PlanetPhysics/Category of C Algebras

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Let ${\displaystyle {\mathcal {A}},{\mathcal {B}}}$ be two C*-algebras. Then a ${\displaystyle *}$-homomorphism


${\displaystyle \phi _{*}:{\mathcal {A}}\longrightarrow {\mathcal {B}}}$ is defined as a C*-algebra homomorphism ${\displaystyle \phi :{\mathcal {A}}\to {\mathcal {B}}}$ which respects involutions, that is:

${\displaystyle \phi (a^{*_{\mathcal {A}}})=\phi (a)^{*_{\mathcal {B}}},\quad {\mbox{ for any }}a\in {\mathcal {A}}.}$

Note: If `by abuse of notation' one uses ${\displaystyle *}$ to denote both ${\displaystyle *_{\mathcal {A}}}$ and ${\displaystyle *_{\mathcal {B}}}$, then any ${\displaystyle *}$-homomorphism ${\displaystyle \phi }$ commutes with ${\displaystyle *}$, i.e., ${\displaystyle \phi *=*\phi }$.

The category ${\displaystyle {\mathcal {C}}}$ whose objects are ${\displaystyle C^{*}}$-algebras and whose morphisms are ${\displaystyle *}$-homomorphisms is called the category of ${\displaystyle C^{*}}$-algebras or the ${\displaystyle C^{*}}$-algebra category.

{\mathbf Remark:} Note that homomorphisms between ${\displaystyle C^{*}}$-algebras are automatically continuous.

References

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.