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category of C*-algebras
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(Definition)
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Definition 0.1 Let $\mathcal{A}, \mathcal{B}$ be two C*-algebras. Then a $*$ -homomorphism $\phi_*:\mathcal{A} \longrightarrow \mathcal{B}$ is defined as a C*-algebra homomorphism $\phi:\mathcal{A} \to \mathcal{B}$ which respects involutions, that is:
$$\phi(a^{*_{\mathcal{A}}}) = \phi(a)^{*_{\mathcal{B}}},\quad\mbox{ for any } a \in \mathcal{A}.$$
Remark 0.1 If `by abuse of notation' one uses $*$ to denote both $*_{\mathcal{A}}$ and $*_{\mathcal{B}}$ , then any $*$ -homomorphism $\phi$ commutes with $*$ , i.e., $\phi*=*\phi$ . Homomorphisms between $C^*$ -algebras are automatically continuous.
Definition 0.2 The category $\mathcal{C}$ whose objects are $C^*$ -algebras and whose morphisms are $*$ -homomorphisms is called the category of $C^*$ -algebras or the $C^*$ -algebra category.
- 1
- Kustermans, J., C*-algebraic Quantum Groups arising from Algebraic Quantum Groups, Ph.D. Thesis, K.U.Leuven, 1997.
- 2
- Sheu, A.J.L., Compact Quantum Groups and Groupoid C*-Algebras, J. Funct. Analysis 144 (1997), 371-393.
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"category of C*-algebras" is owned by bci1.
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Cross-references: morphisms, objects, category, involutions, homomorphism, C*-algebra
There are 17 references to this entry.
This is version 17 of category of C*-algebras, born on 2008-09-20, modified 2009-01-26.
Object id is 11050, canonical name is CategoryOfCAlgebras.
Accessed 2105 times total.
Classification:
| AMS MSC: | 18-00 (Category theory; homological algebra :: General reference works ) | | | 46L05 (Functional analysis :: Selfadjoint operator algebras :: General theory of $C^*$-algebras) | | | 18E05 (Category theory; homological algebra :: Abelian categories :: Preadditive, additive categories) |
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Pending Errata and Addenda
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