= C-star algebra
{c}
{title2=$C^*$-algebra}
{wiki=C\*-algebra}
A <C-star algebra> is a complex <Banach algebra> with an <algebra involution> $a\mapsto a^*$ satisfying the <C-star identity>
$$
\|a^*a\|=\|a\|^2.
$$
The <algebra involution> is conjugate-linear, reverses products and obeys $(a^*)^*=a$. An identity need not be present.<Lectures on C-star algebras by Vahid Shirbisheh>[Definitions 2.1.5 and 2.1.7, printed pp.7–8]{ref}
Examples include complex <matrices> with the <operator norm> and conjugate transpose, <bounded linear operators> on a <Hilbert space> with the operator adjoint, and $C_0(X)$ on a <Hausdorff space> that is a <locally compact space> with pointwise complex conjugation and the <supremum norm>. The identity ties the <norm> to the algebraic product and involution: it implies <spectral radius norm equality for normal elements> and makes an injective <C-star homomorphism> an <isometry>. This rigidity permits <continuous functional calculus> for <normal C-star elements>.
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