  Let M be a closed manifold. We show that the Kontsevich-Vishik trace, which
is defined on the set of all classical pseudodifferential operators on M, whose
(complex) order is not an integer greater than or equal to -dim M, is the
unique functional which (i) is linear on its domain, (ii) has the trace
property and (iii) coincides with the L^2-operator trace on trace class
operators.
  Also the extension to even-even pseudodifferential operators of arbitrary
integer order on odd-dimensional manifolds and to even-odd pseudodifferential
operators of arbitrary integer order on even-dimensional manifolds is unique.
