Norm additive mappings between commutative $C^{*}$-algebras in the range
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866918205555998720 |
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| author | Hirota, Daisuke |
| author_facet | Hirota, Daisuke |
| contents | Let \( A_i \) be a commutative \( C^{*} \)-algebra for \( i = 1, 2 \), and denote by \( A_i^{+} \) its positive cone, consisting of all positive elements of \( A_i \). In this paper, we investigate surjective, not necessarily continuous mappings \( T: A_1^{+} \to A_2^{+} \) that satisfy the norm equality \[ \| T(a + b) \| = \| T(a) + T(b) \| \quad (a, b \in A_1^{+}). \] We prove that such a mapping \( T \) is necessarily additive and positive homogeneous. Furthermore, we show that if the mapping $T:A_{1}^{+}\to A_{2}^{+}$ between the positive cones of two unital commutative $C^{*}$-algebras $A_{i}$ with the unit element \( 1_{A_i} \) for \( i = 1, 2 \), and if \( T \) is also injective, then $T(1_{A_1})^{-1}T$ is a composition operator.
This is the submitted version of a paper currently under minor revision for the Journal of Mathematical Analysis and Applications. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_13083 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Norm additive mappings between commutative $C^{*}$-algebras in the range Hirota, Daisuke Functional Analysis Operator Algebras 46B04, 46E15 Let \( A_i \) be a commutative \( C^{*} \)-algebra for \( i = 1, 2 \), and denote by \( A_i^{+} \) its positive cone, consisting of all positive elements of \( A_i \). In this paper, we investigate surjective, not necessarily continuous mappings \( T: A_1^{+} \to A_2^{+} \) that satisfy the norm equality \[ \| T(a + b) \| = \| T(a) + T(b) \| \quad (a, b \in A_1^{+}). \] We prove that such a mapping \( T \) is necessarily additive and positive homogeneous. Furthermore, we show that if the mapping $T:A_{1}^{+}\to A_{2}^{+}$ between the positive cones of two unital commutative $C^{*}$-algebras $A_{i}$ with the unit element \( 1_{A_i} \) for \( i = 1, 2 \), and if \( T \) is also injective, then $T(1_{A_1})^{-1}T$ is a composition operator. This is the submitted version of a paper currently under minor revision for the Journal of Mathematical Analysis and Applications. |
| title | Norm additive mappings between commutative $C^{*}$-algebras in the range |
| topic | Functional Analysis Operator Algebras 46B04, 46E15 |
| url | https://arxiv.org/abs/2511.13083 |