Extending Characters of Fixed Point Algebras
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arXiv
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| Format: | Preprint |
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2011
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| _version_ | 1866918259724386304 |
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| author | Wagner, Stefan |
| author_facet | Wagner, Stefan |
| contents | A dynamical system is a triple $(A,G,α)$, consisting of a unital locally convex algebra $A$, a topological group $G$ and a group homomorphism $α:G\rightarrow\Aut(A)$, which induces a continuous action of $G$ on $A$. Further, a unital locally convex algebra $A$ is called continuous inverse algebra, or CIA for short, if its group of units $A^{\times}$ is open in $A$ and the inversion $ι:A^{\times}\rightarrow A^{\times},\,\,\,a\mapsto a^{-1}$ is continuous at $1_A$. For a compact manifold $M$, the Fréchet algebra of smooth functions $C^{\infty}(M)$ is the prototype of such a continuous inverse algebra. We show that if $A$ is a complete commutative CIA, $G$ a compact group and $(A,G,α)$ a dynamical system, then each character of $B:=A^G$ can be extended to a character of $A$. In particular, the natural map on the level of the corresponding spectra $Γ_A\rightarrowΓ_B$, $χ\mapstoχ_{\mid B}$ is surjective. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1111_5560 |
| institution | arXiv |
| publishDate | 2011 |
| record_format | arxiv |
| spellingShingle | Extending Characters of Fixed Point Algebras Wagner, Stefan Dynamical Systems 46L55, 22F50 A dynamical system is a triple $(A,G,α)$, consisting of a unital locally convex algebra $A$, a topological group $G$ and a group homomorphism $α:G\rightarrow\Aut(A)$, which induces a continuous action of $G$ on $A$. Further, a unital locally convex algebra $A$ is called continuous inverse algebra, or CIA for short, if its group of units $A^{\times}$ is open in $A$ and the inversion $ι:A^{\times}\rightarrow A^{\times},\,\,\,a\mapsto a^{-1}$ is continuous at $1_A$. For a compact manifold $M$, the Fréchet algebra of smooth functions $C^{\infty}(M)$ is the prototype of such a continuous inverse algebra. We show that if $A$ is a complete commutative CIA, $G$ a compact group and $(A,G,α)$ a dynamical system, then each character of $B:=A^G$ can be extended to a character of $A$. In particular, the natural map on the level of the corresponding spectra $Γ_A\rightarrowΓ_B$, $χ\mapstoχ_{\mid B}$ is surjective. |
| title | Extending Characters of Fixed Point Algebras |
| topic | Dynamical Systems 46L55, 22F50 |
| url | https://arxiv.org/abs/1111.5560 |