*-isomorphism - Knowlpedia

Let $A$ and $B$ be $C^*$ -algebras. A $*$ -isomorphism from $A$ to $B$ is a bijective $*$ -homomorphism $\phi:A\to B$ . Its set-theoretic inverse is automatically complex-linear, multiplicative, and involution-preserving, so it is again a $*$ -homomorphism. The algebras are then called $*$ -isomorphic or $C^*$ -isomorphic. This is the isomorphism notion in the category of $C^*$ -algebras: it preserves the full algebraic and involutive structure, not merely the underlying normed vector spaces.

Automatic norm preservation

Every $*$ -isomorphism is isometric:

\lVert\phi(a)\rVert=\lVert a\rVert\qquad(a\in A).

Consequently it is a homeomorphism for the norm topologies and preserves completeness, spectra, positivity, and continuous functional calculus. No boundedness or isometry hypothesis needs to be added to the definition.

More generally, a faithful $*$ -homomorphism $A\to B$ is a $*$ -isomorphism onto its closed range; surjectivity is the extra condition that identifies the range with all of $B$ .

Units, ideals, and induced structure

If $A$ and $B$ are unital, a surjective $*$ -homomorphism automatically sends $1_A$ to $1_B$ . Thus a $*$ -isomorphism between unital $C^*$ -algebras is automatically unital even when the ambient convention does not require all morphisms to preserve units.

A $*$ -isomorphism sends closed two-sided ideals of $A$ bijectively to closed two-sided ideals of $B$ , preserves inclusion, and induces $*$ -isomorphisms on the corresponding quotients. It also transports states and representations by composition with $\phi$ or $\phi^{-1}$ .

Examples and non-examples

Unitary conjugation,

\operatorname{Ad}_U(T)=UTU^*,

is a $*$ -isomorphism between operator $C^*$ -algebras carried into one another by a unitary $U$ . The transpose map on $M_n(\mathbb C)$ is a linear isometry preserving the involution, but it reverses multiplication, so for $n>1$ it is not a $*$ -isomorphism. A bijective algebra homomorphism that does not preserve involution is likewise not a $C^*$ -isomorphism under this definition.

References

Gerard J. Murphy, C-Algebras and Operator Theory, Academic Press, 1990. DOI record. Relevant: §2.1 on $*$ -homomorphisms, inverse maps, and automatic isometry.
Gert K. Pedersen, C-Algebras and Their Automorphism Groups, 2nd ed., Academic Press, 2018. DOI record. Relevant: §1.2 on $C^*$ -algebra morphisms and isomorphisms.