Manuscript Oai290-A@1 · Oai290-A.tex · section · 7c6e104a

Modular objects and the fixed mean

A separately typed modular bounded-witness module › Modular objects and the fixed mean · lines 1171–1214

Oai290-A@1#eq:modularidentities, Oai290-A@1#eq:scalarcentralizer

Let $M$ be a von Neumann algebra with faithful normal state $φ$ . In standard form, put

H = L^{2} (M), ξ = φ^{1/2}, D = lo g Δ_{φ}, U_{t} = e^{i t D} .

Write $σ_{t} = σ_{t}^{φ}$ and let $E_{D}$ be the spectral resolution of $D$ . We use the standard modular identities

U_{t} (a ξ) = σ_{t} (a) ξ, e^{D /2} a ξ = ξ a, L (a) h = ah, R (a) h = ha .

These are standard-form modular-theory inputs Takesaki. Inner products in this section are linear in the first variable.

The scalar-centralizer assumption is

M_{φ} = {a : σ_{t} (a) = a \forall t} = C 1.

It is not assumed for finite logical codes.

Let $μ_{k}$ be uniform probability measure on $[- k, k]$ . Fix once and for all a free ultrafilter $U$ on $N$ . For bounded continuous scalar $f$ , define

m_{t} f (t) = k \to U lim \int f (t) d μ_{k} (t) .

Boundary estimates make this mean translation invariant and reflection invariant. It is positive and normalized. For bounded continuous $H$ -valued $F$ , define

∥ F ∥_{av} = (m_{t} ∥ F (t) ∥^{2})^{1/2} .

This is a seminorm by the finite-interval triangle inequality and passage to the ultrafilter limit.

Displayed formulas in this passage

Oai290-A@1#eq:modularidentities · line 1182
$U_{t} (a ξ) = σ_{t} (a) ξ, e^{D /2} a ξ = ξ a, L (a) h = ah, R (a) h = ha .$
Oai290-A@1#eq:scalarcentralizer · line 1193
$M_{φ} = {a : σ_{t} (a) = a \forall t} = C 1.$
unlabeled · line 1175
$H = L^{2} (M), ξ = φ^{1/2}, D = lo g Δ_{φ}, U_{t} = e^{i t D} .$
unlabeled · line 1202
$m_{t} f (t) = k \to U lim \int f (t) d μ_{k} (t) .$
unlabeled · line 1208
$∥ F ∥_{av} = (m_{t} ∥ F (t) ∥^{2})^{1/2} .$

Modular objects and the fixed mean

Displayed formulas in this passage

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