Spectral theorems for positive algebra homomorphisms
Fuente:
arXiv
Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2021
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866917737670901760 |
|---|---|
| author | de Jeu, Marcel Jiang, Xingni |
| author_facet | de Jeu, Marcel Jiang, Xingni |
| contents | Let $X$ be a locally compact Hausdorff space, let $A$ be a partially ordered algebra, and let $π\colon \mathrm{C}_{\mathrm c}(X)\to A$ be a positive algebra homomorphism. Under conditions on $A$ that are satisfied in a good number of cases of practical interest, it is shown that $π$ is represented by a unique regular spectral measure $μ$ on the Borel $σ$-algebra of $X$, taking its values in the positive idempotents in $A$. The measure $μ$, which is $σ$-additive in an ordered sense, represents $π$ via the order integral (a generalisation of the Lebesgue integral) that goes back to J.D.M. Wright and which was investigated earlier by the authors.
The positive algebra homomorphism $π$ can be extended from $\mathrm{C}_{\mathrm c}(X)$ to a positive linear map from the accompanying $L^1$-space of $μ$ into $A$. It is shown that, quite often, this $L^1$-space is closed under multiplication, so that it is a vector lattice algebra, and that the extended map from $L^1$ into $A$ is not only an algebra homomorphism but, even when $A$ is not a vector lattice, also a vector lattice homomorphism in a sense that is explained in the paper. When $ A$ has the countable sup property, the image of $L^1$ (or of its positive cone) is described in terms of consecutive ups and downs of the image of ${\mathrm C}_{\mathrm c}(X)$ (or of its positive cone).
The general results are applied in three different contexts, showing how various spectral theorems have a common order-theoretical root: representations on Banach lattices, on Hilbert spaces, and (the algebra need not consist of operators) spectral theory for JBW-algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2109_10690 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Spectral theorems for positive algebra homomorphisms de Jeu, Marcel Jiang, Xingni Functional Analysis Operator Algebras Primary 47B99, Secondary 06F25, 28B15 Let $X$ be a locally compact Hausdorff space, let $A$ be a partially ordered algebra, and let $π\colon \mathrm{C}_{\mathrm c}(X)\to A$ be a positive algebra homomorphism. Under conditions on $A$ that are satisfied in a good number of cases of practical interest, it is shown that $π$ is represented by a unique regular spectral measure $μ$ on the Borel $σ$-algebra of $X$, taking its values in the positive idempotents in $A$. The measure $μ$, which is $σ$-additive in an ordered sense, represents $π$ via the order integral (a generalisation of the Lebesgue integral) that goes back to J.D.M. Wright and which was investigated earlier by the authors. The positive algebra homomorphism $π$ can be extended from $\mathrm{C}_{\mathrm c}(X)$ to a positive linear map from the accompanying $L^1$-space of $μ$ into $A$. It is shown that, quite often, this $L^1$-space is closed under multiplication, so that it is a vector lattice algebra, and that the extended map from $L^1$ into $A$ is not only an algebra homomorphism but, even when $A$ is not a vector lattice, also a vector lattice homomorphism in a sense that is explained in the paper. When $ A$ has the countable sup property, the image of $L^1$ (or of its positive cone) is described in terms of consecutive ups and downs of the image of ${\mathrm C}_{\mathrm c}(X)$ (or of its positive cone). The general results are applied in three different contexts, showing how various spectral theorems have a common order-theoretical root: representations on Banach lattices, on Hilbert spaces, and (the algebra need not consist of operators) spectral theory for JBW-algebras. |
| title | Spectral theorems for positive algebra homomorphisms |
| topic | Functional Analysis Operator Algebras Primary 47B99, Secondary 06F25, 28B15 |
| url | https://arxiv.org/abs/2109.10690 |