Dixmier trace

In mathematics, the Dixmier trace, introduced by Jacques Dixmier (1966), is a non-normal trace on a space of linear operators on a Hilbert space larger than the space of trace class operators.

Some of its applications to noncommutative geometry are described in (Connes 1994).

Definition

If H is a Hilbert space, then L^1,∞(H) is the space of compact linear operators T on H such that the norm

$\|T\|_{1,\infty} = \sup_N\frac{\sum_{i=1}^N \mu_i(T)}{\log(N)}$

is finite, where the numbers μ_i(T) are the eigenvalues of |T| arranged in decreasing order. Let

$a_N = \frac{\sum_{i=1}^N \mu_i(T)}{\log(N)}$.

The Dixmier trace Tr_ω(T) of T is defined for positive operators T of L^1,∞(H) to be

$\operatorname{Tr}_\omega(T)= \lim_\omega a_N$

where lim_ω is a scale-invariant positive "extension" of the usual limit, to all bounded sequences. In other words, it has the following properties:

• lim_ω(α_n) ≥ 0 if all α_n ≥ 0 (positivity)
• lim_ω(α_n) = lim(α_n) whenever the ordinary limit exists
• lim_ω(α₁, α₁, α₂, α₂, α₃, ...) = lim_ω(α_n) (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 L^1,∞(H). If the Dixmier trace of an operator is independent of the choice of lim_ω then the operator is called measurable.

Properties

• Tr_ω(T) is linear in T.
• If T ≥ 0 then Tr_ω(T) ≥ 0
• If S is bounded then Tr_ω(ST) = Tr_ω(TS)
• Tr_ω(T) does not depend on the choice of inner product on H.
• Tr_ω(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 x_α) = sup φ( x_α) for every bounded increasing directed family of positive operators. Any normal trace on $L^{1,\infty}(H)$ is equal to the usual trace, so the Dixmier trace is an example of a non-normal trace.

Examples

A compact self-adjoint operator with eigenvalues 1, 1/2, 1/3, ... has Dixmier trace equal to 1.

If the eigenvalues μ_i of the positive operator T have the property that

$\zeta_T(s)= \operatorname{Tr}(T^s)= \sum{\mu_i^s}$

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).

Connes (1988) showed that Wodzicki's noncommutative residue (Wodzicki 1984) of a pseudodifferential operator on a manifold is equal to its Dixmier trace.

References

• 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, doi:10.1007/BF01218391, ISSN 0010-3616, MR953826, http://projecteuclid.org/euclid.cmp/1104161823 
• Connes, Alain (1994), Non-commutative geometry, Boston, MA: Academic Press, ISBN 978-0-12-185860-5, ftp://ftp.alainconnes.org/book94bigpdf.pdf 
• 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, MR0196508 
• Wodzicki, M. (1984), "Local invariants of spectral asymmetry", Inventiones Mathematicae 75 (1): 143–177, doi:10.1007/BF01403095, ISSN 0020-9910, MR728144