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Dixmier traces as singular symmetric functionals and applications to measurable operators. (English) Zbl 1081.46042
Suppose that \(\psi : [0,\infty) \to [0,\infty)\) is a concave function such that \(\psi(0) = 0\), and \(\lim_{t \to \infty} \psi(t) = \infty\). If \(x\) is a measurable function on \([0,\infty)\), let \(\phi(x)(t) = \int_0^t x^*(s) \, ds/\psi(t)\), where \(x^*\) is the non-increasing rearrangement of \(| x| \). The Marcinkiewicz space \(M(\psi)\) consists of all functions \(x\) on \([0,\infty)\) for which \(\| x\| _{M(\psi)} = \| \phi(x)\| _\infty < \infty\). A positive homogeneous functional \(f : M_+(\psi) \to [0,\infty)\) is symmetric if \(f(x) \leq f(y)\) if \(\int_0^t x^*(s) \, ds \leq \int_0^t y^*(s) \, ds\) for every \(t\), and singular if it vanishes on \(M(\psi) \cap L^1(0,\infty)\).
The authors investigate symmetric linear functionals on the space \(M(\psi)\), and their connections to Banach and Cesàro limits of functions on \([0,\infty)\), and/or sequences. The results obtained are then applied to symmetric spaces of operators. More precisely, suppose that a von Neumann algebra \(N\) is equipped with a semifinite normal faithful trace \(\tau\). For \(a \in N\) and \(t \in [0,\infty)\), define the generalized singular values function by setting \(\mu_t(a) = \inf\{s \geq 0 : \tau(\chi_{(s,\infty)}(a)) \leq t\}\). The Marcinkiewicz ideal \(E\) consists of all \(a \in N\) for which the function \(t \mapsto \mu_t(a)\) belongs to \(M(\psi)\), with \(\psi(t) = \log(1+t)\). Certain symmetric linear functionals on \(M(\psi)\) induce singular traces on \(E\) (that is, traces vanishing on \(E \cap L^1(N,\tau)\)). Two different ways of obtaining such traces are presented. One way leads to Connes–Dixmier traces, while the second one yields a larger class of Dixmier traces. The authors characterize elements of \(E\) for which all Connes–Dixmier traces (or Dixmier traces) coincide.

MSC:
46L52 Noncommutative function spaces
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46L10 General theory of von Neumann algebras
46L07 Operator spaces and completely bounded maps
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