A fundamental tool in noncommutative geometry is Connes' character formula. This formula is used in an essential way in the applications of noncommutative geometry to index theory and to the spectral characterisation of manifolds.

This formula provides 2 equivalent ways to define Hochschild cocycle called Chern character. One of them is by taking commutators with sign of the Dirac operator with components of a Hochschild cycle and then applying the usual trace. The other is by taking commutators of the Dirac operator itself with components of a Hochschild cycle and then applying a singular trace.

Our aim is to establish the Connes character formula for non-unital spectral triples. Until recently, only partial extensions of Connes' character formula to the non-unital case were known.