## Hokkaido Mathematical Journal

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

#### 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

https://projecteuclid.org/euclid.hokmj/1330351338

Digital Object Identifier
doi:10.14492/hokmj/1330351338

Mathematical Reviews number (MathSciNet)
MR2920098

Zentralblatt MATH identifier
1246.22024

#### 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