Concerning the theory of τ-measurable operators affiliated to a semifinite von Neumann algebra. II
Abstract
Let a von Neumann algebra M of operators act on a Hilbert space H, let τ be a faithful normal semifinite trace on M. Let S(M, τ ) be the ∗-algebra of all τ-measurable operators. Assume that X, Y ∈ S(M, τ ). We have (i) if |Y | ≤ |X| then ker(X) ⊂ ker(Y ); (ii) if X is left invertible with X−1 l ∈Mthen ran(X∗) = H. The following generalizes of the Putnam theorem (1951), see also Problem 188 in the book (Halmos P. R. A Hilbert space problem book. D. van Nostrand company, inc., London, 1967): A positive selfcommutator A∗A−AA∗ (A ∈ S(M, τ )) cannot have the inverse in M. Let I be the unit of the algebra M and τ (I) = +∞, let A,B ∈ S(M, τ ) and A = A3. Then the commutator [A,B] cannot have a form λI + K, where λ ∈ C \ {0} and an operator K ∈ S(M, τ ) is τ-compact
About the Author
A. M. Bikchentaev
Kazan Federal University, Volga Region Mathematical Center
Russian Federation
Russian Federation
18 Kremlyovskaya str., Kazan 420008
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Bikchentaev A.M. Concerning the theory of τ-measurable operators affiliated to a semifinite von Neumann algebra. II. Mathematics and Theoretical Computer Science. 2023;1(2):3-11. (In Russ.)
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