A note on traces for the Heisenberg calculus

Abstract

We gave a local formula for the index of Heisenberg elliptic operators on contact manifolds in our earlier work (Gorokhovsky and van Erp, 2022). We constructed a cocycle in periodic cyclic cohomology which, when paired with the Connes–Chern character of the principal Heisenberg symbol, calculates the index. A crucial ingredient of our index formula was a new trace on the algebra of Heisenberg pseudodifferential operators. The construction of this trace was rather involved. In the present paper, we clarify the nature of this trace and thus simplify the index formula from our previous paper (Gorokhovsky and van Erp, 2022).