Hamana's injective envelope as a maximal rigid multiplier cover
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911587276685312 |
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| author | Kania, Tomasz |
| author_facet | Kania, Tomasz |
| contents | Let $A$ be a unital $C^*$-algebra. We call an $A$-multiplier cover a pair $(E,ι)$ consisting of a $C^*$-algebra $E$ and a faithful non-degenerate $*$-homomorphism $ι\colon A\to M(E)$. Ordering such covers by $A$-preserving unital completely positive maps between multiplier algebras, we study those covers for which the inclusion $A\subseteq M(E)$ is rigid in Hamana's sense. We prove that Hamana's injective envelope $I(A)$ is a maximal rigid $A$-multiplier cover and that, conversely, a rigid cover is maximal if and only if its multiplier algebra is canonically $*$-isomorphic to $I(A)$ over $A$. Thus maximal rigid multiplier covers provide an order-theoretic characterisation of the injective envelope. In the commutative case $A=C(X)$, this recovers the familiar realisation $I(C(X))\cong C(G(X))\cong M(C_0(U))$ for a dense cozero set $U$ in the Gleason cover $G(X)$, in a form inspired by Błaszczyk's concise construction of the Gleason cover. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_24441 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hamana's injective envelope as a maximal rigid multiplier cover Kania, Tomasz Operator Algebras General Topology Primary 46L05, Secondary 54G05, 46L30 Let $A$ be a unital $C^*$-algebra. We call an $A$-multiplier cover a pair $(E,ι)$ consisting of a $C^*$-algebra $E$ and a faithful non-degenerate $*$-homomorphism $ι\colon A\to M(E)$. Ordering such covers by $A$-preserving unital completely positive maps between multiplier algebras, we study those covers for which the inclusion $A\subseteq M(E)$ is rigid in Hamana's sense. We prove that Hamana's injective envelope $I(A)$ is a maximal rigid $A$-multiplier cover and that, conversely, a rigid cover is maximal if and only if its multiplier algebra is canonically $*$-isomorphic to $I(A)$ over $A$. Thus maximal rigid multiplier covers provide an order-theoretic characterisation of the injective envelope. In the commutative case $A=C(X)$, this recovers the familiar realisation $I(C(X))\cong C(G(X))\cong M(C_0(U))$ for a dense cozero set $U$ in the Gleason cover $G(X)$, in a form inspired by Błaszczyk's concise construction of the Gleason cover. |
| title | Hamana's injective envelope as a maximal rigid multiplier cover |
| topic | Operator Algebras General Topology Primary 46L05, Secondary 54G05, 46L30 |
| url | https://arxiv.org/abs/2510.24441 |