Gauss polynomials and the rotation algebra

Summary

Newton's binomial theorem is extended to an interesting noncommutative setting as follows: If, in a ring,ba=γab with γ commuting witha andb, then the (generalized) binomial coefficient$\left( {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right)_r$ arising in the expansion

$$\left( {a + b} \right)^n = \sum\limits_{k = 0}^n {\left( {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right)} _\gamma a^{n - k} b^k $$

(resulting from these relations) is equal to the value at γ of the Gaussian polynomial

$$\left[ {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right] = \frac{{\left[ n \right]}}{{\left[ k \right]\left[ {n - k} \right]}}$$

where [m]=(1-x ^m)(1-x ^m−1)...(1-x). (This is of course known in the case γ=1.)

From this it is deduced that in the (universal)C ^*-algebraA _gq generated by unitariesu andv such thatvu=e ^2πiθ uv, the spectrum of the self-adjoint element (u+v)+(u+v)^* has all the gaps that have been predicted to exist-provided that either θ is rational, or θ is a Liouville number. (In the latter case, the gaps are labelled in the natural way-viaK-theory-by the set of all non-zero integers, and the spectrum is a Cantor set.)