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arXiv · 1310.5609

Functional calculus in finite type I von Neumann algebras

Piotr Niemiec
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Abstract

A certain class of matrix-valued Borel matrix functions is introduced and it is shown that all functions of that class naturally operate on any operator T in a finite type I von Neumann algebra M in a way such that uniformly bounded sequences f_1,f_2,... of functions that converge pointwise to 0 transform into sequences f_1[T],f_2[T],... of operators in M that converge to 0 in the *-strong operator topology. It is also demonstrated that the double *-commutant of any such operator T which acts on a separable Hilbert space coincides with the set of all operators of the form f[T] where f runs over all function from the aforementioned class. Some conclusions concerning so-called operator-spectra of such operators are drawn and a new variation of the spectral theorem for them is formulated.

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BibTeXRIS

Piotr Niemiec. 2013-10-21. Functional calculus in finite type I von Neumann algebras. https://doi.org/10.3318/pria.2016.116.10

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