Let $t_1$ and $t_2$ be lower semicontinuous semifinite densely defined traces on a $C^*$-algebra $A$. Let us denote by $\mathcal{R}_1$ and $\mathcal{R}_2$ their ideals of definition, i.e. $\mathcal{R}_i=\left\{x\in A\ \middle|\ t_i(x^*x)<+\infty \right\}$. Next, let $B$ be an involutive dense subalgebra of $A$ such that $B\subset \mathcal{R}_1,\mathcal{R}_2$ and $t_1(b^*b)=t_2(b^*b)$ for any $b\in B$. Is it true that then $t_1=t_2$ on the whole algebra $A$?
I have found a similar statement in “C*-algebras” by J.Dixmier 1977: Lemma 6.5.3 on page 139. But he formulates this claim in terms of bitraces, which look artificial to me. Can anyone provide a straightforward and concise argument without bitraces? Is there a better reference for this?
Best Answer
Yes, this is true.
Let $a\in A_+$. By lower semicontinuity it suffices to show that $t_1((a-\delta)_+) = t_2((a-\delta)_+)$ for all $\delta>0$ (where $(a-\delta)_+$ is the positive part of $a-\delta 1$ in the unitisation of $A$ (which is an element of $A$ and not actually in the unitisation)). Fix $\delta>0$.
The key trick for getting the approximations to work, is to use that $t_i$ extends canonically to a positive linear functional on $\mathrm{span} R_i^\ast R_i$. Hence if $x\in R_i$ and $b,c\in A_+$ then \begin{equation} - \| b-c\| t_i(x^\ast x) \leq t_i(x^\ast(b-c)x) \leq \| b-c\| t_i(x^\ast x) \end{equation} and thus \begin{equation} |t_i(x^\ast b x) - t_i(x^\ast c x) | \leq \| b-c\| t_i(x^\ast x). \end{equation}
This will be used a couple of times (in particular, using that $(a-\delta)_+^{1/2}\in R_i$, since $(a-\delta)_+$ is in the Pedersen ideal; the smallest dense two-sided ideal of $A$).
Let $\epsilon>0$. Pick $e\in A_+$ be such that $(a-\delta)_+ e = (a-\delta)_+$, and let $x\in B$ be close enough to $e^{1/2}$ that $\| e - x^\ast x\| \max\{ t_1((a-\delta)_+) , t_2((a-\delta)_+)\} < \epsilon$. By the trick mentioned above we have \begin{equation} t_i((a-\delta)_+) = t_i((a-\delta)_+^{1/2} e (a-\delta)_+^{1/2}) \approx_\epsilon t_i((a-\delta)_+^{1/2} x^\ast x (a-\delta)_+^{1/2}). \end{equation} Now pick $y\in B$ close enough to $(a-\delta)_+^{1/2}$ so that \begin{equation} \| y y^\ast - (a-\delta)_+ \| \max\{ t_1(x^\ast x) , t_2(x^\ast x)\} < \epsilon. \end{equation} Then by the above trick \begin{equation} t_i((a-\delta)_+^{1/2} x^\ast x (a-\delta)_+^{1/2}) = t_i(x(a-\delta)_+ x^\ast) \approx_\epsilon t_i (xyy^\ast x^\ast). \end{equation} As $xy \in B$ we get $t_1(xyy^\ast x^\ast) = t_2(xyy^\ast x^\ast)$ by assumption, and therefore \begin{equation} | t_1((a-\delta)_+) - t_2((a-\delta)_+) | < 4 \epsilon. \end{equation} As $\epsilon>0$ was arbitrary we get $t_1((a-\delta)_+) = t_2((a-\delta)_+)$, and thus $t_1(a) = t_2(a)$ by lower semicontinuity.