category of C*-algebras

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:

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

Note: 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$ .

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.

Remark: Note that homomorphisms between $ C^*$ -algebras are automatically continuous.

Bibliography

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.



Contributors to this entry (in most recent order):

As of this snapshot date, this entry was owned by bci1.