C*-algebra

For a concrete example, $M_n(\mathbb C)$ uses the operator norm and the conjugate-transpose involution. The identity becomes $\|A^*A\|=\|A\|^2$, the finite-dimensional model for the abstract axiom.

A concrete $C^*$-algebra is a norm-closed self-adjoint subalgebra of $B(H)$ for a complex Hilbert space $H$. The Gelfand–Naimark representation theorem says that every abstract $C^*$-algebra admits an isometric *-representation of this form. Thus the abstract axioms capture exactly the operator-norm structure of closed operator algebras, without choosing a preferred Hilbert space representation.

The $C^*$-identity forces the involution to be isometric:

It also ties the norm to spectral theory; for example, $\lVert a\rVert^2$ is the spectral radius of the positive element $a^*a$. This rigidity distinguishes $C^*$-algebras from general Banach *-algebras, where the algebra norm and involution can carry substantially less spectral information.

The scalar field $\mathbb C$, matrix algebras $M_n(\mathbb C)$, $B(H)$, and $C_0(X)$ for a locally compact Hausdorff space $X$ are $C^*$-algebras. The algebra $C_0(X)$ is unital exactly when $X$ is compact. Real $C^*$-algebras also form a useful theory, but “$C^*$-algebra” without a qualifier normally means a complex one.

1. Gerard J. Murphy, C-Algebras and Operator Theory*, Academic Press, 1990. Elsevier DOI record. Relevant: §2.1, defining paragraph and equation (1), and Theorem 3.4.1.
2. Gert K. Pedersen, C-Algebras and Their Automorphism Groups, 2nd ed., edited by Søren Eilers and Dorte Olesen, Academic Press, 2018. Elsevier DOI record. Relevant: §1.1.