## Kyoto Journal of Mathematics

- Kyoto J. Math.
- Volume 54, Number 1 (2014), 51-73.

### Duality theorem for inductive limit groups

#### Abstract

In this paper, we show the so-called weak duality theorem of Tannaka type for an inductive limit–type topological group $G={lim\hspace{0.17em}}_{n\to \infty}{G}_{n}$ in the case where each ${G}_{n}$ is a locally compact group, and ${G}_{n}$ is embedded into ${G}_{n+1}$ homeomorphically as a closed subgroup. First, we explain what a weak duality theorem of Tannaka type is and explain the difference between the case of locally compact groups and the case of nonlocally compact groups. Then we introduce the concept “separating system of unitary representations (SSUR),” which assures the existence of sufficiently many unitary representations. The present $G$ has an SSUR. We prove that $G$ is complete. We give semiregular representations and their extensions for $G$. Using them, we deduce a fundamental formula about “birepresentation” on $G$. Combining these results, we can prove the weak duality theorem of Tannaka type for $G$.

#### Article information

**Source**

Kyoto J. Math., Volume 54, Number 1 (2014), 51-73.

**Dates**

First available in Project Euclid: 14 March 2014

**Permanent link to this document**

https://projecteuclid.org/euclid.kjm/1394804791

**Digital Object Identifier**

doi:10.1215/21562261-2400274

**Mathematical Reviews number (MathSciNet)**

MR3178546

**Zentralblatt MATH identifier**

1288.22003

**Subjects**

Primary: 22A25: Representations of general topological groups and semigroups

Secondary: 22D35: Duality theorems

#### Citation

Tatsuuma, Nobuhiko. Duality theorem for inductive limit groups. Kyoto J. Math. 54 (2014), no. 1, 51--73. doi:10.1215/21562261-2400274. https://projecteuclid.org/euclid.kjm/1394804791