We examine under which assumptions on a positive normal functional ϕ on a von Neumann algebra, M and a Borel measurable function f : R + → R with f (0) = 0 the subadditivity inequality ϕ(f (A + B)) ϕ(f (A)) + ϕ(f (B)) holds true for all positive operators A, B in M. A corresponding characterization of tracial functionals among positive normal functionals on a von Neumann algebra is presented. 2000 Mathematics Subject Classification: 46L30, 15A45
We present a non-commutative extension of the classical Yosida-Hewitt decomposition of a finitely additive measure into its σ -additive and singular parts. Several applications are given to the characterisation of bounded convex sets in Banach spaces of measurable operators which are closed locally in measure.
Mathematics Subject Classification (2000): Primary 46L52, Secondary 46E30, 47B55The classical Yosida-Hewitt theorem [26] shows that each continuous linear functional on the space of bounded measurable functions on some measure space admits a unique decomposition into its σ -additive and singular parts. This classical result, which goes back to the Lebesgue decomposition theorem, extends naturally to the very general setting of vector lattices where it emerges as a special case of the well known theorem of F. Riesz that every band (order closed lattice ideal) in an order complete vector lattice is a projection band. In the function space setting, the Yosida-Hewitt decomposition has been exploited by Bukhvalov and Lozanovskiȋ [4,5] as a key tool in their study of convex sets which are closed locally in measure in (commutative) spaces of measurable functions. A special case of this study is that a bounded convex set V in the real Banach space X = L 1 (T , , µ) for some σ -finite ✩ Dedicated to the memory of A. C. Zaanen and Y. Ja. Abramovich.
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