##
**Duality theorem for inductive limit groups.**
*(English)*
Zbl 1288.22003

Summary: 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 \(G\).

### MSC:

22A25 | Representations of general topological groups and semigroups |

22D35 | Duality theorems for locally compact groups |

### References:

[1] | T. Edamatsu, On the bamboo-shoot topology of certain inductive limits of topological groups , J. Math. Kyoto Univ. 39 (1999), 715-724. · Zbl 0948.22002 |

[2] | N. Tatsuuma, A duality theorem for locally compact groups , J. Math. Kyoto Univ. 6 (1967), 187-293. · Zbl 0184.17402 |

[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. · Zbl 1154.22007 |

[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. · Zbl 0930.22002 |

[6] | A. Yamasaki, Inductive limit of general linear groups , J. Math. Kyoto Univ. 38 (1998), 769-779. · Zbl 0939.22003 |

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