States and the Gelfand–Naimark–Segal construction

A state is a complex-linear positive functional of norm one. In a unital C*-algebra this is equivalent to positivity together with its value at the identity being one. Positivity means that the value on every adjoint square is nonnegative. It implies a conjugation rule on adjoints and a Cauchy–Schwarz inequality. One proves the latter by applying positivity to the adjoint square of a plus a complex scalar times b and minimizing the resulting quadratic expression.

For matrices, every state has the form Tr(ρa) with a positive density matrix ρ of trace one. Full rank makes this state faithful: a zero expectation of a-star a then forces a to vanish. A rank-deficient density matrix still defines a perfectly valid state, but some nonzero positive elements have zero average. Faithfulness is thus a statement about which positive information the state detects, not merely about being normalized or having nonnegative values.

Define the null space N_phi by the condition that phi of a-star a is zero. Cauchy–Schwarz ensures that the formula for the inner product on quotient classes is independent of representatives. The null space is a left ideal, using the positive inequality a-star c-star c a at most norm-c-squared times a-star a. Complete the quotient in this inner product to obtain H_phi. Completing before removing null vectors would leave a degenerate form instead of a Hilbert space.

Left multiplication defines pi_phi(c)[a]=[ca]. The same positive inequality proves that this action is bounded by the norm of c, so it extends to the completed space. It preserves multiplication and adjoints. The vector Omega represented by the identity is cyclic because its algebra orbit contains every quotient class, dense by construction. The original functional equals the vector average of pi_phi(a) against Omega. This is an explicit realization, not an assumption that the state began as a vector state.

A faithful state has zero null space and a faithful GNS representation: if pi_phi(a) is zero, apply it to Omega and use faithfulness. The converse implication from faithful representation to faithful state is false. In M₂(C), the state reading the top-left entry is not faithful because it annihilates the second coordinate projection. Its GNS representation is nevertheless the usual faithful action on C², with the first basis vector cyclic. Other operators can move that vector into the initially undetected direction.

Taking the Hilbert direct sum of GNS representations from sufficiently many states separates all nonzero algebra elements and gives a faithful representation of the whole C*-algebra. This is the mechanism behind the abstract-to-concrete representation theorem, stated here without its full state-separation proof. One state may discard information, while the full family recovers the algebra. For nonunital algebras, the state definition uses positive norm-one functionals and a modified cyclic construction; the convenient vector [1] must not be inserted when no identity exists.