Results on central sequences, Cuntz comparison, order-zero maps and maximal tensor powers

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.

• The character criterion — Theorem 2.1. Assume (KR) (Kirchberg-Rørdam; not proved in the programme). A unital separable C*-algebra A is Z-stable if and only if its central sequence algebra F_omega(A) has no characters.
• The finite theorem — Theorem 2.2. Every nonzero unital C*-algebra without characters has a finite maximal tensor power that receives a unital homomorphism from I(2,3); proved in the last lesson of the course.
• Infinite tensor powers — Corollary 2.3. Assume (DT) (Dadarlat-Toms; not proved in the programme). The infinite minimal tensor power of a unital separable C*-algebra without characters is Z-stable.
• Maximal tensor products — Lemma 3.1. Universal property, injective commuting legs, regrouping, functoriality, the quotient onto the minimal tensor product, and unital commuting restrictions of irreducible representations.
• Irreducible representations of I(2,3) — Lemma 4.1. They have dimension 2, 3 or 6; I(2,3) and its quotients are subhomogeneous and have no characters.
• Separable reduction — Lemma 4.2. A nonzero unital C*-algebra without characters contains a separable unital C*-subalgebra without characters.
• Commuting copies — Lemma 4.3. For a separable unital subalgebra D of F_omega(A), every finite maximal tensor power of D maps unitally into F_omega(A).
• Exact cuts — Lemma 2.1. If a is Cuntz below b, then (a - delta)_+ = z*z with zz* in the hereditary subalgebra of b.
• Gram maps — Lemma 3.1. Contractions with x_i x_j* = delta_ij h define a completely positive order-zero map on M_q with a supporting homomorphism into the bidual.
• Commuting Gram maps and I(2,3) — Lemma 4.1. Order-zero maps on M_3 and M_2 with commuting ranges and complementary units give a unital homomorphism from I(2,3).
• The Rørdam-Winter criterion — Proposition 1.1. If orthogonal positive contractions a_1 = w*w and a_2 = ww* satisfy 1 - a_1 - a_2 Cuntz below (a_1 - eps)_+, there is a unital homomorphism from I(2,3); in every unital C*-algebra.
• A unitary in the bidual — Lemma 4.3. A unitary path in the unitization of the section algebra D_2 is carried to a unitary W of the bidual that intertwines gamma on both sides.
• The weighted flip — Lemma 5.4. k(1-k) sigma(y) W* = W* phi(Q(d) sigma(y)), which makes the two order-zero maps commute.
• Independence in commuting factors — Lemma 1.1. Products of linearly independent families from commuting factors are linearly independent.
• Represented blocks — Lemma 2.1. In an irreducible representation of a maximal tensor product, the von Neumann algebras generated by blocks of legs are commuting factors.
• A detecting space — Lemma 3.1. An algebra without characters has a finite-dimensional real space of self-adjoint elements that contains a noncommuting pair in every unital representation.
• Real labels — Lemma 4.1. Arrays whose images under all tensor products of rank-2 maps are linearly independent exist, by Sard's theorem.
• Labels — Corollary 5.1. Self-adjoint elements of a tensor power whose images are linearly independent in every irreducible representation of every larger tensor power.
• Full elements — Lemma 1.1. An element is full if and only if it is nonzero in every irreducible representation.
• A full square-zero element — Lemma 2.4. A finite maximal tensor power of an algebra without characters contains a full element w with w^2 = 0.
• The cone homomorphism — Proposition 3.1. A unital homomorphism from the matrix cone into a tensor power with full first corner.
• The cube — Lemma 1.1. The maximal tensor power C^k of the matrix cone is the algebra of continuous M_2^k-valued functions on the cube with scalar factors on the coordinate faces.
• The matrix simplex — Lemma 2.1. A continuous family of unital embeddings of M_2 into M_2^k over the simplex, using only the factors with positive weight.
• From a full cone to I(2,3) — Proposition 3.1. A unital homomorphism from the cone with full first corner gives a unital homomorphism from I(2,3) into a finite tensor power.

Reading order

Six lessons, to be read in order. The finite theorem is proved in full. The main theorem uses the named hypothesis (KR) of Kirchberg and Rørdam, and the corollary on infinite tensor powers the named hypothesis (DT) of Dadarlat and Toms; neither is proved in the programme.