Unital *-homomorphism - Imported Knowlpedia Content

Definition

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

\phi(1_A)=1_B.

Thus $\phi$ is complex-linear, multiplicative, involution-preserving, and unit-preserving. The last condition is additional: a $*$ -homomorphism between unital algebras can instead send $1_A$ to a proper projection in $B$ , or be the zero map. Unital $*$ -homomorphisms are the morphisms in the category of unital $C^*$ -algebras when that category is taken to preserve identities.

Automatic properties

Every unital $*$ -homomorphism is positive, completely positive, and contractive, with norm and completely bounded norm equal to one when the codomain is nonzero. It preserves spectra in the one-sided sense $\sigma_B(\phi(a))\subseteq\sigma_A(a)$ ; equality holds when $\phi$ is injective. It also commutes with continuous and holomorphic functional calculus. These are consequences of the $C^*$ -identity, not extra axioms Murphy, §2.1.

Examples and near-misses

The inclusion of a unital $C^*$ -subalgebra that shares the same identity is a unital $*$ -homomorphism. Evaluation $C(X)\to\mathbb C$ at a point is another. If $p$ is a proper projection in a unital algebra $B$ , the corner map from $\mathbb C$ to $B$ , $\lambda\mapsto\lambda p$ , is a $*$ -homomorphism but is not unital as a map into $B$ ; it is unital only when regarded as a map into the corner $pBp$ .

Conventions and universal properties

References

Gerard J. Murphy, C-Algebras and Operator Theory, Academic Press, 1990. DOI record. Relevant: §2.1 on $*$ -homomorphisms, units, positivity, and contractivity.
Gert K. Pedersen, C-Algebras and Their Automorphism Groups, 2nd ed., Academic Press, 2018. DOI record. Relevant: §1.2 on morphism conventions and quotient maps.