Equivalently, we can define the  C*-enveloping algebra as follows: Define a real valued function on "A" by as "f" ranges over pure states of "A". This is a semi-norm, which we refer to as the "C* semi-norm" of "A". The set I of elements of "A" whose semi-norm is 0 forms a two sided-ideal in "A" closed under involution. Thus the quotient vector space "A" / I is an involutive algebra and the norm  factors through a norm on "A" / I, which except for completeness, is a C* norm on "A" / I (these are sometimes called 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 formula_6 is an isometric *-isomorphism from formula_6 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.