Hokkaido Mathematical Journal

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

Hiroshi ANDO and Yasumichi MATSUZAWA

Full-text: Open access

Abstract

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

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

Dates
First available in Project Euclid: 27 February 2012

Permanent link to this document
https://projecteuclid.org/euclid.hokmj/1330351338

Digital Object Identifier
doi:10.14492/hokmj/1330351338

Mathematical Reviews number (MathSciNet)
MR2920098

Zentralblatt MATH identifier
1246.22024

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

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

Citation

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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