Definition 0.1   The category whose objects are -algebras and whose morphisms are
-homomorphisms is called the category of -algebras or the -algebra category.

Definition 0.2   Let be two C*-algebras. Then a -homomorphism is defined as a C*-algebra homomorphism which respects involutions, that is:

    for any

Note: If `by abuse of notation' one uses to denote both and , then any -homomorphism commutes with , i.e., .

Remark: Note that homomorphisms between -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.