Involutive algebra - Knowlpedia

An involutive algebra, or $*$ -algebra, is a complex vector space $A$ equipped with an associative $\mathbb C$ -bilinear multiplication $A\times A\to A$ and a map $a\mapsto a^*$ such that for all $a,b\in A$ and $\lambda,\mu\in\mathbb C$ ,

(\lambda a+\mu b)^*=\overline\lambda a^*+\overline\mu b^*,\qquad (ab)^*=b^*a^*,\qquad (a^*)^*=a.

Terminology and unity

The operation is called an involution. If $A$ is unital, one normally also requires $1^*=1$ , although this follows from the other axioms when the involution is bijective and $1$ is a two-sided identity. No norm, topology, or completeness is part of this definition.

Morphisms and distinguished elements

A $*$ -homomorphism is a complex-linear map $\phi:A\to B$ satisfying $\phi(ab)=\phi(a)\phi(b)$ and $\phi(a^*)=\phi(a)^*$ . It need not preserve an identity unless this is required separately. An element is self-adjoint when $a=a^*$ , normal when $aa^*=a^*a$ , and unitary in a unital algebra when $a^*a=aa^*=1$ . These algebraic notions become analytic only after a compatible norm or operator representation is supplied.

Examples and additional structure

For a complex Hilbert space $\mathcal H$ , bounded operators form a unital $*$ -algebra under the operator adjoint. Complex matrices use conjugate transpose. A $C^*$ -algebra is a norm-complete $*$ -algebra satisfying $\|a^*a\|=\|a\|^2$ ; an arbitrary involutive algebra need not admit any such norm.

Conventions and scope

More generally, one may work over a field carrying a specified involution and require semilinearity with respect to it. Over $\mathbb R$ , the involution is linear. Some authors build a unit into “algebra” and others do not, so unitality must be stated separately. The symbols $*$ -algebra and involutive algebra are synonymous here.

References

Gerard J. Murphy, $C^*$ -Algebras and Operator Theory, Academic Press, 1990. DOI record. Relevant: §2.1 on involutions and $C^*$ -algebras.
Gert K. Pedersen, $C^*$ -Algebras and Their Automorphism Groups, 2nd ed., Academic Press, 2018. DOI record. Relevant: §1.1 on $*$ -algebra conventions.