# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] Carey, A.; Phillips, J.; Sukochev, F., Spectral flow and Dixmier traces, Adv. math., 173, 68-113, (2003) · Zbl 1015.19003 [2] Connes, A., Noncommutative geometry, (1994), Academic Press New York · Zbl 1106.58004 [3] Dixmier, J., Existence de traces non normales, C. R. acad. sci. Paris, 262, A1107-A1108, (1966) [4] Dodds, P.; de Pagter, B.; Semenov, E.; Sukochev, F., Symmetric functionals and singular traces, Positivity, 2, 47-75, (1988) · Zbl 0915.46021 [5] P. Dodds, B. de Pagter, A. Sedaev, E. Semenov, F. Sukochev, Singular Symmetric Functionals, (Russian) Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 290 (2002) Issled. po Linein. Oper. i Teor. Funkts. 30, 42-71. · Zbl 1090.46020 [6] Dodds, P.; de Pagter, B.; Sedaev, A.; Semenov, E.; Sukochev, F., Singular symmetric functionals with additional invariance properties, Izv. math., 67, 1187-1213, (2003) · Zbl 1075.46028 [7] Fack, T.; Kosaki, H., Generalised s-numbers of τ-measurable operators, Pacific J. math., 123, 269-300, (1986) · Zbl 0617.46063 [8] G. Hardy, Divergent Series, Oxford University Press, Oxford, 1949, pp. 121-135. · Zbl 0032.05801 [9] Krein, S.; Petunin, Y.; Semenov, E., Interpolation of linear operators, () [10] Lorentz, G., A contribution to the theory of divergent sequences, Acta. math., 80, 167-190, (1948) · Zbl 0031.29501 [11] Lindenstrauss, J.; Tzafriri, L., Classical Banach space II, function spaces, (1979), Springer Berlin-Heidelberg-New York · Zbl 0403.46022 [12] Sucheston, L., Banach limits, Amer. math. monthly, 74, 285-293, (1967) · Zbl 0148.12202
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.