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Non-commutative independences

This is a wiki dedicated to non-commutative independences. The objective is to build a catalog with a few informations about each of them and bibliographical reference, with an emphasis on the various relations that may exists between them.

I am listing here some non-commutative independences I am aware of (NG writing). Of course, do not hesitate to add to the list those missings, even if you do not have time/energy to write an entry about it.

You will find [instructions | here]] some advices to write an entry.

One state:

Freeness
Boolean independence
Monotone independence
Orthogonal independence
V Freeness
freeness with subordination
matricial freeness

Mixtures

$\Lambda$ Freeness = $\varepsilon$ freeness
$\Lambda$ Boolean
BM independence
bf independence

Two states:

Conditional freeness
Conditional Monotone
Infinitesimal freeness
Infinitesimal Boolean
Infinitesimal Monotone
Cyclic Monotone
Cyclic Boolean
Cyclic free
$O$ indepedendences

Three states:

Cyclic Conditional
Indented independence

Two faces

Bi-freeness
Bi-Boolean
Type I Bi-monotone
Type II Bi-monotone
Boolean-free