Trace on a C*-algebra
An additive functional on the set of positive elements of that takes values in , is homogeneous with respect to multiplication by positive numbers and satisfies the condition for all . A trace is said to be finite if for all , and semi-finite if for all . The finite traces on are the restrictions to of those positive linear functionals on such that for all . Let be a trace on , let be the set of elements such that , and let be the set of linear combinations of products of pairs of elements of . Then and are self-adjoint two-sided ideals of , and there is a unique linear functional on that coincides with on . Let be a lower semi-continuous semi-finite trace on a -algebra . Then the formula defines a Hermitian form on , with respect to which the mapping of into itself is continuous for any . Put , and let be the completion of the quotient space with respect to the scalar product defined by the form . By passing to the quotient space and subsequent completion, the operators determine certain operators on the Hilbert space , and the mapping is a representation of the -algebra in . The mapping establishes a one-to-one correspondence between the set of lower semi-continuous semi-finite traces on and the set of representations of with a trace, defined up to quasi-equivalence.
|||J. Dixmier, " algebras" , North-Holland (1977) (Translated from French)|
Cf. also -algebra; Trace; Quasi-equivalent representations.
|[a1]||O. Bratteli, D.W. Robinson, "Operator algebras and quantum statistical mechanics" , 1 , Springer (1979)|
Trace on a C*-algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Trace_on_a_C*-algebra&oldid=16757