C*-identity - Imported Knowlpedia Content

Definition

Let $A$ be a complex involutive algebra with a submultiplicative norm. The norm satisfies the $C^*$ -identity when

\lVert a^*a\rVert=\lVert a\rVert^2 \qquad\text{for every }a\in A.

This is an identity for all elements, not merely an inequality or a condition on self-adjoint elements. If $A$ is complete, so that its underlying normed algebra is a Banach algebra, this axiom makes $A$ a $C^*$ -algebra. Unitality is not part of the identity, and no separate compatibility constant between the involution and norm is allowed.

Immediate consequences

Submultiplicativity and the $C^*$ -identity force the involution to be isometric:

\lVert a^*\rVert=\lVert a\rVert.

Indeed, applying the identity to $a$ and $a^*$ gives the two inequalities needed for equality. In the unital case the same identity gives $\lVert 1\rVert=1$ unless the algebra is zero. These consequences explain why the involution need not be declared continuous as a separate $C^*$ -algebra axiom Murphy, Definition 2.1.1 and following remarks.

Spectral rigidity

For every $a$ in a $C^*$ -algebra,

\lVert a\rVert^2=r(a^*a),

where $r$ denotes spectral radius. Thus the algebraic operations and spectrum determine the norm far more rigidly than in a general Banach $*$ -algebra. In particular, an injective $*$ -homomorphism between $C^*$ -algebras is automatically isometric.

Examples and non-examples

The operator norm on $\mathcal B(H)$ satisfies the identity because $\lVert T^*T\rVert=\lVert T\rVert^2$ . A Banach $*$ -algebra whose norm only satisfies $\lVert a^*\rVert=\lVert a\rVert$ is a near-miss: isometric involution does not by itself imply the $C^*$ -identity.

References

Gerard J. Murphy, C-Algebras and Operator Theory, Academic Press, 1990. DOI record. Relevant: Definition 2.1.1 and the norm consequences immediately following it.
Gert K. Pedersen, C-Algebras and Their Automorphism Groups, 2nd ed., Academic Press, 2018. DOI record. Relevant: §1.1 on the defining norm identity.