Characterization of singular traces on the weak trace class ideal generated by exponentiation-invariant extended limits (Q2822636)

On the quasi-Banach operator space \(\mathcal{L}_{1,\infty}\) of all compact operators \(A\in B\left( H\right) \) such that \(\sup\nolimits_{n\geq 0}\left( n+1\right) \mu\left( n,A\right) <\infty\) for the step function \(\mu\left( A\right) \) with \(\mu\left( n,A\right) \) the \(n\)-th singular value of \(A\), this paper studies the space \(\mathcal{D}_{P}\) of non-normal traces \(\mathrm{Tr}_{\omega}\) with \(\mathrm{Tr}_{\omega}\left( A\right) \) for \(0\leq A\in\mathcal{L}_{1,\infty}\) defined as the value of \(\omega\) at the element \(t\mapsto\frac{1}{\log\left( t+1\right) }\int_{0}^{t}\mu\left( s,A\right) ds\) for exponentiation-invariant extended limits \(\omega\) on \(L_{\infty}\equiv L_{\infty}\left( 0,\infty\right) \), or more precisely, positive linear functionals \(\omega\) vanishing at \(\chi_{\left( 0,1\right) }\) and equal to \(1\) at \(\chi_{\left( 0,\infty\right) }\) such that \(\omega\left( P_{a}x\right) =\omega\left( x\right) \) for all \(a>0\) and \(x\in L_{\infty}\), where \(\left( P_{a}x\right) \left( t\right) :=x\left( t^{a}\right) \). We note that \(\omega\) is called a dilation-invariant extended limit if \(P_{a}\) is replaced by \(\sigma_{a}\) in the above condition on \(\omega\), where \(\left( \sigma_{a}x\right) \left( t\right) :=x\left( a^{-1}t\right) \).NEWLINENEWLINEFor operators \(A\in\mathcal{L}_{1,\infty}\), a necessary and sufficient condition for \(A\) to be \(\mathcal{D}_{P}\)-measurable, in the sense that \(\mathrm{Tr}_{\omega}\left( A\right) \) is the same for all \(\mathrm{Tr} _{\omega}\in\mathcal{D}_{P}\), is found as the existence of a uniform limit of some explicit integrals involving \(\mu\left( A\right) \), which if exists coincides with the common value \(\mathrm{Tr}_{\omega}\left( A\right) \).NEWLINENEWLINEA positive unital linear functional \(B\) on \(\ell_{\infty}\) is called a Banach limit if it is invariant under the translation operator \(T\) on \(\ell_{\infty}\) defined by \(T\left( x_{0},x_{1},...\right) =\left( x_{1},x_{2},...\right) \). Via a bijection between positive normalized traces \(\tau\) on \(\mathcal{L}_{1,\infty}\) and Banach limits \(B\) on \(\ell_{\infty}\), it is established that \(\tau\in\mathcal{D}_{P}\) if and only if the corresponding \(B\) equals \(\gamma\circ\pi\circ C\) for some dilation-invariant extended limit \(\gamma\) on \(L_{\infty}\), where \(\pi\) sends \(\left( x_{0},x_{1},...\right) \in\ell_{\infty}\) isometrically to \(\sum_{n=0}^{\infty}x_{n}\chi_{\left[ n,n+1\right) }\in L_{\infty}\) and the Cesàro operator \(C\) on \(\ell_{\infty}\) is defined as \(\left( Cx\right) _{n}:=\frac{1}{n+1} \sum_{k=0}^{n}x_{k}\).NEWLINENEWLINEThe existence of a positive operator in \(\mathcal{L}_{1,\infty}\) that is \(\mathcal{D}_{P}\)-measurable but not Dixmier measurable (and hence non-Tauberian) is proved, answering an open question. It is also shown that \(\mathcal{D}_{P}\subset\mathcal{D}_{M}\) for the class \(\mathcal{D}_{M}\) of traces arising from \(M\)-invariant extended limits where the operator \(M\) is defined by \(\left( Mx\right) \left( t\right) \) equal to \(\frac{1} {\log\left( t\right) }\int_{1}^{t}x\left( s\right) \frac{ds}{s}\) for \(t>1\).