*-homomorphism - Knowlpedia

Let $A$ and $B$ be $C^*$ -algebras. A $*$ -homomorphism is a complex-linear map $\phi:A\to B$ satisfying

\phi(ab)=\phi(a)\phi(b),\qquad \phi(a^*)=\phi(a)^*

for all $a,b\in A$ . If both algebras are unital, $\phi$ is called unital when $\phi(1_A)=1_B$ ; this is an additional condition, not part of the definition. Thus zero maps and inclusions of ideals are legitimate $*$ -homomorphisms. A $*$ -isomorphism is a bijective $*$ -homomorphism, and its inverse automatically preserves all the same structure.

Automatic analytic properties

Every $*$ -homomorphism between $C^*$ -algebras is positive and contractive, so no continuity hypothesis is needed. It is isometric exactly when it is injective. These conclusions are special to the $C^*$ -setting and follow from spectral theory and the $C^*$ -identity. A merely multiplicative complex-linear map need not preserve the involution or enjoy these properties.

Kernels, ranges, and quotients

The kernel of $\phi$ is a closed two-sided self-adjoint ideal. The induced map $A/\ker\phi\to B$ is an isometric $*$ -isomorphism onto $\phi(A)$ ; consequently the range of a $*$ -homomorphism is a closed $C^*$ -subalgebra. This is the $C^*$ -algebraic first isomorphism theorem. Composition of $*$ -homomorphisms is again a $*$ -homomorphism.

Representations and conventions

A representation of $A$ on a Hilbert space $H$ is a $*$ -homomorphism $A\to B(H)$ . For nonunital $A$ , nondegeneracy means that the closed span of $\phi(A)H$ is $H$ . For unital $A$ , a nondegenerate representation is automatically unital, but an arbitrary representation need not be. Authors who define morphisms in the unital category to preserve identity are imposing a category convention and should state it explicitly.

References

Gerard J. Murphy, $C^*$ -Algebras and Operator Theory, Academic Press, 1990. DOI record. Relevant: §2.1 on $*$ -homomorphisms, contractivity, kernels, and ranges.
Gert K. Pedersen, $C^*$ -Algebras and Their Automorphism Groups, 2nd ed., Academic Press, 2018. DOI record. Relevant: §1.2 on morphisms and quotient structure.