Gauss polynomials and the rotation algebra
Newton's binomial theorem is extended to an interesting non-commutative setting as follows: If, in a ring, \(ba=\gamma ab\) with \(\gamma\) commuting with a and b, then the (generalized) binomial coefficient \(\left( \begin{matrix} n\\ k\end{matrix} \right)_{\gamma}\) arising in the expansion \[ (a+b)^ n=\sum^{n}_{k=0}\left( \begin{matrix} n\\ k\end{matrix} \right)_{\gamma}a^{n-k}b^ k \] (resulting from these relations) is equal to the value at \(\gamma\) of the Gaussian polynomial \[ \left[ \begin{matrix} n\\ k\end{matrix} \right]=\frac{[n]}{[k][n-k]} \] where \([m]=(1- x^ m)(1-x^{m-1})...(1-x)\). (This is of course known in the case \(\gamma =1.)\) From this it is deduced that in the (universal) \(C^*\)-algebra \(A_{\theta}\) generated by unitaries u and v such that \(vu=e^{2\pi i\theta}uv\), the spectrum of the self-adjoint element \((u+v)+(u+v)^*\) has all the gaps that have been predicted to exist, - provided that either \(\theta\) is rational, or \(\theta\) is a Liouville number. (In the latter case, the gaps are labelled in the natural way - via K-theory - by the set of all non-zero integers, and the spectrum is a Cantor set.)
scientific article; zbMATH DE number 4088304
scientific article