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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:
[1] Andrews, G.E.: The Theory of Partitions. In: Rota, G.-C. (ed.) Encyclopedia of Mathematics and Its Applications, Vol. 2, Addison-Wesley, Reading Massachusetts, 1976 · Zbl 0371.10001
[2] Andrews, G.E., Baxter, R.J., Forrester, P.J.: Eight-vertex SoS model and generalized Rogers-Ramanujan type identities. J. Statist. Phys.35, 193–266 (1984) · Zbl 0589.60093
[3] Avron, J.E., Simon, B.: Almost periodic Schrödinger operators, II. The density of states. Duke Math. J.50, 369–391 (1983) · Zbl 0544.35030
[4] Bellissard, J.: Almost periodicity in solid state physics andC *-algebras. In: Berg, C., Fuglede, B. (eds.) The Harold Bohr Centenary pp. 35–75, Mat.-Fys. Medd. Danske Vid. Selsk.42:3 (1989)
[5] Bellissard, J., Lima, R., Testard, D.: Almost periodic Schrödinger operators. In: Streit, L. (ed.) Mathematics+Physics, Lectures on Recent Results. Vol. 1 pp. 1–64, World Scientific Publishers, Singapore, 1985 · Zbl 0675.34022
[6] Bellissard, J., Simon, B.: Cantor spectrum for the almost. Mathieu equation. J. Funct. Anal.48, 408–419 (1982) · Zbl 0516.47018
[7] Bhatia, R.: Perturbation Bounds for Matrix Eigenvalues. Pitman Research Notes in Mathematics Series 162, Longman, London, 1987 · Zbl 0696.15013
[8] Chambers, W.G.: Linear-network model for magnetic breakdown in two dimensions. Phys. Rev. A140, 135–143 (1965)
[9] Connes, A.:C * algèbres et géométrie différentielle. C.R. Acad. Sci. Paris290, 559–604 (1980) · Zbl 0433.46057
[10] Elliott, G.A.: Gaps in the spectrum of an almost periodic Schrödinger operator. C.R. Math. Rep. Acad. Sci. Canada4, 255–259 (1982) · Zbl 0516.46048
[11] Elliott, G.A.: Gaps in the spectrum of an almost periodic Schrödinger operator. II. In: Araki, H., Effros, E.G. (eds.) Geometric Methods in Operator Algebras. Pitman Research Notes in Mathematics Series 123, pp. 181–191, Longman, London, 1986
[12] Helffer, B., Sjöstrand, J.: Analyse semi-classique pour l’équation de Harper (avec application à l’équation de Schrödinger avec champ magnétique) (Preprint) · Zbl 0714.34130
[13] Herman, M.: Une méthode pour minorer les exposants de Lyapunov et quelques exemples montrant le caractère local d’un théorème d’Arnol’d et de Moser sur le tore de dimension 2. Comment. Math. Helv.58, 453–502 (1983) · Zbl 0554.58034
[14] Hofstadter, D.R.: Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields. Phys. Rev. B14, 2239–2249 (1976)
[15] Jimbo, M.: Aq-difference analogue ofU (g) and the Yang-Baxter equation. Lett. Math. Phys.10, 63–69 (1986) · Zbl 0587.17004
[16] Peierls, R.: Zur Theorie des Diamagnetismus von Leitungselektronen. Z. Phys.80, 763–791 (1933) · Zbl 0006.19204
[17] Pimsner, M., Voiculescu, D.: Exact sequences forK-groups and Ext-groups of certain crossproductC *-algebras. J. Oper. Theory4, 93–118 (1980) · Zbl 0474.46059
[18] Riedel, N.: On the topological stable rank of irrational rotation algebras. J. Oper. Theory13, 143–150 (1985) · Zbl 0607.46038
[19] Sinai, Ya.G.: Anderson localization for one-dimensional difference Schrödinger operator with quasiperiodic potential. J. Stat. Phys.46, 861–909 (1987) · Zbl 0682.34023
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