Two points, one limit: homogenization techniques for two-point local algebras (Q412465)

This paper studies several algebras generated by convolution, multiplication and flip operators on \(L^p(\mathbb R)\). Let \(\mathcal{A}\) be the smallest closed subalgebra of \(\mathcal{L}(L^p(\mathbb R))\) which contains multiplication operators \(aI\), Fourier convolution operators \(W^0(b)\) with \(a\) and \(b\) piecewise continuous, and the flip operator \(J\). The main result can be stated as follows.NEWLINENEWLINETheorem. There is a family of algebra homomorphisms \(Y_{s,t}\) labeled by the points in \(([0,\infty]X\{\infty\})\times (\{\infty\}X[0,\infty])\) such that an operator \(a\in \mathcal{A}\) is Fredholm on \(L^p(\mathbb R)\) if and only if all operators \(Y_{s,t}(A)\) are invertible.NEWLINENEWLINEThe result explores the properties of the Fourier transform in \(L^p(\mathbb R)\), when \(p\neq 2\).