Noncommutative Mathematics for Quantum Systems

92 Noncommutative Mathematics for Quantum Systems

Voiculescu’s[VDN92]free productφ 1 ∗φ 2 :A 1 ‰A 2 →Cof two
unital linear functionalsφ 1 :A 1 → Candφ 2 : A 2 → Ccan be
defined recursively by

(φ 1 ∗φ 2 )(a 1 a 2 ···am) (1.8.2)

= ∑

I${1, ...,m}

(− 1 )m−]I+^1 (φ 1 ∗φ 2 )

(

→

∏

k∈I

ak

)

∏

k6∈I

φek(ak)

for a typical elementa 1 a 2 ···am ∈ A 1 ‰A 2 , with ak ∈ Aek,
e 16 =e 26 =··· 6=em, that is, neighboringa’s do not belong to the
same algebra. Here]Idenotes the number of elements ofIand
∏→k∈Iakmeans that thea’s are to be multiplied in the same order in
which they appear on the left-hand side. We use the convention
(φ 1 ∗φ 2 )

(

∏→k∈∅ak

)
=1.
The formula in Equation (1.8.2) can be derived from Definition
(1.7.2); it is the only way to define a functionalΦ:A 1 ‰A 2 →C
such that the restrictions ofΦto the subalgebrasAi∼=jAi(Ai)agree
withφifori=1, 2, and that these two subalgebras are free w.r.t.Φ.

Exercise 1.8.12 Derive Formula (1.8.2) from Definition 1.7.2 by
expanding

0 = (φ 1 ∗φ 2 )

(

k

∏

i= 1

(

ai−φεi(ai) 1

)

)

fork≥1,ε∈Ak, anda 1 ∈Aε 1 ,... ,ak∈Aεk.

Ifφ 1 andφ 2 are states, then their free product is a again a state,
see also [VDN92, BNT05, NS06] and the references given there.

Example 1.8.13 (Boolean, monotone, and anti-monotone
independence)
Ben Ghorbal and Schurmann[BG01, BGS99] and Muraki[Mur03] ̈
also considered the category of non-unital algebraic probability
spaces, which we will denote byNuAlgProb, consisting of pairs
(A,φ)of a not necessarily unital algebraAand a linear functional
φ, in their classifications. On this category we can define three
more products, namely the boolean product , the monotone
product.and the anti-monotone product/, of states. They are
defined by setting,