## Hokkaido Mathematical Journal

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

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

Hiroshi ANDO and Yasumichi MATSUZAWA

#### 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