Orbit misbehavior, isotropy discontinuity, and large isotypic components
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917346099068928 |
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| author | Chirvasitu, Alexandru |
| author_facet | Chirvasitu, Alexandru |
| contents | Let $\mathbb{G}$ be a compact Hausdorff group acting on a compact Hausdorff space $X$, $α$ an irreducible $\mathbb{G}$-representation, and $C(X)$ the $C^*$-algebra of complex-valued continuous functions on $X$. We prove that the isotypic component $C(X)_α$ is finitely generated as a module over the invariant subalgebra $C(X/\mathbb{G})\subseteq C(X)$ precisely when the map sending $x\in X$ to the dimension of the space of vectors in $α$ invariant under the isotropy group $\mathbb{G}_x$ is locally constant. This (a) specializes back to an observation of De Commer-Yamashita equating the finite generation of all $C(X)_α$ with the Vietoris continuity of $x\mapsto \mathbb{G}_x$, and (b) recovers and extends Watatani's examples of infinite-index expectations resulting from non-free finite-group actions.
We also show that the action of a compact group $\mathbb{G}$ on the maximal equivariant compactification on the disjoint union of its Lie-group quotients has tubes about all orbits precisely when $\mathbb{G}$ is Lie. This is the converse (via a canonical construction) of the well-known fact that actions of compact Lie groups on Tychonoff spaces admit tubes. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_08121 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Orbit misbehavior, isotropy discontinuity, and large isotypic components Chirvasitu, Alexandru Operator Algebras Functional Analysis Group Theory Representation Theory 22C05, 22D10, 57S10, 55R10, 13C10, 46M20, 54C60, 43A85, 22F30 Let $\mathbb{G}$ be a compact Hausdorff group acting on a compact Hausdorff space $X$, $α$ an irreducible $\mathbb{G}$-representation, and $C(X)$ the $C^*$-algebra of complex-valued continuous functions on $X$. We prove that the isotypic component $C(X)_α$ is finitely generated as a module over the invariant subalgebra $C(X/\mathbb{G})\subseteq C(X)$ precisely when the map sending $x\in X$ to the dimension of the space of vectors in $α$ invariant under the isotropy group $\mathbb{G}_x$ is locally constant. This (a) specializes back to an observation of De Commer-Yamashita equating the finite generation of all $C(X)_α$ with the Vietoris continuity of $x\mapsto \mathbb{G}_x$, and (b) recovers and extends Watatani's examples of infinite-index expectations resulting from non-free finite-group actions. We also show that the action of a compact group $\mathbb{G}$ on the maximal equivariant compactification on the disjoint union of its Lie-group quotients has tubes about all orbits precisely when $\mathbb{G}$ is Lie. This is the converse (via a canonical construction) of the well-known fact that actions of compact Lie groups on Tychonoff spaces admit tubes. |
| title | Orbit misbehavior, isotropy discontinuity, and large isotypic components |
| topic | Operator Algebras Functional Analysis Group Theory Representation Theory 22C05, 22D10, 57S10, 55R10, 13C10, 46M20, 54C60, 43A85, 22F30 |
| url | https://arxiv.org/abs/2402.08121 |