 By the Krein extension theorem for positive linear functionals, there is a state "f" on "A" such that "f"("z") ≥ 0 for all non-negative z in "A" and "f"(−"x"* "x") < 0.Consider the GNS representation π"f" with cyclic vector ξ.Since

it follows that π"f" (x) ≠ 0, so π (x) ≠ 0, so π is injective.