diff 8ea744b..242af4d tracial_property_gamma_v2.txt
diff --git a/tracial_property_gamma_v2.txt b/tracial_property_gamma_v2.txt
index 59f66e9..085f05a 100644
--- a/tracial_property_gamma_v2.txt
+++ b/tracial_property_gamma_v2.txt
@@ -328,14 +328,14 @@ Standing hypotheses for Part I:
The only hypothesis Route 1 uses beyond these is: D has real rank zero
in its quotient C*-norm. Route 1 also produces a finite-set statement
-(Theorem 4.6) in which D need not be nuclear or simple. That
+(Theorem 4.3) in which D need not be nuclear or simple. That
generality is useful because D = B^omega is not, strictly speaking,
known to be nuclear.
GOAL OF PART I. Prove Theorem 5.2 (real rank zero implies uniform
property Gamma) and Corollary 5.3 (all traces on B are designated).
-The finite-set statement Theorem 4.6 is proved first, in Section 4,
+The finite-set statement Theorem 4.3 is proved first, in Section 4,
using only the transferable facts of Section 2-3. Section 5 then uses
a diagonal argument along omega to obtain one projection for all of B
at once, and identifies all traces of B.
@@ -3102,7 +3102,8 @@ finite at every sigma (k_n finite matrix; c_n finite matrix; H_n in
A). The comparison hypothesis in (comparison) gives k_n <~ b.
Cutdown with trace control. Write b := H_n + c_n. Choose x with
-||x* b x - k_n|| < 1/4 and set
+||x* b x - k_n|| < eta_n, where 0 < eta_n -> 0 along omega (Cuntz
+comparison permits arbitrary positive tolerance), and set
d := (k_n - 1/2)_+^{1/2}, y := b^{1/2} x d.
@@ -3122,18 +3123,57 @@ Traciality transfers these estimates from w* w = f(k_n) to
r_n := w w*, so | sigma(r_n) - sigma(a_n) | -> 0 and
sigma(r_n - r_n^2) -> 0 uniformly in sigma (both along omega).
-Compress to the H-block. In the H-block of the sum b = H_n + c_n,
-compressed r_n becomes a positive contraction p_n in A (identified
-with its H-block component). The complementary corner is supported by
-c_n, so the compression loses at most 10 gamma_n of trace. The defect
+Compress to the H-block. The bidual r_n controlled sigma(r_n) and
+sigma(r_n - r_n^2), but r_n itself is in the bidual of the ambient
+matrix algebra, and the class (r_n) in l^inf(A) is not available.
+The positive contraction in A is instead
+
+ Y_n := 2 y y* (in P_H M_N(A) P_H identified with A),
+
+which satisfies 0 <= Y_n <= 1 because ||y* y|| = ||d x* b x d||
+<= (1/2) ||x* b x|| <= 1/2. Its trace mirrors sigma(r_n):
+
+ sigma(Y_n) = 2 sigma(d x* b x d)
+ = 2 sigma(d k_n d) + 2 sigma(d (x* b x - k_n) d)
+ = 2 sigma((k_n - 1/2)_+ k_n) + O(eta_n),
+
+which equals sigma(a_n) up to O(sqrt(gamma_n)) along omega
+(projection-nearness of k_n bounds sigma(k_n - (2k_n - 1)_+ k_n)
+by O(sqrt(gamma_n)) using the small-mass argument on
+[(1/2) + eps, 1 - eps] with eps := sqrt(gamma_n)).
+
+Projection defect. Comparing Y_n to the bidual r_n,
+
+ Y_n - r_n = 2 v (d k_n d - d^2 + d (x* b x - k_n) d) v* =: e_n,
+
+and sigma(|e_n|) <= 2 sigma((1 - k_n)(k_n - 1/2)_+) + 2 eta_n -> 0
+along omega; since ||r_n|| <= 1,
+
+ |sigma(Y_n - Y_n^2) - sigma(r_n - r_n^2)| <= 4 sigma(|e_n|) -> 0,
+
+so sigma(Y_n - Y_n^2) -> 0 also. Compressing to the H-block:
+
+ p_n := P_H Y_n P_H in A,
+
+and the complementary corner is supported by c_n, so
+
+ sigma((1 - P_H) Y_n (1 - P_H)) <= 2 sigma(c_n) <= 20 gamma_n,
+ |tr((1 - P_H) Y_n P_H)| <= ||c_n||_1^{1/2} ||2 x d^2 x*||
+ <= O(sqrt(gamma_n)).
+
+Hence |sigma(p_n) - sigma(a_n)| -> 0 uniformly in sigma. The defect
identity
- P r_n P - (P r_n P)^2 = P (r_n - r_n^2) P + P r_n (1 - P) r_n P,
+ p_n - p_n^2 = P_H (Y_n - Y_n^2) P_H + P_H Y_n (1 - P_H) Y_n P_H,
+
+with tr(P_H Y_n (1 - P_H) Y_n P_H) <= O(sqrt(gamma_n)), gives
-with tr(P r_n (1 - P) r_n P) <= d_sigma(c_n), gives bounds
-<= 12 gamma_n on both |sigma(p_n) - sigma(a_n)| and
-sigma(p_n - p_n^2), uniformly on T. Also p_n in Her(H_n), so
-||(1 - h_n) p_n|| <= delta_n, and this vanishes along omega.
+ sigma(p_n - p_n^2) <= sigma(Y_n - Y_n^2) + O(sqrt(gamma_n)) -> 0.
+
+h-coupling. On the H-block, 1 - h_n <= delta_n (operator order,
+from H_n = (h_n - 1 + delta_n)_+). So
+
+ ||(1 - h_n) p_n|| <= delta_n -> 0 along omega.
Conclusion. p := [(p_n)] in D satisfies the properties: as a
projection, its positive-contraction defects have vanishing trace,
@@ -3848,3 +3888,148 @@ are proved.
===========================================================================
END OF DOCUMENT
===========================================================================
+
+===========================================================================
+ A readers' map (short)
+===========================================================================
+
+If you are reading this for the first time, three things are worth
+holding in mind.
+
+ (1) The whole document is about one shape of problem: MAKE A
+ PROJECTION THAT SPLITS EVERY TRACE, WHILE RESPECTING EVERY
+ COEFFICIENT YOU INTRODUCE. The three parts are three different
+ ways to do this, under three different hypotheses.
+
+ (2) The three parts are INDEPENDENT. Route 1 needs real rank zero
+ of a completion ultrapower. Route 2 does not use central
+ sequence tricks and does not need a real rank zero assumption;
+ it starts from strict comparison. Route 3 is nonunital and
+ works under comparison restricted to finite target ranks.
+ Coupling between the parts is only the shared machinery:
+ Section 2 (projection tools, real rank zero) and Section 7
+ (rank approximation and comparison cutdowns).
+
+ (3) THE SHARED OBSTACLE. In every route, moving from "trace close
+ to half" to "exactly half" requires moving a projection off an
+ occupied corner. In Route 1, this is done by finding a
+ partition of the residual projection into finitely many small
+ pieces, in Section 4. In Route 2, it is done by passing to a
+ completing limit in D = A^omega along the diagonal, in Sections 8
+ and 9. In Route 3, it is done by repeated small-splitting
+ selection in a compact complement, in Sections 13 and 14.
+
+The document does not assume K-theory or UCT. It does assume the
+standard equivalence between "uniform property Gamma" in the
+completion form and in the all-m form (Section 1.4), which cites
+[CCEGSTW].
+
+===========================================================================
+ REFERENCES
+===========================================================================
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+
+===========================================================================
+ END OF DOCUMENT
+===========================================================================