GAUSS POLYNOMIALS AND THE ROTATION ALGEBRA

Newton's binomial theorem is extended to an interesting noncommutative setting as follows: If, in a ring, ba=γab with γ commuting with a and b, then the (generalized) binomial coefficient {Mathematical expression} arising in the expansion {Mathematical expression} (resulting from these relations) is equal to the value at γ of the Gaussian polynomial {Mathematical expression} where [m]=(1-xm)(1-xm-1)...(1-x). (This is of course known in the case γ=1.) From this it is deduced that in the (universal)C*-algebra Agq generated by unitaries u and v such that vu=e2πiθuv, 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-via K-theory-by the set of all non-zero integers, and the spectrum is a Cantor set.) © 1996 Springer-Verlag.