## Gauss polynomials and the rotation algebra.(English)Zbl 0665.46051

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.)
Reviewer: G.A.Elliott

### MSC:

 46L55 Noncommutative dynamical systems 46L80 $$K$$-theory and operator algebras (including cyclic theory)
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### References:

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