## Kyoto Journal of Mathematics

### Duality theorem for inductive limit groups

Nobuhiko Tatsuuma

#### Abstract

In this paper, we show the so-called weak duality theorem of Tannaka type for an inductive limit–type topological group $G=\lim_{n\to\infty}G_{n}$ 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

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

#### References

• [1] T. Edamatsu, On the bamboo-shoot topology of certain inductive limits of topological groups, J. Math. Kyoto Univ. 39 (1999), 715–724.
• [2] N. Tatsuuma, A duality theorem for locally compact groups, J. Math. Kyoto Univ. 6 (1967), 187–293.
• [3] N. Tatsuuma, “Duality theorem for inductive limit group of direct product type” in Representation Theory and Analysis on Homogeneous Spaces, RIMS Kôkyûroku Bessatsu B7, Res. Inst. Math. Sci. (RIMS), Kyoto, 2008, 13–23.
• [4] N. Tatsuuma, Duality theorem for inductive limit groups (in Japanese), RIMS Kôkyûroku, 1722 (2010), 48–67.
• [5] N. Tatsuuma, H. Shimomura, and T. Hirai, On group topologies and unitary representations of inductive limits of topological groups and the case of the group of diffeomorphisms, J. Math. Kyoto Univ. 38 (1998), 551–578.
• [6] A. Yamasaki, Inductive limit of general linear groups, J. Math. Kyoto Univ. 38 (1998), 769–779.