Dixmier Trace

In mathematics, the Dixmier trace, introduced by , 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 .

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

:$\, T\, _ = \sup_N\frac$

is finite, where the numbers ''μ''_''i''(''T'') are the eigenvalues of , ''T'', arranged in decreasing order. Let
:$a_N = \frac$.
The Dixmier trace Tr_''ω''(''T'') of ''T'' is defined for positive operators ''T'' of ''L''^1,∞(''H'') to be

:$\operatorname_\omega(T)= \lim_\omega a_N$

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''(''α''₁, ''α''₁, ''α''₂, ''α''₂, ''α''₃, ...) = lim_ω(''α''_''n'') (scale invariance)

There are many such extensions (such as a Banach limit of ''α''₁, ''α''₂, ''α''₄, ''α''₈,...) 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 $L^(H)$ 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
:$\zeta_T(s)= \operatorname(T^s)= \sum$
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).

showed that Wodzicki's noncommutative residue of a pseudodifferential operator on a manifold ''M'' of order ''-dim(M)'' 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.
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*{{Citation , last1=Wodzicki , first1=M. , title=Local invariants of spectral asymmetry , doi=10.1007/BF01403095 , mr=728144 , year=1984 , journal=Inventiones Mathematicae , issn=0020-9910 , volume=75 , issue=1 , pages=143–177

See also

* Singular trace

Von Neumann algebras
Hilbert space
Operator theory
Trace theory