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Viewing Version
2
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'$C^*$-algebra homomorphisms have closed images'
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| Title of object: |
$C^*$-algebra homomorphisms have closed images |
| Canonical Name: |
CAlgebraHomomorphismsHaveClosedImages |
| Type: |
Theorem |
| Created on: |
2008-01-14 20:19:19 |
| Modified on: |
2008-01-14 20:21:43 |
| Classification: |
msc:46L05 |
| Synonyms: |
$C^*$-algebra homomorphisms have closed images=image of $C^*$-homomorphism is a $C^*$-algebra |
Revision comment (for changes between this and next version):
| Two A's changed to \mathcal{A} |
Preamble:
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\usepackage{amssymb}
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Content:
\PMlinkescapeword{image}
\PMlinkescapeword{closed}
{\bf Theorem -} Let $f: \mathcal{A} \longrightarrow \mathcal{B}$ be a *-homomorphism between the \PMlinkname{$C^*$-algebras}{CAlgebra} $\mathcal{A}$ and $\mathcal{B}$. Then $f$ has \PMlinkname{closed}{ClosedSet} \PMlinkname{image}{Function}, i.e. $f(A)$ is closed in $\mathcal{B}$.
Thus, the image $f(A)$ is a $C^*$-subalgebra of $\mathcal{B}$. |
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