 The Gelfand–Naimark representation π is the direct sum of representations π"f"
of "A" where "f" ranges over the set of pure states of A and π"f" is the irreducible representation associated to "f" by the GNS construction.Thus the Gelfand–Naimark representation acts on 
the Hilbert direct sum of the Hilbert spaces "H""f" by

π("x") is a bounded linear operator since it is the direct sum of a family of operators, each one having norm ≤ ||"x"||.Theorem.