Manuscript En05-C@1 · En05-C.tex · proposition · aa213ea6

Proposition: Finite-dimensional expectation

Conditional expectations are optional · lines 809–839 · proof included or linked

En05-C@1#prop:finiteexpect, En05-C@1#eq:finiteexpect

Suppose $M = B (H)$ and

H = α ⨁ (C^{d_{α}} \otimes C^{m_{α}}), A = α ⨁ (M_{d_{α}} \otimes 1_{m_{α}}) .

Then

E (x) = α ⨁ [(id \otimes τ_{m_{α}}) (e_{α} x e_{α}) \otimes 1_{m_{α}}], τ_{m} = Tr / m,

is a trace-preserving conditional expectation.

Proof. Central-block pinching, partial normalized trace, and identity extension are CP. The formula is unital and fixes $A$ . Moving the left and right matrix factors through the partial trace gives bimodularity. The ordinary trace of the resulting block equals that of the original diagonal block, and off-diagonal blocks have zero trace.

Displayed formulas in this passage

unlabeled · line 812
$H = α ⨁ (C^{d_{α}} \otimes C^{m_{α}}), A = α ⨁ (M_{d_{α}} \otimes 1_{m_{α}}) .$
En05-C@1#eq:finiteexpect · line 818
$E (x) = α ⨁ [(id \otimes τ_{m_{α}}) (e_{α} x e_{α}) \otimes 1_{m_{α}}], τ_{m} = Tr / m,$

Proposition: Finite-dimensional expectation

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