Gauss polynomials and the rotation algebra

Abstract

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.)