Continuous functional calculus

Summary

Let $a$ be a normal element of a C*-algebra, and let $f$ be a continuous function on the spectrum of $a$. The continuous functional calculus gives a meaning to $f(a)$. It extends the evaluation of polynomials at $a$ to all continuous functions, and it respects sums, products, adjoints and norms. It is built from the commutative Gelfand–Naimark theorem. It is used to define square roots, absolute values, positive and negative parts, and exponentials of operators.