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. Dixmier traces are examples of singular traces.

Some applications of Dixmier traces 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, MR 953826
• Connes, Alain (1994), Non-commutative geometry (PDF), Boston, MA: Academic Press, ISBN 978-0-12-185860-5
• 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, doi:10.1007/BF01403095, ISSN 0020-9910, MR 728144

See also

• Singular trace