w_{2m}\), \[ \phi(w_{i(1)}\cdots w_{i(n)})= \prod^n_{k< l} \phi(w_{i(k)} w_{i(l)}),\;i(1),\ldots,i(n)\in\mathbb{N},\;1\leq i(1),\ldots,i(n)\leq2m, \] is required. Let \(T^{i(1)\cdots i(t)}\) be the set of monomials formed by the generators \(\mathbf{1}\), \(z^{(i(1)},\overline z^{i(1)}, \ldots, z^{i(t)}\), \(\overline z^{i(t)}\). If there exists a monomial \(Y\in T^{i(1)\cdots i(t)}\) such that \(X=\lambda\overline YY\), then \(X\) is said to be regular (in \(T^{i(1)\cdots i(t)}\)) (Def. 7). Then, the linear functional \(\psi^{i(1)\cdots i(t)}\) defined by \(\psi^{i(1)\cdots i(t)}(X)= \phi(X)\), if \(X\) is regular in \(T^{i(1)\cdots i(t)}\) (\(\psi^{i(1)\cdots i(t)}(X)= 0\) otherwise), is shown to be a tracial state (Th. 21). A generalization \(\psi^{i(1)\cdots i(t)}_{x_1\cdots x_t}\) of \(\psi^{i(1)\cdots i(t)}\) is also introduced (Def. 22), and it is shown that, if \(\theta_{ij}\) are irrational numbers, then this type of tracial states gives the extremal points of the tracial state space (Prop. 31). The paper is concluded giving non-trivial pure states on \(C^{\text{alg}}(\mathbb{R}^{2m}_\theta)\) by the same way (Def. 32 and Th. 33).
extremal points of the tracial state space
scientific article; zbMATH DE number 5734880