Results on quasitraces, expanding bundle maps, Chern class obstructions and weighted shifts

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.

• Basic properties of quasitraces — Lemma 1.1. A 1-quasitrace is positive and monotone on commuting self-adjoint elements; every 1-quasitrace on M_2 extending a normalized 2-quasitrace takes the value 2 at the unit; every tracial state is a normalized 2-quasitrace.
• Properly infinite algebras have no quasitraces — Lemma 2.1. A properly infinite unital C*-algebra has no normalized 1-quasitrace.
• A uniform additivity defect — Lemma 4.1. If x_1, ..., x_N in a unital C*-algebra B satisfy sum x_i* x_i = 1 and sum x_i x_i* <= c 1 with c < 1, explicit positive contractions a, b in M_2N(B) satisfy tau(a + b) - tau(a) - tau(b) >= (1 - c)/(2N) for every normalized 1-quasitrace tau on M_2N(B).
• A counterexample to the quasitrace conjecture — Theorem 3.1. Assume (CBH) (Cuntz, Blackadar and Handelman; not proved in the programme) and (SM) (facts on smooth manifolds; not proved in the course). There are a separable unital C*-algebra A with normalized 2-quasitraces and positive contractions a, b in A with tau(a + b) - tau(a) - tau(b) >= 1/144 for every normalized 2-quasitrace tau; A has no tracial state. Without (CBH) the inequality holds for every normalized 1-quasitrace.
• Smooth images of smaller manifolds are null — Lemma 3.1. Assume (SM). The image of a smooth manifold of dimension a under a smooth map to R^n with a < n has Lebesgue measure zero.
• The incidence lemma — Lemma 4.1. Assume (SM). For smooth Hermitian bundles E, F of ranks d, r over a compact smooth manifold of real dimension 2h and finitely many pairs (s, t) with (qs - t)(r - t) - s(d - s) > h, there are smooth bundle maps T_1, ..., T_q from E to F such that the images of every s-dimensional subspace of a fiber span a space of dimension greater than t.
• The min-max principle — Lemma 5.1. The k-th largest eigenvalue of a Hermitian matrix is the largest mu for which some k-dimensional subspace has quadratic form at least mu; eigenvalues are monotone and 1-Lipschitz in the matrix.
• A determinant expansion — Lemma 5.2. det(I + CC*) is the sum over all sets J of columns of C of the Gram determinants of the columns in J.
• The normalization lemma — Lemma 6.1. If bundle maps T_1, ..., T_n from E to F over a compact Hausdorff space double the dimension of every subspace of every fiber, there are bundle maps X_i with sum X_i* X_i = I and sum X_i X_i* <= (2/3) I; by Remark 6.2 every bound c > 1/2 can replace 2/3.
• Subbundles and orthogonal complements — Lemma 1.1. If p is a bundle map of a Hermitian bundle G whose values are orthogonal projections, the images of p and of I - p are vector bundles and G is isomorphic to their direct sum.
• Cohomology of a product of projective spaces — Lemma 2.1. For Y = CP^K_1 x ... x CP^K_R and x_i = -c_1 of the pulled back tautological line bundles, the monomials x_1^e_1 ... x_R^e_R with 0 <= e_i <= K_i form a basis of H*(Y; Z).
• A Chern class obstruction — Proposition 2.2. A Hermitian bundle over Y isomorphic to L_0^h_0 + L_1^K_1 + ... + L_R^K_R, with L_0 trivial and L_i the pulled back tautological line bundles, contains no trivial subbundle of rank greater than h_0.
• Bundle tests — Theorem 6.1. Assume (SM). For m, R >= 1 there are a compact smooth manifold Y, Hermitian bundles E_j with E_0 trivial of rank d_0 = 64 m 4^R, and 24 bundle maps from E_j to E_(j+1) at every level with sum X* X = I and sum X X* <= (2/3) I, such that (E_0 + ... + E_R)^m contains no trivial subbundle of rank 2 m d_0. Seven maps suffice (Remark 6.2).
• No properly infinite matrix algebra — Theorem 5.1. Assume (SM). The C*-algebra B generated by the weighted shifts of all bundle tests is separable, its 24 generators satisfy sum x_i* x_i = 1 and sum x_i x_i* <= (2/3) 1, and no matrix algebra M_m(B) is properly infinite.
• A sharper constant — Corollary 6.1. Assume (SM). For every c in (1/2, 1) there is a separable unital C*-algebra with seven generators, sum x_i* x_i = 1, sum x_i x_i* <= c 1 and no properly infinite matrix algebra; every normalized 1-quasitrace on its 14 x 14 matrices has additivity defect at least (1 - c)/14 on an explicit pair.

Reading order

Four lessons. The second and the third can be read before the first, and the fourth uses all three. The theorem is proved from two named hypotheses that are not proved in the programme: the existence theorem for quasitraces of Cuntz, Blackadar and Handelman (only for the existence of a quasitrace) and standard facts on smooth manifolds (for the incidence lemma). Chern classes are taken from the course Characteristic classes, and the facts on C*-algebras from the course Foundations of von Neumann algebras.