We define the central extension \widetilde{Aut}+(R) of the automorphism group Aut+(R) of the extended affine root system. We give the action of \widetilde{Aut}+(R) on the flat theta invariants (theta functions). This describes the modular property for the flat theta invariants.
The types of von Neumann algebras generated by quasifree representations of infinite dimensional Clifford algebras are studied in terms of spectral properties of positive operators parametrizing quasifree states.
Methods from noncommutative harmonic analysis are used to develop an abstract theory of orthonormal wavelets. The relationship between the existence of an orthonormal wavelet and the existence of a multi-resolution is clarified, and four theorems guaranteeing the existence of wavelets are proved. As a special case of the fourth theorem, a generalization of known results on the existence of smooth wavelets having compact support is obtained.
We introduce the Fréchet differential of operator functions on C*-algebras obtained via spectral theory from ordinary differentiable functions. In the finite-dimensional case this differential is expressed in terms of Hadamard products of matrices. A perturbation formula with applications to traces is given.