Abstract
New properties of the space of integrable (with respect to the faithful normal semifinite trace) operators affiliated with a semifinite von Neumann algebra are found. A trace inequality for a pair of projections in the von Neumann algebra is obtained, which characterizes trace in the class of all positive normal functionals on this algebra. A new property of a measurable idempotent are determined. A useful factorization of such an operator is obtained; it is used to prove the nonnegativity of the trace of an integrable idempotent. It is shown that if the difference of two measurable idempotents is a positive operator, then this difference is a projection. It is proved that a semihyponormal measurable idempotent is a projection. It is also shown that a hyponormal measurable tripotent is the difference of two orthogonal projections.
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Original Russian Text © A.M. Bikchentaev, 2016, published in Doklady Akademii Nauk, 2016, Vol. 466, No. 2, pp. 137–140.
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Bikchentaev, A.M. Trace and integrable operators affiliated with a semifinite von Neumann algebra. Dokl. Math. 93, 16–19 (2016). https://doi.org/10.1134/S1064562416010075
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DOI: https://doi.org/10.1134/S1064562416010075