On the “quantum disk” and a “non-commutative circle”

Abstract

Let T _q be the universal C*-aIgebra generated by an element zsatisfying the equation:

$$z{{z}^{*}} = q{{z}^{*}} + \left( {1 - q} \right)I$$

where q ∈ [-1, 1] is a parameter. We show that, in contrast to T _q for -1 < q < 1, which is known to be the Toeplitz algebra, T _-1 is a (stably) finite C*-algebra, embeddedable into ${{M}_{2}} \otimes C\left( {\bar{D}} \right)$. We prove the continuity of the field (T _q )_-1≤q≤1, by using a remarkable continuous field of states, $\left( {{{\phi }_{q}}} \right) - 1{{ \leqslant }_{q}} \leqslant 1$. $\left( {{{\phi }_{q}}} \right) - 1{{ \leqslant }_{q}} \leqslant 1$. is a faithful trace-state of T _-1. For q > 0, $\phi - 1$ is the restriction to T _q of the Haar measure on the quantum group S _√q U(2). The distribution (with respect to ${{\phi }_{q}}$ of the real part of the generator z is deformed from the semicircle law $\frac{2}{\pi }\sqrt {{1 - {{t}^{2}}dt}}$, at q = 0, to |t|dt, at q = -1.