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Krein's trace formula is probably one of the earliest trace formula which says that if H and H_0 are two self-adjoint operators such that their difference is trace class, then for a large class of functions phi, phi(H) -- phi(H_0) is trace class and there exists a unique L^1(R)- function Xi such that Tr{ phi(H)--phi(H_0) } can be expressed as an integral of phi' and Xi. One can think of the Trace as a kind of generalised integral on Non-commutative "functions" ( Operators ), which is being related to the conventional integral. The next important formula is the so-called Helton-Howe formula for the trace of the commutator of "functions " of non-commutative Hyponormal Operators being expressed as classical integral of the Jacobian of the functions over the complex plane and again a kind of Xi-function ( this time in L^1( C, leb) ). These formulae encourage a picture with a larger canvas: Non-commutative Geometry and Topology . |