pre-C*-norms). Taking the completion of A / I relative to this pre-C*-norm produces a C*-algebra B. By the Krein–Milman theorem one can show without too much difficulty that for x an element of the Banach *-algebra A having an approximate identity: It follows that an equivalent form for the C* norm on A is to take the above supremum over all states. The universal construction is also used to define universal C*-algebras of isometries. Remark. The Gelfand representation or Gelfand isomorphism for a commutative C*-algebra with unit is an isometric *-isomorphism from to the algebra of continuous complex-valued functions on the space of multiplicative linear functionals, which in the commutative case are precisely the pure states, of A with the weak* topology. See also GNS construction Stinespring factorization theorem Gelfand–Raikov theorem Tannaka–Krein duality References (also available from Google Books) , also available in English from North Holland press, see in particular sections 2.6 and 2.7. Operator theory Theorems in functional analysis C*-algebras