diff f6bf760..6fbd3ac tracial_property_gamma_v2.txt
diff --git a/tracial_property_gamma_v2.txt b/tracial_property_gamma_v2.txt
index 869544f..70cb468 100644
--- a/tracial_property_gamma_v2.txt
+++ b/tracial_property_gamma_v2.txt
@@ -54,7 +54,7 @@ Throughout, A is a separable C*-algebra. A C*-algebra is unital
unless stated otherwise. All ideals are closed and two-sided.
--------------------------------------------------
-1.1 Traces and their norm inequalities
+1.1 Traces and their norm ineualities
--------------------------------------------------
T(A) is the set of tracial states: linear functionals tau: A -> C with
@@ -68,7 +68,7 @@ For tau in T(A) put
||a||_{2,tau} := tau(a*a)^{1/2}, ||a||_{1,tau} := tau(|a|).
For a subset X of T(A) write ||a||_{2,X} := sup_{tau in X} ||a||_{2,tau}.
-The following inequalities are used throughout:
+The following ineualities are used throughout:
|tau(b*a)| <= ||b||_{2,tau} ||a||_{2,tau} (Cauchy-Schwarz)
|tau(a b)| <= ||a|| ||b||_{1,tau} (duality)
@@ -720,7 +720,7 @@ property,
tau(a q b) <= ||a q||_{2,tau} ||q b||_{2,tau} <= ||a|| ||b|| tau(q).
-Applying this map to the inequality (*) and multiplying by the unitary
+Applying this map to the ineuality (*) and multiplying by the unitary
x*, we obtain
||Phi(q) - x Phi(q) x*||_{1,tau} <= mu tau(q). End. []
@@ -863,9 +863,9 @@ gives, for a unitary x,
<= 2 sqrt(tau(b)) ||[x, sqrt(b)]||_{2,tau}
<= 2 sqrt(tau(b) ||b - x b x*||_{1,tau}). (#)
-Middle inequality: factor |s - t| = |sqrt(s) - sqrt(t)| (sqrt(s) + sqrt(t))
+Middle ineuality: factor |s - t| = |sqrt(s) - sqrt(t)| (sqrt(s) + sqrt(t))
and use Cauchy-Schwarz; the second squared sum is at most
-2 sum_{s,t} (s + t) ||e_s x e_t||^2 = 4 tau(b). Last inequality uses
+2 sum_{s,t} (s + t) ||e_s x e_t||^2 = 4 tau(b). Last ineuality uses
Lemma 3.2.
For b := b_q, using (mass)-type estimates plus (centrality) of
@@ -1430,7 +1430,7 @@ Proof. Every tau in T(A) gives a functional [a] |-> d_tau(a) on Cu(A)
by ERS Prop 4.2 (applied to the lower semicontinuous trace tau (x) Tr).
Only this direction of the trace-functional correspondence is used.
-Let u, v in (A (x) K)_+ satisfy the functional inequalities. If v = 0,
+Let u, v in (A (x) K)_+ satisfy the functional ineualities. If v = 0,
any tau in T(A) gives d_tau(u) = 0, and the faithful stabilized trace
then gives u = 0, so u <~ v.
@@ -1468,7 +1468,7 @@ Define
I := { (a_l) in l^inf(N, A) : lim_{l to omega} ||a_l||_{2, T(A)} = 0 }
D := l^inf(N, A) / I.
-The tracial inequalities xy, yx, x* all obey the correct 1-Lipschitz
+The tracial ineualities xy, yx, x* all obey the correct 1-Lipschitz
bound in ||.||_{2, tau}; so I is a closed two-sided *-ideal and D is a
unital C*-algebra. Faithfulness of every tau makes a |-> (a, a, ...)
injective, so we identify A with its image in D.
@@ -1576,7 +1576,7 @@ For example:
||[x, a] - [x, b]||_{2, Lambda} <= 2 ||x|| ||a - b||
sup_{tau in Lambda} |tau((a - b) x)| <= ||a - b|| ||x||.
-So the identities and inequalities extend from the dense subset to
+So the identities and ineualities extend from the dense subset to
every coefficient by continuity. In particular a vanishing uniform
2-norm commutator gives exact commutation in D.
@@ -1734,7 +1734,7 @@ A (x) K with ||X* b X - v|| < t and set
w := x* b x = P_L X* b X P_L in M_L(A), ||w - v|| < t.
Put alpha := ||w - v||, gamma := t - alpha > 0. In M_L(A)** put
-E := 1_{(3 t, inf)}(v), F := 1_{(t, inf)}(w). The order inequalities
+E := 1_{(3 t, inf)}(v), F := 1_{(t, inf)}(w). The order ineualities
(w - 2 t)_+ >= w - 2 t 1 and w >= v - alpha 1 give
E (w - 2 t)_+ E >= E (v - (2 t + alpha) 1) E >= gamma E,
@@ -1745,7 +1745,7 @@ Neumann algebra and Q a projection with Q a Q >= gamma Q for gamma > 0.
Then the polar decomposition of a^{1/2} Q has initial projection Q
(absorbs through Q a Q >= gamma Q) and final projection at most
supp(a) (its range is inside the range of a^{1/2}). Applying that
-observation to the two displayed inequalities, and taking normal
+observation to the two displayed ineualities, and taking normal
traces, yields
d_tau((v - 3 t)_+) <= d_tau((w - 2 t)_+),
@@ -1776,7 +1776,7 @@ Taking traces and using equality of cut ranks:
tau(p_0 - p_0^2) <= d_tau((w - t)_+) - d_tau((w - 2 t)_+).
Combine with (rank-xfer) and (uniform-cut) to get the stated
-inequalities. v = 0 is included (the rank approximation then gives
+ineualities. v = 0 is included (the rank approximation then gives
tau(b) < delta for all tau, and x = 0 is a witness). End. []
--------------------------------------------------
@@ -1808,7 +1808,7 @@ Let p be the class of (p_l) in D. Since p_l is a positive contraction,
so the projection defect has vanishing uniform 2-norm along omega;
p = p* = p^2 in D. For an arbitrary lambda = lim_omega tau_l on
-represented sequences, the first inequality gives
+represented sequences, the first ineuality gives
lambda(p) = lim_{l to omega} tau_l(b_l) = lambda((h - eta)_+).
@@ -1889,8 +1889,8 @@ and tau in Lambda, t := tau(u - v). Since u - v is a projection:
||u - v||_{2, tau} = sqrt(t), ||u x u - v x v||_{2, tau} <= 2 sqrt(t).
-The second inequality writes the difference as
-(u - v) x u + v x (u - v) and uses the tracial 2-norm inequalities.
+The second ineuality writes the difference as
+(u - v) x u + v x (u - v) and uses the tracial 2-norm ineualities.
Telescope powers (all contractive):
||(u x u)^k - (v x v)^k||_{2, tau} <= 2 k sqrt(t) (k >= 1).
@@ -2168,7 +2168,7 @@ projection e with orthogonality of phi(e) and phi(1_F - e),
strongly as t -> 0. Because projections span F linearly, pi_t has a
pointwise strong limit pi: F -> D**, CP, pi(1_F) = q, and pi takes
projections to projections with range in qD**q. As a UCP map into
-that corner, pi is multiplicative: equality in the Schwarz inequality
+that corner, pi is multiplicative: equality in the Schwarz ineuality
at a projection e (Schwarz applied to e + z x for all z in C) forces
pi(e x) = pi(e) pi(x) and pi(x e) = pi(x) pi(e); taking linear
combinations gives multiplicativity. Also
@@ -2191,7 +2191,7 @@ projection p in D with
p <= 1_{[eta, 1]}(a) <= pi(e_{11}),
tau(p) = tau((a - eta)_+) for every tau in Lambda.
-The last inequality follows from pi(e_{11}) = supp(a). Define on [0, 1]:
+The last ineuality follows from pi(e_{11}) = supp(a). Define on [0, 1]:
k(s) := min(s / eta, 1),
g(s) := k(s) / s on (0, 1], g(0) := 1 / eta.
@@ -2242,7 +2242,7 @@ commutation with pi(c) give
||rho(c) - pi(c)||_{2, tau}^2 <= tau(q - P) <= 1 - tau(h) + eta.
Here q - h is a positive contraction and q - P a projection, and both
-commute with pi(c). Squared triangle inequality finishes. End. []
+commute with pi(c). Squared triangle ineuality finishes. End. []
--------------------------------------------------
9.3 An observable with exact coefficient moments
@@ -2292,7 +2292,7 @@ Lemma 9.2:
sum_i beta_i ||T_i(a) - rho_i psi_i(a)||_{2, tau}^2
<= 4 sum_i beta_i (1 - tau(h_i)) + 2 eta <= 6 eta. (six)
-Second triangle inequality with (three) and (six):
+Second triangle ineuality with (three) and (six):
sum_i beta_i ||a - rho_i psi_i(a)||_{2, tau}^2
<= 2 (3 eta) + 2 (6 eta) = 18 eta. (eighteen)
@@ -2401,7 +2401,7 @@ finite positive measure with mu_a(U) := < E_z(U) a, a >. Since g_R is
This proves (comm).
-Scalar inequalities. For real s,
+Scalar ineualities. For real s,
|s - g_R(s)| <= |s|^4 / R^3,
|s^2 - g_R(s)^2| <= |s|^4 / R^2,
@@ -2420,7 +2420,7 @@ which follows from the polar decomposition x = u|x| in D**, with y :=
<= tau(y^2) = tau(|x|).
So (tail) and (fourth) hold without a commutation assumption on b.
-Combining with Prop 9.3 gives the four displayed inequalities. End. []
+Combining with Prop 9.3 gives the four displayed ineualities. End. []
--------------------------------------------------
9.5 Local bounded moments in A
@@ -2582,7 +2582,7 @@ norm <= 1. Write d := d_{N,j}. Scalar Taylor with integral remainder:
| e^{ix} - 1 - ix + x^2 / 2 | <= |x|^3 / 6 (x in R).
-Functional calculus followed by the tracial inequality
+Functional calculus followed by the tracial ineuality
| tau(b y) | <= tau(|y|) for ||b|| <= 1 (from 9.4 (comm)'s
polar-decomposition argument) gives
@@ -2591,7 +2591,7 @@ polar-decomposition argument) gives
<= |t|^3 / (6 N^{3/2}) tau(|d|^3)
<= 3^{3/4} |t|^3 / (6 N^{3/2}) (taylor)
-The last inequality is Holder's inequality for the probability measure
+The last ineuality is Holder's ineuality for the probability measure
associated with d at tau, together with tau(|d|^3) <= tau(d^4)^{3/4}
and (row-moments). Inserting the two conditional moment bounds from
(row-moments) into (taylor), for every 1 <= j <= N:
@@ -2880,7 +2880,7 @@ Define
D := A^omega := l^inf(A) / I_T,
I_T := { (a_n) in l^inf(A) : lim_{n to omega} ||a_n||_{2, T} = 0 }.
-The tracial inequalities make I_T a closed two-sided *-ideal; D is a
+The tracial ineualities make I_T a closed two-sided *-ideal; D is a
C*-algebra. We regard A as a subalgebra of D through constant
sequences; simplicity and faithfulness of members of T make this
injective.
@@ -3299,7 +3299,7 @@ By Hahn-Banach separation there is a nonzero bounded linear functional
phi on C(T, R) with phi <= phi_on_O on the hull of {L}. Since O is
closed under adding ARBITRARY negative functions, phi is positive:
if not, adding a strongly negative function would push < phi, f >
-arbitrarily far down avoiding the separation inequality. So phi
+arbitrarily far down avoiding the separation ineuality. So phi
normalizes to a probability measure mu on T.
Barycenter. Let sigma_mu be the barycenter of mu in compact convex T.
@@ -3603,7 +3603,7 @@ Lemma 11.1 for trace comparison and 2-norm detection):
Commutators with the dense sequence (x_k)_k of zero uniform 2-norm
give exact commutation, so e in E'. Norm continuity extends the
-trace inequality from the dense self-adjoint unit ball to the whole
+trace ineuality from the dense self-adjoint unit ball to the whole
ball, and scaling to every self-adjoint element of E. End. []
--------------------------------------------------
@@ -3635,7 +3635,7 @@ Define
These are positive contractions of norm at most one, commuting with E
and with one another. For a self-adjoint contraction x in E, the
element P_{j-1} x is a permitted test at step j (it lies in the
-enlarged algebra and has norm <= 1). The trace inequality of Lemma
+enlarged algebra and has norm <= 1). The trace ineuality of Lemma
14.1 gives, uniformly in tau in Lambda,
tau(P_j x) = tau(P_{j-1} x) - tau(e_j P_{j-1} x)