Results on pure states, excision, free products, HNN adjunctions and the bridge, with their assumptions

The course proves the statements below; each link leads to the statement with its assumptions and its proof. The lessons are self-checked by the writing AI; independent mathematical review is not complete.

• Equivalent pure states are unitarily conjugate — Lemma 2.2. For a pure state h and a unit vector eta in its GNS space there is a unitary u of the algebra with pi_h(u) xi_h = eta; equivalent pure states are conjugate by a unitary, and so are their unique extensions. No hypothesis.
• Peaking under restriction — Lemma 2.5. If a decreasing sequence of projections of B peaks to a state H of a larger algebra C, it peaks in B to the restriction of H, and H is the only extension of that restriction. No hypothesis.
• Decreasing excision — Theorem 3.2. For a pure state h there is a decreasing net of positive contractions x with h(x) = 1, converging strongly to the support r_h, with ||x (a - h(a)) x|| -> 0 for every a; r_h is minimal and r_h X r_h = h(X) r_h. No hypothesis.
• Projection excision — Lemma 3.3. A peaking sequence of projections for h decreases strongly to r_h and excises h. No hypothesis.
• Cross-excision — Lemma 4.1. For inequivalent pure states h, k with excising nets x, y, ||x a y|| -> 0 for every a. No hypothesis.
• Trace-null supports — Lemma 4.2. If B has no finite-dimensional irreducible representation, every tracial state vanishes on the support of every pure state. No hypothesis.
• The expectation onto a peaking algebra — Proposition 2.1. A peaking sequence with p_0 = 1 and traces strictly decreasing to 0 generates a copy of C(N_0 u {infinity}), and E(a)(n) = tau(d_n a d_n)/tau(d_n), E(a)(infinity) = h(a) is the trace-preserving conditional expectation onto it. No hypothesis.
• Simple extensions — Proposition 3.1. If B in C has unique extensions of pure states, B is simple and C has a faithful conditional expectation onto B, then C is simple. No hypothesis.
• Moments determine norms — Lemma 4.1. For a faithful state t, ||x|| = lim t((x*x)^r)^(1/2r). No hypothesis.
• Limits of simple and tracial algebras — Lemma 4.3. The closure of an increasing union of simple C*-algebras is simple, and compatible faithful tracial states extend to a faithful tracial state. No hypothesis.
• The free trace — Proposition 3.2. On the reduced amalgamated free product of two tracial algebras over D, the vacuum expectation onto D vanishes on centered alternating products and tau_D composed with it is a faithful tracial state extending both traces. No hypothesis.
• The expectation onto a factor — Proposition 4.2. There is a faithful trace-preserving conditional expectation of F *_D G onto F that vanishes on products a_0 w a_m with w a core; F and these products span a dense subspace. No hypothesis.
• Reduced HNN adjunctions — Theorem 1.1. For trace-preserving data D_1, D_2, E_i, theta over a tracial algebra A there is a faithfully tracial algebra generated by A and a unitary v with v d v* = theta(d), with a faithful expectation onto A vanishing on reduced words. No hypothesis.
• Uniqueness of the adjunction — Proposition 2.2. The relation, the trace on A and the vanishing of the trace on reduced words determine the adjunction up to an isomorphism fixing A and v. No hypothesis.
• Interchanging two adjunctions — Proposition 4.1. Two adjunctions with data in a common base can be performed in either order: H_k(H_j(O)) and H_j(H_k(O)) are isomorphic, fixing O and both stable unitaries. No hypothesis.
• Adjunctions and limits — Lemma 5.1. Over an increasing union of bases with fixed data, the closure of the union of the adjunctions is the adjunction over the closure. No hypothesis.
• The bridge (OpenAI) — Theorem 1.1. For two inequivalent pure states with peaking sequences of equal traces strictly decreasing to 0, over a faithfully tracial algebra without finite-dimensional irreducible representations, the adjunction gives every pure state a unique extension and joins exactly the two designated classes; simplicity is preserved. No hypothesis.
• Decay of compressed words — Proposition 3.4. For the tested pairs of pure states, x W y -> 0 in norm for every reduced word W of the two-by-two model. No hypothesis.
• Unique extension along chains — Lemma 1.1. In a continuous chain whose successor inclusions have unique extensions of pure states, every inclusion has them, and equivalent states have equivalent extensions. No hypothesis.
• Inherited iterations — Lemma 3.1. A dependency-closed set of steps of a bridge iteration is itself a bridge iteration; the resulting algebra is simple with a faithful trace, separable for countable sets. No hypothesis.
• Closed subiterations (OpenAI) — Theorem 4.1. For every dependency-closed set S of steps, every pure state of the algebra generated by the steps in S has exactly one state extension to the whole iteration. No hypothesis.
• Countable determination (OpenAI) — Lemma 5.1. Every pure state of a bridge iteration is the unique extension of its pure restriction to a separable subalgebra generated by a countable dependency-closed set of steps. No hypothesis.
• Counting pure states — Corollary 5.2. A bridge iteration of length at most c has at most c pure states. No hypothesis.
• The successor cardinal of the continuum — Lemma 1.1. There is an ordinal kappa = c+ not injecting into c with every smaller ordinal injecting into c; |kappa x kappa| = |kappa|, and every countable subset of kappa is bounded. No hypothesis.
• A counterexample to Naimark's problem in ZFC (OpenAI) — Theorem 4.1. There is a unital, infinite-dimensional, simple C*-algebra with a faithful tracial state whose nonzero irreducible representations are all unitarily equivalent; it is not isomorphic to any K(H). Assumes the named hypothesis (KOS).

Reading order

Seven lessons, to be read in order. The first two collect tools on states, excision and expectations; the third and fourth build free products and HNN adjunctions; the fifth proves the bridge; the sixth and seventh iterate it. The final theorem assumes the homogeneity theorem (KOS) of Kishimoto, Ozawa and Sakai, stated in the seventh lesson; everything else is proved in the programme.