Dixmier traces

It is a bit awkward to deal with the index $N$ of $\sg_N(A)$ as a discrete variable, but we can fix this by a simple linear interpolation.

If $N \leq \la \leq N + 1$, so that $\la = N + t$ with $0 \leq t \leq 1$, we put

$$\sg_\la(A) := (1 - t) \sg_N(A) + t \sg_{N+1}(A).$$

Note that $\sg_\la(A + B) \leq \sg_\la(A) + \sg_\la(B)$ now holds for all $\la \geq 0$: every $\sg_\la$ is a norm on $\sK$.

(The $e$ here is by convention: any constant $> 1$ would do. Also, the notation $\sL^{1+}$ is not universally accepted: some authors prefer the clumsier notation $\sL^{1,\infty}$, or even $\sL^{(1,\infty)}$, which comes from the historical origin of these operator ideals in real interpolation theory: see [Con, IV.C] for that.)

Since each $\sg_\la$ is a norm on $\sK$, so also is this supremum whenever it is finite. Thus $\sL^{1+}$ has a natural (¡unitarily invariant!) norm

$$\Vert T\Vert _{1+} := \sup_{\la\geq e} \frac{\sg_\la(T)}{\log\la} \word{for} T \in \sL^{1+}.$$

As stated, the norm depends on the chosen constant $e$, but the ideal $\sL^{1+}(\sH)$ does not.

Note that $T \in \sK$ is traceclass if and only if $\sg_\la(T)$ is bounded (by $\Vert T\Vert _1$, for instance) without need for the factor $(1/\log\la)$. Thus $\sL^{1}(\sH) \subset \sL^{1+}(\sH)$.

We get an additive functional defined on $\sL^{1+}$ in three more steps. First, we dampen the oscillations in $\sg_\la(T)/\log\la$ by taking a Cesàro mean with respect to the logarithmic measure on an interval $[\la_0,\infty)$ for some $\la_0 > e$. For definiteness, we choose $\la_0 = 3$. Our treatment closely follows the appendix of the local-index paper of Connes and Moscovici [CM].

Since the failure of additivity of $\tau_\la$ vanishes as $\la\to\infty$, the second step is to quotient out by functions vanishing at infinity. For that we consider the ``corona'' $C^*$-algebra

$$B_\infty := \frac{C_b([3,\infty))}{C_0([3,\infty)}.$$

The function $\la \mapsto \tau_\la(A)$, for $A \geq 0$ in $\sL^{1+}$, lies in $C_b([3, \infty))$, and its image $\tau(A)$ in $B_\infty$ defines an additive map, that is,

$$\tau(A + B) = \tau(A) + \tau(B) \word{for} A \geq 0, B \geq 0 \word{in} \sL^{1+}.$$

The final step is to compose this map with a state on $B_\infty$.

This definition has a drawback: since $B_\infty$ is a non-separable $C^*$-algebra, there is no way to exhibit even one such state. However, Dixmier traces are still computable in a special case: if $\lim_{\la\to\infty} \tau_\la(T)$ exists, then $\tau(T)$ coincides with the image of a constant function in $B_\infty$, and since the state $\omega$ is normalized, $\omega(1) = 1$, the value $\omega(\tau(T))$ equals this limit:

$$\Trw T = \lim_{\la\to\infty} \tau_\la(T)$$

is independent of $\omega$, provided that the limit exists. Such operators are called measurable. When this happens, we shall suppress the label $\omega$ and write $\Tr^+ T$ for the common value of all Dixmier traces.

The use of the Cesàro mean () simplifies the original definition that Dixmier [Dix1] gave of these traces. A detailed analysis of these (and other related) functionals was made recently by Lord, Sedaev and Sukochev [LSS], who called them ``Connes-Dixmier traces''. As an unexpected consequence of their work, they have shown that a positive operator $A \in \sL^{1+}(\sH)$ is measurable if and only if the original sequence $\set{\sg_N(A)/\log N : N \in \bN}$ is already convergent. Thus it is not necessary to compute $\tau_\la(A)$, since

$$\Tr^+ A = \lim_{N\to\infty} \frac{\sg_N(A)}{\log N} \word{for positive, measurable} A \in \sL^{1+}.$$