$C^{\ast }$-algebras of $b$-pseudodifferential operators and an ${\mathbb{R}}^k$-equivariant index theorem

We compute $K$-theory invariants of algebras of pseudodifferential operators on manifolds with corners and prove an equivariant index theorem for operators invariant with respect to an action of ${\mathbb{R}}^k\mathrm{.}$ We discuss the relation between our results and the $\eta$-invariant.