The K-inductive structure of the noncommutative Fourier transform


  • Samuel G. Walters



The noncommutative Fourier transform $\sigma (U)=V^{-1}$, $\sigma (V)=U$ of the irrational rotation C*-algebra $A_\theta $ (generated by canonical unitaries $U$, $V$ satisfying $VU = e^{2\pi i\theta } UV$) is shown to have the following K-inductive structure (for a concrete class of irrational parameters, containing dense $G_\delta $'s). There are approximately central matrix projections $e_1$, $e_2$, $f$ that are σ-invariant and which form a partition of unity in $K_0$ of the fixed-point orbifold $A_\theta ^\sigma $, where $f$ has the form $f = g+\sigma (g) +\sigma ^2(g)+\sigma ^3(g)$, and where $g$ is an approximately central matrix projection as well.


Buck, J. and Walters, S., Connes-Chern characters of hexic and cubic modules, J. Operator Theory 57 (2007), no. 1, 35–65.

Echterhoff, S., Lück, W., Phillips, N. C., and Walters, S., The structure of crossed products of irrational rotation algebras by finite subgroups of $\mathrm SL_2(\mathbb Z)$, J. Reine Angew. Math. 639 (2010), 173–221.

Elliott, G. A. and Evans, D. E., The structure of the irrational rotation $C^*$-algebra, Ann. of Math. (2) 138 (1993), no. 3, 477–501.

Lin, H., Classification of simple tracially AF $C^*$-algebras, Canad. J. Math. 53 (2001), no. 1, 161–194.

Polishchuk, A., Holomorphic bundles on $2$-dimensional noncommutative toric orbifolds, in “Noncommutative geometry and number theory”, Aspects Math., E37, Friedr. Vieweg, Wiesbaden, 2006, pp. 341--359.

Walters, S. G., Chern characters of Fourier modules, Canad. J. Math. 52 (2000), no. 3, 633–672.

Walters, S. G., $K$-theory of non-commutative spheres arising from the Fourier automorphism, Canad. J. Math. 53 (2001), no. 3, 631–672.

Walters, S. G., The AF structure of non commutative toroidal $\mathbb Z/4\mathbb Z$ orbifolds, J. Reine Angew. Math. 568 (2004), 139–196.

Walters, S. G., Decomposable projections related to the Fourier and flip automorphisms, Math. Scand. 107 (2010), no. 2, 174–197.

Walters, S. G., Toroidal orbifolds of $\mathbb Z_3$ and $\mathbb Z_6$ symmetries of noncommutative tori, Nuclear Phys. B 894 (2015), 496–526.

Walters, S. G., Continuous fields of projections and orthogonality relations, J. Operator Theory 77 (2017), no. 1, 191–203.

Walters, S. G., Semiflat orbifold projections, Houston J. Math. 44 (2018), no. 2, 645–663.



How to Cite

Walters, S. G. (2019). The K-inductive structure of the noncommutative Fourier transform. MATHEMATICA SCANDINAVICA, 124(2), 305–319.