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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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References:
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