Hokkaido Mathematical Journal

Lie group-Lie algebra correspondences of unitary groups in finite von Neumann algebras

Hiroshi ANDO and Yasumichi MATSUZAWA

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We give an affirmative answer to the question whether there exist Lie algebras for suitable closed subgroups of the unitary group U($¥mathcal{H}$) in a Hilbert space $¥mathcal{H}$ with U($¥mathcal{H}$) equipped with the strong operator topology. More precisely, for any strongly closed subgroup G of the unitary group U($¥mathfrak{M}$) in a finite von Neumann algebra $¥mathfrak{M}$, we show that the set of all generators of strongly continuous one-parameter subgroups of G forms a complete topological Lie algebra with respect to the strong resolvent topology. We also characterize the algebra $¥overline{¥mathfrak{M}}$ of all densely defined closed operators affiliated with $¥mathfrak{M}$ from the viewpoint of a tensor category.

Article information

Hokkaido Math. J., Volume 41, Number 1 (2012), 31-99.

First available in Project Euclid: 27 February 2012

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 22E65: Infinite-dimensional Lie groups and their Lie algebras: general properties [See also 17B65, 58B25, 58H05]
Secondary: 46L51: Noncommutative measure and integration

finite von Neumann algebra unitary group affiliated operator measurable operator strong resolvent topology tensor category infinite dimensional Lie group infinite dimensional Lie algebra


ANDO, Hiroshi; MATSUZAWA, Yasumichi. Lie group-Lie algebra correspondences of unitary groups in finite von Neumann algebras. Hokkaido Math. J. 41 (2012), no. 1, 31--99. doi:10.14492/hokmj/1330351338. https://projecteuclid.org/euclid.hokmj/1330351338

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