Index of families of convolution operators on Abelian groups (Q803913)

Let H be an arbitrary locally compact (but not compact) Abelian group with Haar measure and let \(L^ n_ p(H)\) \((1<p<\infty)\) be the space of n-dimensional vector functions with elements from \(L_ p(H)\). Further, denote by End \(L^ n_ p(H)\) the Banach algebra of all linear bounded operators in \(L^ n_ p(H)\). The symbol \(W^ n_ p(H)\) stands for the Banach algebra of \(n\times n\) quadratic matrices over \(W_ p(H)\), where \(W_ p(H)\) is a closed subalgebra of End \(L^ n_ p(H)\) generated by the operators of the form \[ (Bf)(h)=\lambda f(h)+\sum^{s}_{j=1}\phi_ j(h)\int_{H}a_ j(h-g)f(g)dg+(Tf)(h). \] In the paper under review a topological formula for the index of families of Noether convolution operators acting in \(W^ n_ p(H)\) is obtained.