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The Radon-Nikodym theorem for weights on semifinite JBW-algebras. (English) Zbl 0567.46036

The purpose of the paper is to obtain a version of the Radon-Nikodym theorem in JBW-algebras - Jordan analogues of von Neumann algebras. The main result can be formulated as follows.
3.6. Corollary. Let A be a JBW-algebra with a faithful normal semifinite trace \(\tau\). if \(\phi\) is a normal semifinite weight on A then there is a unique positive spectral family \((e_{\lambda})\) in A such that \[ \phi (a)=\lim_{\epsilon \to 0}\tau (a\int^{+\infty}_{0}\lambda (1+\epsilon \lambda)^{-1}de_{\lambda}) \] for all \(a\in A^+\). Conversely, every positive spectral family \((e_{\lambda})\) in A defines a normal semifinite weight \(\phi\) by the above formula. Furthermore, the weight \(\phi\) is bounded if and only if \(\int^{+\infty}_{0}\lambda d\tau (e_{\lambda})<+\infty;\) \(\phi\) is a trace if and only if all idempotents \(e_{\lambda}\) are central; \(\phi\) is dominated by \(\tau\) if and only if \(e_{\mu}=1\) for some \(\mu >0\).

MSC:

46L99 Selfadjoint operator algebras (\(C^*\)-algebras, von Neumann (\(W^*\)-) algebras, etc.)
17C65 Jordan structures on Banach spaces and algebras
46L51 Noncommutative measure and integration
46L53 Noncommutative probability and statistics
46L54 Free probability and free operator algebras
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References:

[1] Ajupov, Sh.A.: Extension of traces and type criterions for Jordan algebras of self-adjoint operators. Math. Z.181, 253-268 (1982) · Zbl 0487.46045 · doi:10.1007/BF01215023
[2] Ajupov, Sh.A.: On the construction of Jordan algebras of self-adjoint operators. Soviet Math. Dokl.26, 623-626 (1982) · Zbl 0528.46046
[3] Ajupov, Sh.A.: Integration on Jordan algebras. (Russian) Izv. Akad. Nauk SSSR Ser. Mat.47, 3-25 (1983)
[4] Ajupov, Sh.A.: Locally measurable operators for JW-algebras and representation of ordered Jordan algebras. (Russian) Izv. Akad. Nauk SSSR Ser. Mat.48, 211-236 (1984) · Zbl 0547.46042
[5] Alfsen, E.M., Shultz, F.W., Stormer, E.: A Gelfand-Neumark theorem for Jordan algebras. Advances in Math.28 11-56 (1978) · Zbl 0397.46065 · doi:10.1016/0001-8708(78)90044-0
[6] Berdikulov, M.A.:L 1- andL 2-spaces for semi-definite JBW-algebras. (Russian) Dokl. Akad. Nauk UzSSR6, 3-4 (1982) · Zbl 0631.46063
[7] Haagerup, U.: Normal weights onW *-algebras. J. Functional Analysis19, 302-317 (1975) · Zbl 0304.46043 · doi:10.1016/0022-1236(75)90060-9
[8] Iochum, B.: Cones autopolaires et Algebres de Jordan. These, Provence 1982
[9] King, W.P.C.: Semi-finite traces on JBW-algebras. Math. Proc. Cambridge Philos. Soc.93, 503-509 (1983) · Zbl 0543.46044 · doi:10.1017/S0305004100060813
[10] Pedersen, G.K., Takesaki, M.: The Radon-Nikodym theorem for von Neumann algebras. Acta Math.130, 53-87 (1973) · Zbl 0262.46063 · doi:10.1007/BF02392262
[11] Sarymsakov, T.A., Ajupov, Sh.A.: Partially ordered Jordan algebras. Soviet Math. Dokl.20, 1352-1355 (1979) · Zbl 0442.46037
[12] Sarymsakov, T.A., Ajupov, Sh.A., Hadjiev, D?., Chilin, V.I.: Ordered Algebras. (Russian) Tashkent: Fan 1983
[13] Shultz, F.W.: On normed Jordan algebras which are Banach dual spaces. J. Functional Analysis31, 360-376 (1979) · Zbl 0421.46043 · doi:10.1016/0022-1236(79)90010-7
[14] Stacey, P.J.: TypeI 2 JBW-algebras. Quart. J. Math.33, 115-127 (1982) · Zbl 0449.46058 · doi:10.1093/qmath/33.1.115
[15] Stormer, E.: Jordan algebras of typeI. Acta Math.115, 165-184 (1966) · Zbl 0139.30502 · doi:10.1007/BF02392206
[16] Stormer, E.: Irreducible Jordan algebras of self-adjoint operators. Trans. Amer. Math. Soc.130, 153-166 (1968) · Zbl 0164.44602
[17] Stormer, E.: Real structure in the hyperfinite factor. Duke Math. J.47, 145-153 (1980) · Zbl 0462.46044 · doi:10.1215/S0012-7094-80-04711-0
[18] Topping, D.: Jordan algebras of self-adjoint operators. Mem. Amer. Math. Soc.53 (1965) · Zbl 0149.09801
[19] Topping, D.: An isomorphism invariant for spin factors. J. Math. Mech.15, 1055-1064 (1966) · Zbl 0154.38802
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