Real scalars and the Loewner order #
Multiplication by a nonnegative real scalar preserves positivity (it is the
conjugation by √δ • 1).
Multiplication by a nonnegative real scalar is monotone.
For 0 ≤ x and real α ≤ β, α • x ≤ β • x.
A real number is a self-adjoint complex number.
Corners #
An element of the corner r M r is fixed by left multiplication by r.
An element of the corner r M r is fixed by right multiplication by r.
r y r lies in the corner of r.
The adjoint of an element of the corner r M r lies in the corner.
The center of the corner p M p #
An element z r with z ∈ Z(p M p) and r ≤ p commutes with every element of
r M r.
Step 1: the spectral projection χ_{[δ, 1-δ]}(e) #
The clip t ↦ max 0 (min t 1) of the real line onto [0, 1].
Equations
- Orthogonalization.MvN.clip t = max 0 (min t 1)
Instances For
The spectrum of 0 ≤ e ≤ 1 lies in [0, 1].
clip(e) = e when the spectrum of e lies in [0, 1].
bump(e) = e (1 - e) when the spectrum of e lies in [0, 1].
A self-adjoint contraction 0 ≤ e ≤ 1 which is not a projection has a nonzero
spectral projection χ_{[δ, 1-δ]}(e) for some 0 < δ < 1/2: otherwise
‖e (1 - e)‖ ≤ δ for every such δ, so e² = e.
e χ_I(e) = (clip · 1_I)(e).
δ r ≤ e r for r = χ_{[δ, 1-δ]}(e).
e r ≤ (1 - δ) r for r = χ_{[δ, 1-δ]}(e).
r ≤ δ⁻¹ e r for r = χ_{[δ, 1-δ]}(e).
A projection r ≤ s • (e r) (as r = χ_{[δ, 1-δ]}(e) with s = δ⁻¹) lies below every
projection c commuting with it such that c e = e.
Step 1. For 0 ≤ e ≤ 1 in c M c (c a projection), commuting with a and not a
projection, some spectral projection r = χ_{[δ, 1-δ]}(e) (0 < δ < 1/2) is nonzero;
it lies in M, below c, commutes with a and e, and δ r ≤ e r ≤ (1 - δ) r.
Step 2: a self-adjoint element of r M r commuting with a, not in Z(p M p) r #
i (x - x*) is self-adjoint.
If every self-adjoint element of r M r is of the form z r with z ∈ Z(p M p)
(r ≤ p), then r M r is commutative.
Step 2. If r M r is not abelian (r ≤ p a projection of M) and r commutes with
the self-adjoint a ∈ M, there is a self-adjoint b ∈ r M r commuting with a which is not
of the form z r with z ∈ Z(p M p): r a if r a is not of that form, and otherwise any
self-adjoint element of r M r not of that form (all of r M r then commutes with a).
Step 3: normalization to 0 ≤ b ≤ r #
A self-adjoint b = r b r satisfies b ≤ ‖b‖ r.
A self-adjoint b = r b r satisfies -‖b‖ r ≤ b.
Step 3. A self-adjoint b₁ ∈ r M r commuting with a and not of the form z r
(z ∈ Z(p M p)) is renormalized to b₂ = (2 ‖b₁‖)⁻¹ (b₁ + ‖b₁‖ r), which satisfies
0 ≤ b₂ ≤ r and keeps the other properties.
Step 5: the perturbation stays in [0, 1] #
Step 5. If r is a projection commuting with 0 ≤ e ≤ 1 with
δ r ≤ e r ≤ (1 - δ) r, and b = r b r is self-adjoint of norm ≤ 2, then
e + t b ∈ [0, 1] for |t| ≤ δ / 2: decompose e = e (1 - r) + e r with
0 ≤ e (1 - r) ≤ 1 - r, and -δ r ≤ t b ≤ δ r.
The perturbation step of Lemma 3.1 #
The type II₁ perturbation of Lemma 3.1 (paper, Section 3, "The latter case is easier"):
if e ∈ c M c, 0 ≤ e ≤ 1, commutes with a and is not a projection, then there is a nonzero
self-adjoint b ∈ c M c commuting with a with E b = 0 such that e + t b ∈ [0, 1] for all
small real t. Here c is a central projection of the finite corner p M p whose corner
c M c has no nonzero abelian projection (type II₁), and E is the center-valued trace of
p M p.