数理解析研究所講究録
第 1820 巻 2012 年 37-4237

Free independence and its generalization
Takahiro Hasebe*
Graduate School of Science, Kyoto University,
Kyoto 606-8502, Japan

1Introduction

1.1

Algebraic probability space

The mathematical description of quantum theory can be seen as a probability theory on an
non-commutative algebra. This viewpoint has led to the development of non-commutative
probability theory (or quantum probability theory). Many probabilistic concepts can beextended to non-commutative algebras: for example, independence of random variables,
moments of random variables and probability distributions of random variables. In particular, it tums out that independence of random variables, the basic concept of probabihty
theory, is not uniquely determined. The most famous independence, which differs from theusual independence, is free independence [8]. On the other hand, monotone independence
was introduced by Muraki [5] as another possible independence of random variables in a
non-commutative algebra.
The main purpose of this article is to explain a key idea to unify free and monotone
independencesl, with a new look at free independence. Let us start from basic conceptson non-commutative probability theory.
Letbe a unital -algebra over , that is, a unital algebra with an involution :which is anti-linear.
state on is a linear functional from towhich isunital and positive: $\varphi(1_{A})=1$ andfor any
typical example of isthe set of all bounded linear operators
on a Hilbert space $H$ . If we denote by
the inner product of, an involution is the usual adjoint operation:$(a\in \mathbb{B}(H), x, y\in H)$ .
typical state is a vector state defined by, whereis a unit vector of $H.$
An algebraic probabihty space is a pairof $a*$ -algebra and a state on it. An
element
is called a random variable.
$\mathcal{A}$$\mathbb{C}$

$*$

$A$

$\mathcal{A}arrow \mathcal{A}$$*$$\mathcal{A}$$\mathcal{A}$$\mathbb{C}_{\}}$$\varphi$$\varphi(a^{*}a)\geq 0$$a\in \mathcal{A}.$$A$$\mathcal{A}$$\mathbb{B}(H)$$H^{2}$$\langle\cdot,$$*$$\langle x,$$A$$\varphi(a)$$:=\langle v,$$av\rangle$$(\mathcal{A}, \varphi)$$X\in \mathcal{A}$$*$The author is supported by Grant-in-Aid for JSPS Fellows. Email: hsb@kunms.kyoto-u.ac.jp

lThis article is based on a preprint [4].
2The inner product here is linear with respect to the right component.

$\cdot\rangle$$ay\rangle=\langle a^{*}x,$ $y\rangle$

$v$

38
From now on we assume that is a von Neumann algebra, i.e., is $a*$ -subalgebra, the set of bounded linear operators on a Hilbert space $H$ , and is closed underof$\mathcal{A}$$\mathcal{A}$$\mathcal{A}$$\mathbb{B}(H)$the strong topology.
The probability distribution of a self-adjoint random variable$\mu_{X}(B):=\varphi(E_{X}(B))$$X\in \mathcal{A}$is defined by,where $E_{X}(B)$ is the spectral projection associated to $X$ and $B$ is a Borel set ofself-adjoint operator $X$ isWe can also consider some unbounded operators on $H.$
said to be affiliated to if its spectral projections $E_{X}(B),$ Borel sets, all belong toin the same way.
Then we can define$\mathbb{R}.$

$A$

$B$

$\mathcal{A}$$\mathcal{A}.$$\mu_{X}$

Example 1.1. The usual probability theory is recovered if we take a probability space. For a real, acting on the Hilbert spaceand letcoincides with the distribution, the probabihty measurevalued random variableself-adjoint random variables.
Theofsetofsets
Borel
forof $X:\mu_{X}(B)=P(X\in B)$
affiliated to is now equal to the set of real-valued random variables.
$L^{2}(\Omega, \mathcal{F}, P)$$\mathcal{A}:=L^{\infty}(\Omega, \mathcal{F}, P)$$(\Omega, \mathcal{F}, P)$$X\in \mathcal{A}$$\mu_{X}$$B$$\mathbb{R}$$\mathcal{A}$1.2Free and monotone independences

Independence can be understood as a rule for calculating mixed moments. We now define
two kinds of independences. In probability theory, independence can be defined forwhich correspond to -subalgebras in non-commutative probability. Hence we formulate
independence in terms $of*$ -subalgebras.

$\sigma-fields$$*$, where thecontaining the unit
be -subalgebras ofDefinition 1.2. Let
followingindependent)
ifthe(orfreefreebetoindex set is arbitrary. They are said
property holds:$\{\mathcal{A}_{\dot{\eta}}\}_{i\in I}$$1_{A}$$\mathcal{A}$

$*$

$I$

(F) $\varphi(a_{1}\cdots a_{n})=0$ if$i_{1}\neq i_{2},$ $i_{2}\neq i_{3},$$\cdots,$$i_{n-1}\neq i_{n},$$a_{k}\in A_{k}$and $\varphi(a_{k})=0$ for any$k.$Using this definition, we can calculate mixed moments of random variables which
belong to different free subalgebras.
Example 1.3. Let$c,$$c’\in C$

and$d\in \mathcal{D}$$\mathcal{B},$$\mathcal{C},$$\mathcal{D}$be free -subalgebras of$*$$\mathcal{A}$. For random variables$b,$ $b’\in \mathcal{B},$, we have$\varphi(bc)=\varphi(b)\varphi(c) , \varphi(bcb’)=\varphi(bb’)\varphi(c)$,$\varphi(bcb’c’)=\varphi(bb’)\varphi(c)\varphi(c’)+\varphi(b)\varphi(b’)\varphi(cc’)-\varphi(b)\varphi(b’)\varphi(c)\varphi(c’)$$\varphi(bcd)=\varphi(b)\varphi(c)\varphi(d) , \varphi(bcdc^{l}b’)=\varphi(bb’)\varphi(cd)\varphi(d)$

,

(1.1)

.

Now we define monotone independence, another famous independence in non-commutative
probability theory.
sequence $of*$ -subalgebras
Definition 1.4. Let be a totally ordered set.
is said to be monotonically independent if the following property holds:
$I$

$A$

$(\mathcal{A}_{\eta})_{i\in I}$of$\mathcal{A}$39(M)$\varphi(a_{1}\cdots a_{n})=\varphi(a_{j})\varphi(a_{1}\cdots a_{j-1}a_{j+1}\cdots a_{n})$$i_{j+1}$if$a_{k}\in\lambda_{k},$$1\leq k\leq n$

and $i_{j-1}<i_{j}>$$(if j=1 or j=n, one of the$ inequalities $are$ eliminated) .

$:=\{1,2,3, \cdots\}.$now on we only consider
The definition of monotone independence shows asymmetry of independent subalge$\mathbb{R}om$$I=\mathbb{N}$

bras: the monotone independence ofdoes not imply the monotone independenceof. Hence we need to consider a sequence of subalgebras, not a set of subalgebras.
$(\mathcal{A}_{1}, \mathcal{A}_{2})$$(\mathcal{A}_{2}, \mathcal{A}_{1})$Example 1.5. Let
dom variables

be monotonically independent -subalgebras of$c’\in C$ and, we have$(\mathcal{B}, C, \mathcal{D})$$b,$ $b’\in \mathcal{B},$

$c,$

$*$

$\varphi(bc)=\varphi(b)\varphi(c) , \varphi(bcb’)=\varphi(bb’)\varphi(c)$. For ran-,$\varphi(cbc’)=\varphi(c)\varphi(b)\varphi(c’) , \varphi(bcb’c’)=\varphi(bb’)\varphi(c)\varphi(c’)$

(1.2)

,

$\varphi(bcd)=\varphi(b)\varphi(c)\varphi(d) , \varphi(bcdc’b’)=\varphi(bb’)\varphi(cc’)\varphi(d)$A difference can be observed in the calculations ofasymmetry of monotone independence.

2$\mathcal{A}$$d\in \mathcal{D}$$\varphi(bcb’)$and$\varphi(cbc’)$.
. This reflects theGeneralization of free independence

The original definition of free independence requires every element to have a zero expectation in order that the product of elements has a zero expectation. However, Example 1.3
indicates that the assumption of the zero expectation of every element is too much. For
example, if $\varphi(c)=0$ , then $\varphi(bcb’)=0$ ; the assumptions $\varphi(b)=\varphi(b’)=0$ are not needed;
if $\varphi(b)=0,$ $\varphi(c)=0$ or $\varphi(d)=0$ , then $\varphi(bcd)=0$ . Thus, we can weaken the assumptions
on the conditions of zero expectations.
Ifare free and, a crucial structure in thisarticle is the graph ofas in Fig. 1. If we regard as a function of , then it attains
local extrema at some points. For example in Fig. 1, the function attains local extrema
at $k=1,3,6,7,8,9,10,11,12,13$ . Let
denote the points at which the function
attains local extrema.3 Then we have an equivalent definition of free independence.
$a_{k}\in \mathcal{A}_{i_{k}}(i_{1}\neq i_{2}, \cdots, i_{n-1}\neq i_{n})$$\{\mathcal{A}_{\eta}\}_{i\in \mathbb{N}}$$(i_{k})_{k=1}^{n}$$k$$i_{k}$$i_{k}$$E(i_{1}, \cdots, i_{n})$$i_{k}$

Proposition 2.1. Let-subalgebras of
onlyandfollowing
theholds:
if
if$\mathcal{A}_{\eta}be*$(F’) $\varphi(a_{1}\cdots a_{n})=0$ if$\varphi(a_{k})=0$

for$i_{1}\neq i_{2},$ $i_{2}\neq i_{3},$$k\in E(i_{1}, \cdots , i_{n})$Proof. The implication
us assume that.4 are free and$(F’)\Rightarrow(F)$$\mathcal{A}$$\cdots,$

containing the unit
$i_{n-1}\neq i_{n},$$1_{\mathcal{A}}$$a_{k}\in \mathcal{A}_{i_{k}}$.

. Then they are free

for $1\leq k\leq n$ andis immediate, and hence we will prove $(F)\Rightarrow(F’)$ . Let$a_{k}\in \mathcal{A}_{\tau_{k}},$$i_{1}\neq i_{2},$ $i_{2}\neq i_{3},$$\cdots,$$i_{n-1}\neq i_{n}$and$\varphi(a_{k})=0$

for $k\in E(i_{1}, \cdots, i_{n})$ . We have to prove that $\varphi(a_{1}\cdots a_{n})=0$ . Ifthen $i_{k-1}<i_{k}<i_{k+1}$ or $i_{k-1}>i_{k}>i_{k+1}$ . In both cases, let us decompose. By iterating this procedure, we can write
3We include the edge points 1, in.

$k\not\in E(i_{1}, \cdots , i_{n})$$a_{k-1}a_{k}a_{k+1}$$\varphi(a_{k})a_{k-1}a_{k+1}+a_{k-1}(a_{k}-\varphi(a_{k})1_{\mathcal{A}})a_{k+1}$$n$$E(i_{1}, \cdots, i_{n})$,as$a_{1}\cdots a_{n}$40Figure 1: $i_{1}=i_{6}=i_{12}=1,$ $i_{2}=i_{5}=i_{10}=2,$ $i_{4}=i_{8}=i_{11}=i_{13}=3,$ $i_{3}=i_{7}=i_{9}=4.$

as a sum$i_{n_{j}}(j)$$\sum_{j=1}^{p}a_{1}^{(j)}\cdots a_{n_{j}}^{(j)}$and$\varphi(a_{k}^{(j)})=0$, wherefor any$k=1,$
Take
where the function$i_{k}\in \mathbb{N},$$j,$ $k$$\cdots,$ $n$$a_{k}^{(j)}\in \mathcal{A}_{\tau_{k}(j)}(1\leq k\leq n_{j}),$$i_{1}(j)\neq i_{2}(j),$ $\cdots,$ $i_{n_{j}-1}(j)\neq$. Then the ffee independence imphes

so that$i_{1}\neq i_{2},$$\cdots,$$i_{n-1}\neq i_{n}$$\varphi(a_{1}\cdots a_{n})=0.$$\square$. Now we know that the points

takes local extrema are important to define free independence.
takes
Moreover, such points can be classified into two subsets: the points at which
local maxima and the points at which takes local minima.4 Let us denote the former. In Fig. 1, ${\rm Max}(i_{1}, \cdots, i_{13})=$and the latter bypoints by$k\mapsto i_{k}$$i_{k}$$i_{k}$${\rm Max}(i_{1}, \cdots, i_{n})$$\{3,7,9,11,13\}$${\rm Min}(i_{1}, \cdots, i_{n})$and ${\rm Min}(i_{1}, \cdots, i_{13})=\{1,6,8,10,12\}.$

To distinguish these two classes, we introduce another state
new definition of independence.

$\psi$

on$\mathcal{A}$and arrive at a

be a sequence
be an algebraic probability space. Let
Definition 2.2. Let$of*$ -subalgebras of. Assume that there is another state on
containing the unit
is said to be ordered free independent if the following condition. The sequenceholds:$(\mathcal{A}_{i})_{i\in N}$$(\mathcal{A}, \varphi)$$\mathcal{A}$$\mathcal{A}$$\psi$$1_{\mathcal{A}}$$(\mathcal{A}_{\eta}\cdot)$. for$i_{n-1}\neq i_{n},$( $OF$ ) $\varphi(a_{1}\cdots 0_{n})=0,$ $\psi(a_{1}\cdots a_{n})=0$ if$1\leq k\leq n,$ $\varphi(a_{k})=0$ for $k\in{\rm Max}(i_{1}, \cdots, i_{n})$ and $\psi(a_{k})=0$ for $k\in{\rm Min}(i_{1}, \cdots, i_{n})$ .
$i_{1}\neq i_{2},$ $i_{2}\neq i_{3},$ $\cdots,$$a_{k}\in \mathcal{A}_{\eta_{k}}$This definition enables us to calculate mixed moments by using both
examples are shown below.
$\mathcal{D}Ex$

ample 2.3. Let$(\mathcal{B}, C, \mathcal{D})$be ordered free independent. Then, for$\varphi$

and$\psi$. Some$d\in C,$ $d\in$$b,$ $b’\in \mathcal{B},$ $c,$,$\varphi(bc)=\varphi(b)\varphi(c) , \psi(bc)=\psi(b)\psi(c) , \varphi(bcd)=\varphi(b)\varphi(c)\varphi(d)$$\varphi(bcb’)=\varphi(c)\varphi(bb’) , \varphi(cbc’)=\psi(b)\varphi(cc’)+\varphi(c)(\varphi(b)-\psi(b))\varphi(c’)$$\psi(bcb’)=\varphi(c)\psi(bb’)+\psi(b)(\psi(c)-\varphi(c))\psi(b’) , \psi(cbc’)=\psi(b)\psi(cc’)$,$\varphi(cbc’b’)=\varphi(cc’)\psi(b)\varphi(b’)+\varphi(c)\varphi(c’)\varphi(bb’)-\varphi(c)\psi(b)\varphi(c’)\varphi(b’)$ ,$\varphi(bcb’c’)=\varphi(bb’)\varphi(c)\varphi(c’)+\varphi(b)\psi(b’)\varphi(cc’)-\varphi(b)\varphi(c)\psi(b’)\varphi(c’)$,$\psi(cbc’b’)=\psi(cc’)\psi(b)\psi(b’)+\psi(c)\varphi(c’)\psi(bb’)-\psi(c)\psi(b)\varphi(c’)\psi(b’)$ .

$\psi(bcb’c’)=\psi(bb’)\varphi(c)\psi(c’)+\psi(b)\psi(b’)\psi(bb’)-\psi(b)\varphi(c)\psi(b’)\psi(c’)$

include edge points also. For example, if$\overline{4We}$$i_{1}>i_{2}$, then $1\in{\rm Max}(i_{1}, \cdots, i_{n})$ .

,

,
(2.1)

41
This new independence extends free independence and also monotone independence.
A connection to free independence can be observed as follows: if we set, the above$OF$condition ( ) is equivalent to the free independence. For monotone independence, wehave to assume an additional structure on the algebra$\varphi=\psi$$\mathcal{A}.$

Proposition 2.4. Let
be an algebraic probability space. If has a decompositionas von Neumann algebras, we can define a canonical state
whereand. Let-subalgebms of . Then the sequencewith respect to is monotonically independent if and only if the sequenceis ordered free independent with respect to the states.
$(\mathcal{A}, \varphi)$$\mathcal{A}$$\mathcal{A}=\mathbb{C}1_{A}\oplus \mathcal{A}^{0}$$\delta(\lambda 1_{\mathcal{A}}+a^{0})$$a^{0}\in \mathcal{A}^{0}$$\lambda\in \mathbb{C}$$\mathcal{A}^{0}$$\mathcal{A}_{i}^{0}be*$$:=\lambda,$$(\mathcal{A}_{i}^{0})_{i\in N}$$(\mathbb{C}1_{A}\oplus \mathcal{A}_{i}^{0})_{i\in N}$$\varphi$$(\varphi, \delta)$

Proof. We prove only one implication. Assume that the sequenceis ordered
free independent with respect to the states. Take natural numbers
suchthatand. If we use the associative law of ordered freeindependence [4], it is sufficient to consider only two subalgebras: $i_{k}\in\{1,2\},$ $1\leq k\leq n.$
If $i_{k}=2$ , let us define $a_{k}’;=a_{k}-\varphi(a_{k})1_{A}$ and if $i_{k}=1,$ $a_{k}’;=a_{k}$ . Then we have
. This implies that$(\mathbb{C}1_{\mathcal{A}}\oplus \mathcal{A}_{i}^{0})_{i\in N}$$(\varphi, \delta)$$i_{1}\neq i_{2},$$\cdots,$$i_{n-1}\neq i_{n}$$i_{1},$$\cdots,$$i_{n}$$a_{k}\in \mathcal{A}_{i_{k}}^{0}$$\varphi(a_{1}’\cdots a_{n}’)=0$$\varphi(a_{1}\cdots a_{n})=\varphi(\prod_{k.\cdot i_{k}=1}a_{k})\prod_{k.i_{k}=2}\varphi(a_{k})$where the product symbol
independence of.

$\prod_{i}$,preserves the order of the index . This means the monotone$i$$(\mathcal{A}_{1}^{0}, \mathcal{A}_{2}^{0})$$\square$

Therefore, if one set $\psi(b)=\psi(c)=\psi(d)=\psi(b’)=\psi(c’)=0$ in Example 2.3, resultsof Example 1.5 can be obtained. We mention below further information on ordered freeindependence. See [4] for detalis.

Remark 2.5. (1) The independence ( $OF$ ) can be generalized more to include Boolean
independence [1, 7]. In that case, we need three states.
(2) Ordered free independence of two subalgebras
dence [2, 3].

$(\mathcal{A}_{1}, \mathcal{A}_{2})$is related to -free indepen$c$(3) We can formulate central limit theorem as follows. Let
be a sequence of i.i.
self-adjoint random variables with $\varphi(X_{n})=\psi(X_{n})=0$ and(independence is in the sense of ( $OF$ )). Then the distribution ofconverges
to a hmit distribution
with respect to the states. In fact the limit distributions
are Kesten distributions.
$d.$$(X_{n})_{i\in \mathbb{N}}$$\varphi(X_{n}^{2})=\alpha^{2},$ $\psi(X_{n}^{2})=\beta^{2}$$X_{1}+\cdots+X_{N}$$\overline{\sqrt{N}}$$(\mu, \nu)$$\mu,$$(\varphi, \psi)$$v$(4) Anti-monotone independence [6] is also included: it is realized by the ordered freeindependence with respect to the states.
$(\delta, \varphi)$42References[1] M. Bozejko, Positive definite functions on the free group and the noncommutative
Riesz product, Bull. Un. Mat. Ital. (6) 5-A (1986), 13-21.
[2] M. Bozejko, Uniformly bounded representations of free groups, J. Reine Angew.
Math. 377 (1987), 170-186.

[3] M. Bozejko and R. Speicher, -independent and symmetrized white noises, Quantum
Probability and Related Topics (L. Accardi, ed.), World Scientific, Singapore VI(1991), 219-236.
$\psi$[4] T. Hasebe, New associative product of three states generahzing free, monotone,anti-monotone, Boolean, conditionally free and conditionally monotone products,
arXiv: 1009. 1505vl.
[5] N. Muraki, Monotonic independence, monotonic central limit theorem and monotonic law of small numbers, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 4(2001), 39-58.

[6] N. Muraki, The five independences as natural products, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 6, No. 3 (2003), 337-371.
[7] R. Speicher and R. Woroudi, Boolean convolution, in Free Probability Theory, Fields
Inst. Commun. 12 (Amer. Math. Soc., 1997), 267-280.
[8] D. Voiculescu, Symmetries of some reduced free product algebras, Operator algebras
and their connections with topology and ergodic theory, Lect. Notes in Math. 1132,Springer, Berlin (1985), 556-588.