The canonical trace of nuclear operators, which is the extension of the matrix trace in finite linear algebra to the infinite dimensional linear algebra of Hilbert space, is often considered the only trace on linear operators. There are, however, a multitude of ideals between the nuclear operators and the compact operators not present in the finite case. In the finite case the matrices are the nuclear, compact (and bounded) linear operators. Are there
any new traces on these ideals? What are their properties? Are they extensions of the matrix trace or arise uniquely in the infinite case? Is there any use to these other traces ?
We describe some general results on traces from the work of Nigel Kalton. We also consider the interesting case of the ideal of compact operators generated by the harmonic sequence. This ideal is, of course, strictly larger than the nuclear operators and it is has some interesting uniquely infinite properties. Finally we display our generalisation of Connes Trace Theorem, showing the link between traces on the ideal generated by the harmonic sequence and the Wodzicki residue of Riemannian differential geometry.