A factorization theorem for smooth crossed products (Q1325803)

The author generalizes a result of Dixmier and Malliavin about the factorization property of the convolution algebra of compactly supported \(C^ \infty\)-functions on a Lie group \(G\). In this work, the convolution algebra is replaced by the smooth crossed product \(G\times {\mathcal S} (M)\) where \(G\) is a Lie group acting on the locally compact space \(M\) and \({\mathcal S}\) the Schwartz-type functions on \(M\) vanishing rapidly with respect to a continuous proper map \(\sigma: M\to [0, \infty)\). The main result is that any differentiable \(G\times {\mathcal S} (M)\)-module \(E\) is the finite span of elements \(ae\) where \(a\in G\times {\mathcal S} (M)\) and \(e\in E\).