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.
References
[1] | J. Dixmier, "![]() |
Comments
Cf. also -algebra; Trace; Quasi-equivalent representations.
References
[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