In mathematics, the Dixmier trace, introduced by Jacques Dixmier (1966), is a non-normal[clarification needed] trace on a space of linear operators on a Hilbert space larger than the space of trace class operators. Dixmier traces are examples of singular traces.
If H is a Hilbert space, then L1,∞(H) is the space of compact linear operators T on H such that the norm
is finite, where the numbers μi(T) are the eigenvalues of |T| arranged in decreasing order. Let
The Dixmier trace Trω(T) of T is defined for positive operators T of L1,∞(H) to be
where limω is a scale-invariant positive "extension" of the usual limit, to all bounded sequences. In other words, it has the following properties:
- limω(αn) ≥ 0 if all αn ≥ 0 (positivity)
- limω(αn) = lim(αn) whenever the ordinary limit exists
- limω(α1, α1, α2, α2, α3, ...) = limω(αn) (scale invariance)
There are many such extensions (such as a Banach limit of α1, α2, α4, α8,...) so there are many different Dixmier traces. As the Dixmier trace is linear, it extends by linearity to all operators of L1,∞(H). If the Dixmier trace of an operator is independent of the choice of limω then the operator is called measurable.
- Trω(T) is linear in T.
- If T ≥ 0 then Trω(T) ≥ 0
- If S is bounded then Trω(ST) = Trω(TS)
- Trω(T) does not depend on the choice of inner product on H.
- Trω(T) = 0 for all trace class operators T, but there are compact operators for which it is equal to 1.
A trace φ is called normal if φ(sup xα) = sup φ( xα) for every bounded increasing directed family of positive operators. Any normal trace on is equal to the usual trace, so the Dixmier trace is an example of a non-normal trace.
A compact self-adjoint operator with eigenvalues 1, 1/2, 1/3, ... has Dixmier trace equal to 1.
If the eigenvalues μi of the positive operator T have the property that
converges for Re(s)>1 and extends to a meromorphic function near s=1 with at most a simple pole at s=1, then the Dixmier trace of T is the residue at s=1 (and in particular is independent of the choice of ω).
- Albeverio, S.; Guido, D.; Ponosov, A.; Scarlatti, S.: Singular traces and compact operators. J. Funct. Anal. 137 (1996), no. 2, 281—302.
- Connes, Alain (1988), "The action functional in noncommutative geometry", Communications in Mathematical Physics, 117 (4): 673–683, ISSN 0010-3616, MR 953826, doi:10.1007/BF01218391
- Connes, Alain (1994), Non-commutative geometry (PDF), Boston, MA: Academic Press, ISBN 978-0-12-185860-5[permanent dead link]
- Dixmier, Jacques (1966), "Existence de traces non normales", Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences. Séries a et B, 262: A1107–A1108, ISSN 0151-0509, MR 0196508
- Wodzicki, M. (1984), "Local invariants of spectral asymmetry", Inventiones Mathematicae, 75 (1): 143–177, ISSN 0020-9910, MR 728144, doi:10.1007/BF01403095