C*-algebra

Summary

A C*-algebra is a complex Banach algebra with an involution $a \mapsto a^*$ whose norm satisfies the C* identity $\norm{a^* a} = \norm{a}^2$. One basic example is the algebra of bounded operators on a Hilbert space. Another is the algebra of continuous functions vanishing at infinity on a locally compact space. By the Gelfand–Naimark theorems these examples are universal: commutative C*-algebras are function algebras, and every C*-algebra is a closed algebra of operators. C*-algebras are the setting of operator K-theory and of noncommutative topology.