A dichotomy for derivations and automorphisms of C*-algebras
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915470422048768 |
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| author | Lupini, Martino |
| author_facet | Lupini, Martino |
| contents | Building on previous work of Kadison--Ringrose, Elliott, Akemann--Pedersen, and this author, we prove a dichotomy for the relation of outer equivalence of derivations and unitary equivalence of derivable automorphisms for a separable C*-algebra $A$: either such relations are trivial, or the relation $E_{0}^{\mathbb{N}}$ of tail equivalence of countably many binary sequences is reducible to them. When $A$ is furthermore \emph{unital}, this implies that $A$ has no outer derivation if and only if the group $\mathrm{Inn}\left( A\right) $ of inner automorphisms is $\boldsymbol{Σ}_{2}^{0}$ in $\mathrm{Aut}\left( A\right) $, if and only if it is $\boldsymbol{Σ}_{3}^{0}$ in $\mathrm{Aut}\left( A\right) $. Furthermore, one has that the space of inner derivations is norm-closed if and only if \textrm{Inn}$\left(A\right) $ is norm-closed, if and only if $\mathrm{Inn}\left( A\right) $ is $\boldsymbol{Π}_{3}^{0}$ in $\mathrm{\mathrm{Aut}}\left( A\right) $. This provides a complexity-theoretic characterization of C*-algebras with only inner derivations, which as a by-product rules out $D(\boldsymbol{Π}_{2}^{0})$ as a possible complexity class for $\mathrm{Inn}\left( A\right) $ in $\mathrm{\mathrm{Aut}}\left( A\right) $ for a separable unital C*-algebra $A$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_21726 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A dichotomy for derivations and automorphisms of C*-algebras Lupini, Martino Operator Algebras Logic 46L40, 46L57 (Primary), 03E15, 54H05 (Secondary) Building on previous work of Kadison--Ringrose, Elliott, Akemann--Pedersen, and this author, we prove a dichotomy for the relation of outer equivalence of derivations and unitary equivalence of derivable automorphisms for a separable C*-algebra $A$: either such relations are trivial, or the relation $E_{0}^{\mathbb{N}}$ of tail equivalence of countably many binary sequences is reducible to them. When $A$ is furthermore \emph{unital}, this implies that $A$ has no outer derivation if and only if the group $\mathrm{Inn}\left( A\right) $ of inner automorphisms is $\boldsymbol{Σ}_{2}^{0}$ in $\mathrm{Aut}\left( A\right) $, if and only if it is $\boldsymbol{Σ}_{3}^{0}$ in $\mathrm{Aut}\left( A\right) $. Furthermore, one has that the space of inner derivations is norm-closed if and only if \textrm{Inn}$\left(A\right) $ is norm-closed, if and only if $\mathrm{Inn}\left( A\right) $ is $\boldsymbol{Π}_{3}^{0}$ in $\mathrm{\mathrm{Aut}}\left( A\right) $. This provides a complexity-theoretic characterization of C*-algebras with only inner derivations, which as a by-product rules out $D(\boldsymbol{Π}_{2}^{0})$ as a possible complexity class for $\mathrm{Inn}\left( A\right) $ in $\mathrm{\mathrm{Aut}}\left( A\right) $ for a separable unital C*-algebra $A$. |
| title | A dichotomy for derivations and automorphisms of C*-algebras |
| topic | Operator Algebras Logic 46L40, 46L57 (Primary), 03E15, 54H05 (Secondary) |
| url | https://arxiv.org/abs/2508.21726 |