 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