数理解析研究所講究録 1227 巻 2001 年 96-113

96

Deformed free probability of Voiculescu

MAREK BO\.{Z}EJKO
Institute Mathematics, Wroclaw University
Pl. Grunwaldzki 2/4, 50384 Wroclaw, Poland*

Abstract. We introduce r-ffee product of states on the ffoe product of C -algebras and r-free
convolution probability measures on real line. This makes unification ofthe ffee andBoolean probabiity. New classes ofassociative convolution measures are consideredrelated to Muraki-Lou examples.
*

The plan ofthis paper is following:
1. Introduction.
2. -free product $(0\leq r\leq 1)$ of states.
. $r=1$ -free product ofVoiculescu. $r=0$ -Boolean product3. -Fock Space and -Gaussian random variables.
4. r-free convolution ofprobability measures on R.
5. Central limit theorem for r-convolution.
6. Remarks to Muraki-Lou convolution and A-convolution ofmeasures.
$\mathrm{r}$$\mathrm{a}$$\mathrm{b}$$r$.Partially supported by KBN grant$r$$2\mathrm{P}\mathrm{O}3\mathrm{A}05415$

97

1. Introduction
As each discrete group$G$with $N$ generators is ahomorphism image ofthe ffee groupthe same manner we would like to say thatthe free probability$\mathrm{e}\mathrm{a}\mathrm{c}\mathrm{h}" \mathrm{n}\mathrm{a}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{a}\mathrm{l}$$\mathrm{F}_{\mathrm{N}}$in”probability is adeformation of

Voiculescu. In the papers [BS] [BKS] we considered defomedclassical probability and we get so called -deformed Fock space, -second quantization
$q$

$q$

and -Gaussian processes. In this note we propose some versions of deformation ofthe$q$Voiculescu using our technique coming ffom the conditional ffoe

ffee probability

product construction [BLS],[BW]. We use one parameter defomation$0\leq r\leq 1$

and we getfor $r=1$ the ffee probability and for $r=0$ the Boolean probability.
One ofthe main result this paper is the construction on the ffee product of non-unital, (we recall that by astate on anon-unital algebras-algebras4with states :we mean positive ffinctional norm 1), anew examples of statessuch that$\mathrm{C}^{*}$$4arrow \mathbb{C}$$\varphi_{i}$$\varphi:*4arrow \mathbb{C}$$\mathrm{i}$$\mathrm{i}\mathrm{i}$.

$\varphi|=\varphi_{i}4$.

(Voiculescu property If $\varphi(a_{l})=0$ for $i=1$ ,$a_{j}\in 4_{j}$

then

,

$i_{1}\neq i_{2}\neq\ldots$$\ldots$, $n$ , and,$\varphi(a_{1}a_{2}\ldots a_{n})=0$

In the case $r=1$ we get the construction of the ffee product of states of Voiculescu. Ifr $=0$,(called also Boolean product). It has
then we have the regular ffee product of states, than,.
the property that if$[\mathrm{B}1,\mathrm{B}2]$$a_{j}\in 4_{j}$$\varphi(a_{1}a_{2}\ldots a_{n})=\varphi(a_{1})\ldots\varphi(a_{n})$$i_{1}\neq i_{2}\neq\ldots$

Using the construction-ffee product of states $(0\leq r\leq 1)$ we can form the r-ffee
convolution ofprobability measures on R. Then we introduce the analogue of.
transform. The main ideas comes ffom the our paper$\mathrm{o}\mathrm{f}r$$\mathrm{R}\{\mathrm{j}\mathrm{j})-\mathrm{R}-$$[\mathrm{B}\mathrm{L}\mathrm{S},\mathrm{B}\mathrm{W}2]$-transfom we obtain central limit theorem for rAs an example of application
convolution. Our central limit measure , is the “symmetrization” ofthe MarcenkO-Pastur:
measure (the ffee Poisson measure) which Cauchy transform is ofthe$\mathrm{o}\mathrm{f}\mathrm{R}$$\mu$$\mathrm{f}\mathrm{o}\mathrm{m}$1$G_{h}(z)=$

1

$z-$

$r$

$z-$

$z-$

1

.

$z- \frac{r}{z-}..$

98
which is 2-periodic continued
and the measureis supported on two intervals if
05 $r<1$ .
In the section 6we propose some generalization of our construction so we can get someresults Muraki and Lou concerning monotonic convolution and then in central limit we$\mathrm{f}\mathrm{f}\mathrm{f}\mathrm{a}\alpha \mathrm{i}\mathrm{o}\mathrm{n}$$\mu_{r}$have the arcsinus low that means the.

$\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{e}\frac{1}{\pi}\sqrt{1-x^{2}}\$

2. -Free Product of States
Let 4be anon-unital -algebra4with states:) and we define the extension of4(i.e.

$4arrow \mathbb{C}$define anew state$\mathrm{f}\mathrm{f}\mathrm{i}\mathrm{n}\alpha \mathrm{i}\mathrm{o}\mathrm{f}\mathrm{f}\mathrm{i}$

$r$

. Let

$\mathrm{C}^{*}$$\varphi_{i}$$\tilde{4}=4$$\psi_{i}=r\varphi_{\mathrm{i}}+(1-r)\delta_{1}$where$\delta_{1}(x)=\{$$\delta_{1}$is the0if
if$\lambda$$\varphi_{t}$$x$be the unitalization of
Moreover let$\overline{\varphi_{i}}(1)=1,\tilde{\varphi_{i}}|=\varphi_{i}4^{\cdot}$

defined as$\neq\lambda 1$$x$ $=\lambda 1$

is also astate on unital algebra 4and we can fom the conditional ffee product stateon the ffee product -algebra A:

then$\tilde{\varphi}$as$+\mathbb{C}1$$\tilde{4}$$\psi_{i}$$\sim=*\tilde{4.}=\overline{*4}$$\mathrm{C}^{\cdot}$$\tilde{\varphi}^{=}*(\tilde{\varphi_{i}},\psi_{i})$By [BLS] we knew that$\tilde{\varphi}$

is astate on.

$\sim$$\mathrm{C}^{\cdot}$-algebra A . Hence also we get state

the ffoe product non-unital algebra $A=*4.$ .We call
From the construction of we have the folowing properties:$\varphi^{=}*,\varphi_{i}$-the rfree$\varphi=\tilde{\varphi}|_{A}$$pr\ovalbox{\tt\small REJECT} ct$o$\mathrm{n}$state.

$\varphi$(i)$\varphi|=\varphi_{i}4$(ii) if $a_{j}\in 4,$ ,$i_{1}\neq i_{2}\neq\ldots$, then$\tilde{\varphi}[(a_{1}-r\varphi(a_{1})1(a_{2}-r\varphi(a_{2})1\ldots(a_{n}-r\varphi(a_{n})1]=(1-r)^{n}\varphi(\mathrm{q})\ldots\mu a_{n})$The formula (i) is equivalent to:$\vee$(iii)$\vee$$\varphi(a_{1}a_{2}\ldots a_{n})=r\sum_{J}\varphi(a_{j})\varphi(a_{1}\ldots a_{j}\ldots a_{n})-r^{2}\sum_{i<j}\varphi(a_{i})\varphi(a_{j}w(a_{1}\ldots a_{i}\ldots a_{j}\ldots a_{n})$$+\ldots+[(-1)^{n+1}r^{n}+(1-r)^{n}]\varphi(\mathrm{q})\ldots\varphi(a_{n})$.

We see that in the case $r=0$ we get the regular ffee product of states (or Boolean), i.e.
, if,
.The class such as states we foundedin our paper ffom 1986
which is ageneralization Haagerup states on the ffae
product ofyoup. [Haal].
$\varphi(a_{1}a_{2}\ldots a_{n})=\varphi(a_{1})\ldots\varphi(a_{n})$$a_{j}\in 4_{J}.$$i\neq i_{2}\neq\ldots$$[\mathrm{B}1,\mathrm{B}2]$

The most natural state on the group algebra ofthe ffae group,,is the Haagerup stat$x_{1}$$x_{2},\ldots$$x_{N}$$H_{q}(g)=q^{l(g)}$,$g\in \mathrm{F}_{\mathrm{N}}$$\mathrm{F}_{\mathrm{N}}$with the ffee generators

99
if g

;

$\ovalbox{\tt\small REJECT}$\ldots $x_{::}$$x\ovalbox{\tt\small REJECT}_{X_{\ovalbox{\tt\small REJECT}}}^{\ovalbox{\tt\small REJECT}};:$, g t- e,i.

$\ovalbox{\tt\small REJECT} i^{\ovalbox{\tt\small REJECT}}$$i_{2}+i_{3}+$ $\ldots,n_{j}e$$\mathit{7}!t$,$j(g)\ovalbox{\tt\small REJECT} \mathrm{p}\ovalbox{\tt\small REJECT} 1^{\mathrm{n}_{\mathrm{j}}}1^{l(e)\ovalbox{\tt\small REJECT} \mathrm{Q}}$. Since the foil C-

$J$

$N$

algebra$\mathrm{C}’(\mathrm{F}_{\mathrm{N}})^{\ovalbox{\tt\small REJECT}}[\mathrm{J}\mathrm{C}^{\cdot}(\mathrm{Z})^{\ovalbox{\tt\small REJECT}}$, where the product is the free product of C’algebras and$j_{\ovalbox{\tt\small REJECT}}\mathit{1}$$H_{q}=P_{q\mathrm{o}}*\ldots*{}_{0}P_{q}$$=d]$ ,
is the Boolean ffee product, whereis the classical Fourier transform ofthe
Poisson kernel.
One can see that in the case $r=1$ our construction give Voiculescu ffae product of states in
the case when the algebras 4are unital.
$P_{q}(n)$$(n\in \mathbb{Z})$Remark 2.1. If $(A,\varphi)=*,(4,\varphi_{i})$ is the rfree
different algebras , then$A_{i}$$pr\ovalbox{\tt\small REJECT} ct$

as defined above then$\varphi(\mathrm{q}a_{2}\ldots a_{n})=\varphi(\mathrm{q})\varphi(a_{2})\ldots\varphi(a_{n})$Moreover if 4,$a_{2}\in 4$$\mathrm{i}\mathrm{f}a_{i}$are in.

, $b\in A_{j}$ , $i\neq j$ , then

(2.1)$\varphi(a_{1}ba_{2})=r\varphi(\mathrm{q}a_{2})\varphi(b)+(1-r)\varphi(a_{1})\varphi(b)\varphi(a_{2})$Remark 2.2. From the formula (2.1) we can infer that for $r\neq 0,1$ our -ffoe product is not
associative . ifthen $r=0$ or $r=1$ .
$r$$\mathrm{i}.\mathrm{e}$$(\varphi_{1r}*\varphi_{2})_{r}*\varphi_{3}=\varphi$ $*r(\varphi_{2r}*\varphi_{3})$$[]$Remark 2.3. From the formula (iii) we see that the -ffoe product of states
Voiculescu property:$r$

If$\varphi(a_{j})=0$

for allj and$a_{j}\in 4_{j}$,$\mathrm{t}$$\neq i_{2}\neq\ldots$$(\varphi^{(r)}=*,\varphi_{i})$has, then $\varphi(a_{1}a_{2}\ldots a_{n})=0$

Also foris different ffom the ffee product ofVoiculescu.
Problem 1. Find other examples of states $F$ onsuch that:$r\neq 1\varphi^{(\prime)}$$*(4, \varphi_{i})$(i)$F|_{A}=\varphi_{i}$(ii)$\mathrm{F}$satisfies Voiculescu property

Problem 2.

2.

$r$

-free product of states is again astate for r $>1$ ?

$\mathrm{I}\mathrm{f}r$-Fock space and -Gaussian random variables$\mathrm{r}$Let $H$ be areal Hilbert space andspace$\mathbb{C}\mathrm{O}\mathrm{O}\bigoplus_{n=1}^{\infty}H_{\mathrm{c}}^{\otimes n}$$\mathrm{F}(\mathrm{H}\mathrm{C})$For$x_{n},y_{n}\in H_{\mathbb{C}}^{\otimes n}$$\langle x_{n},y_{n}\rangle,$

We can see (product$\langle,\rangle_{r}$$\mathrm{x},$. Now we make deformation ofthe scalar product as follows:

ifn $=2k$ or $n=2k+1$ , $k=0,1,2,3$ ,$\langle \mathrm{q}\Omega\rangle_{r}=\langle \mathrm{q}\Omega\rangle=1$$x\rangle_{r}=\langle x,x\rangle$$\ldots$.

.

for $x\in H$. The completion$\mathrm{o}\mathrm{f}F(H_{\mathrm{c}})$with respect the scalar

we called -Fock space and will be denoted $F(H, r)$ . Moreover$r$define the -creation operation
operator $A(f)$ such that$r$be its complexification. We define the ffee Fockwe put$=r^{k}\langle x_{n},y_{n}\rangle$

Moreover$H_{\mathrm{C}}\mathrm{w}\mathrm{i}\mathrm{U}$$A^{+}\zeta f$$\mathrm{A}(\mathrm{f})\mathrm{Q}$)$x_{1}\otimes\ldots\otimes x_{n}=f\otimes\triangleleft\otimes\ldots\otimes x_{n}$and$\mathrm{f}\mathrm{o}\mathrm{r}f\in H$

weand the r-annihilation$A(f)\eta\otimes\ldots\otimes x_{n}=\lambda_{n}\langle f,x_{1}\rangle_{h}\otimes\ldots\otimes\chi_{n}$

,

100

,.

$\ovalbox{\tt\small REJECT}$

if
if$\mathrm{C}$$\mathrm{n}\ovalbox{\tt\small REJECT}$$n\ovalbox{\tt\small REJECT}$$2k+1$

2k

Proposition 3.1.
(i) $A(f).=A^{+}(f),f\in H$(\"u)$|A(f)||=||A^{+}(g)||= \max(1r)|\mathrm{k}||$(iii) $A(f)A^{+}(g)=\lambda(N)\langle f,g\rangle$ where(iv)$\mathrm{I}\mathrm{f}P$@$\mathrm{X}(\mathrm{N})$$\ldots$$x_{n}=\lambda_{n}x_{1}\otimes\ldots x_{n}$is the orthogonal projection ofF(H, ) onto ,$r$$\mathrm{X}(\mathrm{N})=rP+(I-P)=I+(r-1)P$

If

(v)

$\bigoplus_{\triangleright 0}^{\infty}H^{\emptyset 2n}$.

, then.

$4=4(\mathrm{e},)$, where$\{\mathrm{e},\}$

is an orthonormal basis$| \sum a_{j}\otimes 4||^{2}=\max(1r)||\sum a_{i}a_{i}.||$, then$\mathrm{o}\mathrm{f}H$.

Proof of (i) to (i) follows directly ffom the definition. To get (v) let us observe that$| \sum a_{l}\otimes 4|^{2}=||(\sum a_{l}\otimes 4.)(\sum aj\otimes A_{j}^{+})\#=|\sum a_{l}aj\otimes\lambda(N)\delta_{t}|=|\sum a_{i}a_{t}.\otimes\lambda(N)|=$$=| \sum a_{l}a_{j}.|||\lambda(N)[$.

is the diagonal operator, therefore $[ \lambda(N)||=\max(1,r)$ .
$G(f)=A(f)+A^{+}(f)$ and for
random variables. For
Now we define.
we define the vacuum statebounded operator on
Since$\lambda(N)$$\mathrm{r}- Msg\dot{|a}l$$T$$\mathrm{f}\in H$$\mathrm{f}(H, r)$$\epsilon(T)=\{T\mathrm{q}\Omega\rangle$

Corollary 3.2.
$\max\{||\sum a_{i}a_{i}.|,|\sum a_{i}.a_{i}||\}\leq\#\sum a_{i}\otimes G_{i}|\leq 2\max(r,1)\max\{|\sum a_{l}a_{i}.\int,|\sum a_{i}.a_{i}|\}$.

We can now state the generalization of classical Wick formula (see []).
Let us recall that $NC_{2}(1,2n)$ denote the set of all non-crossing 2-partitions on {1, 2,$2\mathrm{n}\}$$\ldots$,,:
is even number}. Herepartition as was defined in .

$e(V)=\#${$B_{j}\in V$$d_{v}(B_{j})$$V$$B_{j}$

in the$[]$Theorem 3.3 .
(3.1)) is the depth ofthe block$d\langle B_{j}$$\mathrm{I}\mathrm{f}f_{j}\in H$

then.

$\epsilon(G(f_{1})G\mathrm{C}f_{2})\ldots G(f_{2n}))=\sum_{V\in NC_{2}(1\ldots,2n)}\langle f_{\mathrm{i}},f_{j_{1}}\rangle\ldots\langle f_{j_{*}},f_{j_{*}}\rangle r.(V)$The proof of the formula (3.1) folows ffom general result which was proven by us in
the paper with Accardi [AB]101
Remark 3.4. In the case $r=1$ (the ffee Gaussian random variable) this formula$\mathrm{w}j$obtained by R. Speicher,[Spl].
$\mathrm{I}\mathrm{f}r=0$(the Boolean Gaussian random variable) we have the following simple formula:$\epsilon(G(f_{1})G(f_{2})\ldots G(f_{2n}))=\langle f_{1},f_{2}\rangle\langle f_{3},f_{4}\rangle\ldots\langle f_{2\sim 1},f_{2n}\rangle$.

In the special case when $f_{i}=f$ we have$\epsilon(G(f)^{k})=\{\begin{array}{l}|f||^{2n}\mathrm{i}\mathrm{f}k=2n0\mathrm{i}\mathrm{f}k=2n+\mathrm{l}\end{array}$Hence, we see that the distribution ofthe Boolean Gaussian random variables $G\eta$$\mathrm{i}\mathrm{f}||f||=1$

in the vacuum state$\epsilon$

is the Bernoulli law$\mu_{0}=\frac{1}{2}(\delta_{1}+\delta_{-1})$.

Later on we will calculate the distribution ofthe -free Gaussian random variables.
Moreover in the Boolean case we have much more that corollary 3.2.
$r$

Corollary 3.5.
(3.2)$\#\sum a_{i}\otimes G_{\mathrm{i}}\#=\max\{||\sum a_{i}a_{i}.||,|\sum a_{i}^{*}a_{i}||\}$The proof of (3.2) follows ffom the following observation for the block matrices:$(\begin{array}{ll}0 T\Gamma 0\end{array})(\begin{array}{ll}0 TT^{*} 0\end{array})=(\begin{array}{ll}\Pi^{*} 00 T^{*}T\end{array})$0$a_{1}$...

$a_{n}$$a\mathrm{i}$and.

$T= \sum a_{i}\otimes G_{i}=\{\begin{array}{llll}0 \mathrm{q} .\cdot a_{n}\dot{\mathrm{q}} \vdots 0 a_{\dot{n}} \end{array}\}$$.\cdot$0$a_{\dot{n}}$Problem 3Let VNO(N)in the -Fock space $F(H, r)$ .

$=\mathrm{I}_{r}(G_{1}, \ldots, G_{n})$$G_{1},\ldots,G_{n}$$\mathrm{I}\mathrm{f}r=0$will be the von Neumann algebra generated by$r$, then VNO(N) $=M_{N}(\mathbb{C})$ .

, thenIf
Try to verify if
When does exist a$\mathrm{r}=1$$\mathrm{V}\mathrm{N}\mathrm{i}(\mathrm{N})$is the ffoe group.
is also afactorial von Neumann algebra for $0<r<1$ .
$\mathrm{f}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{o}\mathrm{r}-\mathrm{V}\mathrm{N}(\mathrm{F}_{\mathrm{N}})$$\mathrm{V}\mathrm{N}\mathrm{r}(\mathrm{N})$3.

$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$on $mXN$)?

-Free Convolution of Probability Measures on R.

$r$

In this section we will work mainly with probability measures$(\mu\in \mathcal{F}^{\mathrm{p}})$. Let$m_{k}( \mu)=\int_{\mathrm{R}}x^{k}d\mu(x)$, $\mathrm{k}=0,1,2,\ldots$

$\mu$

on

$\mathbb{R}$with compact support

102
as astate on the algebra of polynomials
we define their
Ifwe take two probability measures
as follows:-ffee convolution
and we treat the measure$\mu$$\mathbb{C}\langle X\rangle:\mu[X^{k}]=\mu_{k}(\mu)$$l4,\mu_{2}\in P$$(n^{[eggR]}\mu_{2})$

$r$

(4.1)

$(\mathrm{A}^{[eggR]}h)[X^{\iota}]=(kh^{*}, \mu_{2})[(X_{1}+X_{2})^{\iota}]$, $k=0,1,\ldots$ ,here $(l4 *\mu_{2}r)$ is the -free product of states on the algebra ofnon-commutative polynomials$r$$\mathbb{C}\langle X_{1},X_{2}\rangle$.

On the other hand using our conditionally ffee product ofpairs probability measure as wasis the measure denoted asdone in [BLS]. The convolution measure ,can be obtained in the following way:
, where is the Voiculescu ffee product$r$$\mathrm{A}$$(\mathrm{A}, V_{r}(\mathrm{A}))\mathrm{f}\mathrm{f}1(\mu_{2},V,(\mu_{2}))=(\mu,v)$$V,(/\eta)\mathrm{f}\mathrm{f}1V_{r}(\mu_{2})=\nu$$n^{[eggR]}\mu_{2}$$\mu$$\mu_{2}$

$\nu$

.

Here $V,(\mu)=r\mu+(1-r)\delta_{0}$ .
This implies that$\int x^{k}dV,(\mu \mathrm{X}x)=r\int x^{\iota}d(\mu)(x)$, $k\geq 1$

and therefore using the conditional -transform
following formula for calculation ofmoments for $\mu\in P$ :
$R$

(4.2)

where

$R_{\mu}(k)=R(\mu,V_{r}(\mu)\mathrm{X}k)$$\int x^{n}d\mu(x)=\sum_{V\in \mathfrak{X}(n)}R_{l’}(V)r^{*(\psi)}$$R_{\mu}(V)= \prod_{B\in V}R_{\mu}(\# B)$

we have the,and is asuitable function on the set ofnon crossing partitions$e$.
The important property ofthe function is that $e(V_{0})=1$ , where $V_{0}=\{\{1, \ldots, n\}\}$ .
The formula (4.2) is obtained directly ffom the formula (4.3) ffom the paper [BLS]$NC(\mathit{2}n)$

$e$

(4.3)

$m_{n}( \mu)=\sum_{\succ- 1}^{n}$$l(1)+..i_{(} \iota \mathrm{F}n-k\sum_{l(1),,.\mathit{1}(k)\geq 0}R_{\mu}(k)m_{l(1)}(\mu)\ldots m_{l(k-1)}(\mu)m_{l(k)}(\mu)r^{\prime\vdash k-l(l)}$The formula (4.2) implies that -ffoe convolution of probability measure is associative.
.
, thenMoreover if is Dirac measure at point$r$$x\in \mathbb{R}$$\delta_{x}$$\delta_{x}\copyright\delta_{y}=\delta_{(x+y)}$

Problem 4
From theorem (3.3) we know that for 2-n0n-cr0ssing partition ,{$B\in V:d_{V}(B)$ is even}. Find description ofthe ffinction for$V$$\mathrm{e}(\mathrm{V})$partitions.
After this consideration we can now formulate our result:
Proposition 4.1
If $\mu\in P$ and for(4.4)$z\in \mathbb{C}^{+}=\{z\in \mathbb{C}:\mathrm{h}\mathrm{n}(z)>0\}$then$\frac{1}{G_{\mu}(z)}=z-R_{\mu}(ae_{\mu}(z)+(1-r)\frac{1}{z})$, where$\underline{\mathrm{a}\mathrm{u}}$non-crossing

.

103
,

$\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}$$G.(z)\ovalbox{\tt\small REJECT}$z-x$R_{\ovalbox{\tt\small REJECT}}\ovalbox{\tt\small REJECT}^{)}(zi^{\ovalbox{\tt\small REJECT}}$$R.(z)\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT} R.(k)z$’$1\ovalbox{\tt\small REJECT} 1$

The proof of (4.4) is the reformulation ofthe corresponding formula ffom the theorem 5.2 in.
in the particular case where the measure$[]$$\gamma=\gamma_{\rho\ell}+(1-r)\delta_{0}$Therefore $G_{V}(z)=ae_{\mu}(z)+(1-r) \frac{1}{z}$ .
The details are left to the reader.
Remark 4.2

the fact (4.4) the Voiculescu theorem for the ffee cumulant.
Boolean cumulant formula of Speicher and Wourudi[SW]:$\mathrm{I}\mathrm{f}r=1$$\mathrm{i}\grave{\mathrm{s}}$$\mathrm{I}\mathrm{f}r=0$

then we have$\frac{1}{G_{\mu}(z)}=z-R_{\mu}^{(0)}(\frac{1}{z})$4. Central Limit Theorem
This section is devoted to the main result ofthis paper.
Theorem 5.1

are free
algebra with astate and ,, where A is aLetand. That means that $A=*r4$ ,random variables in the probabiistic system. Assume that:$C^{*}$$X_{j}=X_{i}^{\cdot}\in(A,\varphi)$$\varphi$$X_{1}$$X_{2},\ldots$$(A,\varphi)$$\varphi^{=}*\varphi_{j}r$$X_{i}=X_{i}^{\cdot}\in(4,\varphi_{i})$(i) $\varphi(X_{i})=0$(ii)$\varphi(X_{i}^{2})=1$(iii)$||X_{i}\#<C$

If we take.

$S_{N}= \frac{1}{\sqrt{N}}\sum_{1}^{N}X_{j}$, then$\lim_{Narrow\infty}\varphi(S_{N}^{k})=\int x^{k}d\mu_{r}(x)$, where the probability measure$\mu_{r}=\frac{1}{2}(f_{r}(x)\chi_{I_{r}}+f_{r}(-x)\chi_{(-I,)})\ and$ $f_{r}(x)$ $= \frac{1}{\pi x}\sqrt{4r-(x^{2}-(1+r))^{2}}$Moreover the Cauchy transform ofthe measure

has the following continued ffaction$\mu_{r}$1$G_{\mu}(z)=$

1

$z-$

$r$

$z-$

$z-$

.

1

.

$z- \frac{r}{z-}..$$\mathrm{f}\mathrm{o}\mathrm{m}$:104
Proof. The limit measure , is such that
$\mu$

(5.1)

$R_{\hslash}^{(r)}(z)=z$.

The argument is almost the same as in the proof ofthe ffee probabiity central limit theorem
so we omit it (see [VDN]).

measureHence the Cauchy transform $G(z)=G_{h}(z)$(5.2)$\frac{1}{G(z)}=z-(rG(z)+(1-r)\frac{1}{z})$satisfies the equation:$\mu_{r}$.

1

Now we will show that $G(z)=H(z)$, where $H(z)=$

1

$z-$

$r$

$z-$

1
$z-z- \frac{r}{z-}..$.

First we see that $H(z)=z- \frac{1}{z-rH(z)}$ . So we see that $H=H(z)$ satisfies the following$\eta \mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$:$zrH^{2}+(1-z^{2}-r)H+z=0$ .

(5.3)

But ffom the formula (5.2) follows that $G(z)$ is also the root ofthe equation (5.3). Therefore$G(z)=H(z)$ .
Now we want to calculate explicit form ofthe limit measure
the folowing
holds:$\mu_{r}$. For this let us observe that$\hslash\alpha$(5.4)$m_{2n}( \mu,)=\frac{1}{r}m_{n}(p,)$,

is the ffee Poisson measure with intensivity .
To show (5.4) let us recall that

where $n>0$ and$r$$p_{r}$(5.5)$m_{n}(p,)= \sum_{V\in NC(n)}r^{\mathrm{K}V)}$

For the proof
limit measure. On the other side let us calculate the moments of ourthat fact see
and the ffae Poisson measure using our theorem ffom [AB]:$[\mathrm{S}\mathrm{p}1,\mathrm{B}\mathrm{L}\mathrm{S}]$$\mu_{r}$$p_{r}$$m_{2}(\mu,)=1$$n\}(p_{r})=r$$m_{4}(\mu_{r})=1\dagger r$$m_{2}(p_{r})=r(1+r)$$m_{6}(\mu,)=1+3r\dagger r^{2}$$m_{6}(p_{r})=r(1+3r+r^{2})$.

and by the induction argument we get the proof $M$ $(5.4)$

Hence by small calculation we obtain the equation:(5.6)$G_{pr}(z)= \frac{1}{r}zG_{p},(z^{2})+(1-\frac{1}{r})\frac{1}{z}$Now in the proofwe need the following simple lemma.

105Lemma 5.2 Let, and$f\in L^{1}(\mathbb{R})$and$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}ff\succ^{-}\mathrm{I}\subset \mathbb{R}^{+}$$\tilde{f}(x)=\frac{1}{2}(f(x)\chi_{l}(x)+f(-x)\chi_{(-l)}(x))$

then the Cauchy transform of(5.7)$\tilde{f}$is ofthe$\mathrm{f}\mathrm{o}\mathrm{m}$:$G_{\tilde{f}}(z)=zG_{F}(z^{2})$Since $=(1-r)\delta_{0}+F_{r}(x)\$ , where
This implies that:
$P,$

(5.8)

, where$0\leq r\leq 1$$F(x)= \frac{f(\sqrt{x})}{2\sqrt{x}}$.

.

$G_{h}(z)=zG_{F},(z^{2})=G_{\tilde{f}},(z)$.

Therefore by lemma 5.2 we get that
(5.9)

As we knew (see$[\mathrm{V}\mathrm{D}\mathrm{N},\mathrm{B}\mathrm{L}\mathrm{S}]$)$F,(x)= \frac{1}{2\pi x}\sqrt{4r-(x-(1+r))^{2}}$

Since
and

&.

$\mu_{r}=\tilde{f}_{r}(x)\ = \frac{1}{2}(f(x)\chi_{I}(x)+f(-x)\chi_{(-I)}(x))$$/\mathrm{r}(\mathrm{x})$$2xF_{r}(x^{2})= \frac{1}{\pi x}\sqrt{4r-(x^{2}-(1+r))^{2}}$, where$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}f_{r}=I_{r}$.

$I_{r}$

is interval ofthe form $I_{r}=[1-\sqrt{r},1+\sqrt{r}]$ , therefore this completes

the proof of theorem 5.1.
Remark 5.3 Ifr $=1$ , we have

Therefore$\mu_{1}=\frac{1}{2\pi}\sqrt{4-x^{2}}\chi_{[-2,2]}$$f_{1}(x)$ $= \frac{1}{\pi}\sqrt{4-x^{2}}$.

&-so this is semicircle low ofWigner (free Gaussian

random variables).
Remark 5.4 It is also possible to calculate the measure
for $r>1$ and then we can seethat measure has aone atom at 0(see [K]). It
be interesting to see why that measure isconnected with quasi-ffee ffee state considered by Shlyakhtenko$\mu_{r}$$\mathrm{w}\mathrm{i}\mathrm{u}$$[\mathrm{S}\mathrm{h}]^{\eta}$

5. Remarks on Muraki-Lou convolution and A-convolution.
In this chapter we present some generalization-ffee convolution ofprobability measures.
For this let :Prob(E) Prob(E) will be some map and Prob(E) is the set of all$\mathrm{o}\mathrm{f}r$$C$

probability measures on the real line. We will define -ffee convolution ofmeasures asfollows:
$C$

(6.1)

$(\mathrm{A}, C(\mu_{1}))\mathrm{f}\mathrm{f}\mathrm{l}(\mu_{2},C(\mu_{2}))=(\mu,C(\mu_{1})\mathrm{f}\mathrm{f}1C(\mu_{2}))$,where the convolution ofthe pairs of measure is conditionallyfiee convolution ([BLS]).
The formula (6.1) defines -ffoe convolution of$C$$\mu_{1}[eggc]\mu_{2}=\mu$.

106
In the special case when $C(\mu)=V_{r}(\mu)=r\mu+(1-r)\delta_{0}=(r\delta_{1}+(1-r)\delta_{0})\square \mu$ by above method
we obtain again -free convolution. Here denote the multiplicative convolution of$\square$$r$

probability measures on real line.
Another example of deformed ffee convolution was presented by Wysoczanski and myself). This corresponds to C-convolution, where $C=U_{t}(t \geq 0)$ is defined by the(see$[\mathrm{B}\mathrm{W}1,\mathrm{B}\mathrm{W}2]$$\eta \mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$$\frac{1}{G_{\mu(t)}(z)}=\frac{t}{G_{\mu}(z)}+(1-t)z$, where$\mu(t)=U_{t}(\mu)\mathrm{N}\mathrm{C}(\mathrm{n})$. In that example the central

is the Kesten measure which is the spectral measure for the random walks on. The Cauchy transfom ofthe measure has
the ffoe group and the parameter
limit measure$K_{t}$$K_{t}$$t=1- \frac{1}{2N}$$\mathrm{F}_{N}$following continued ffaction form:

1$G_{K_{t}}(z)=$

1

$z-$

$t$

$z-$
$t$

$z-$

.

$z- \frac{t}{z-}..$The rest that chapter will be devoted to the special class ofconvolution-called
Prob(R) Prob(R) done by the-convolution which corresponds to the map
multiplicative convolution on the real he by the suitable measure, thenor in another words if
so we define $C(\mu)=\mu$$m_{n}(C(\mu))=\delta_{n}m_{n}(\mu)$ , $n=0,1$ ,. In that case our A-convolution is associative, since wewhich make linearization of our -convolution. That exactly
have
$C$

$arrow$$\square$$[]$$a),\cdot$$\delta_{n}=\int x^{n}d\varpi(x)$$\varpi$$\ldots$$C$$R-\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{s}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}-R_{\mu}^{\Delta}=R^{\Delta}(\mu)$means that

$R^{\Delta}(\mathrm{A}\copyright\mu_{2})=R^{\Delta}\omega)+R^{\Delta}(\mu_{2})$. Also there is nice connection between$R^{\Delta}-$cumulants and moments done by fomula:$\int_{\mathrm{B}}x^{n}d\mu(x)=\sum_{\nu^{\mathrm{r}}\epsilon \mathfrak{X}(n)}R_{\mu}^{\Delta}(V)t(V,\mathrm{A})$on non-crossing partition set
for proper
generalization of central limit theorem for$\mathrm{f}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{c}\dot{\mathrm{u}}\mathrm{o}\mathrm{n}$$t(\cdot,\mathrm{A})$$\mathrm{N}\mathrm{C}(\mathrm{n})$$\mathrm{A}-\omega \mathrm{n}\mathrm{v}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$We recall that dilatation$E\subset \mathbb{R}$and$s\neq 0$$D_{s}$

ofthe measure$\mu$

Now we can present a.

is defined as$D_{s}( \mu\cross E)=\mu(\frac{1}{s}E)$.

Theorem 6.1

Let$\mu_{l}\in \mathrm{P}\mathrm{r}\mathrm{o}\mathrm{b}(\mathrm{R})$and all moments ofmeasures(i) $\int d\mu_{j}(x)=0$(ii)$\int x^{2}d\mu_{j}(x)=1$(iii)$| \int x^{\iota}d\mu_{j}(x)|\leq B_{k}$, for$\mathrm{a}\mathrm{U}\mathrm{j}$,$\mu_{i}$are finite. Assume thatfor Borel set

107then the measures$S_{N}\ovalbox{\tt\small REJECT}$

D$(_{\ovalbox{\tt\small REJECT}}\mathrm{q})$$\ovalbox{\tt\small REJECT}$O...

$\mathrm{O}_{\ovalbox{\tt\small REJECT}}D$)1$D_{s}(\ovalbox{\tt\small REJECT} \mathrm{u}\mathrm{X}\mathrm{E})\ovalbox{\tt\small REJECT}$$\mathrm{p}(,\mathrm{A}\ovalbox{\tt\small REJECT})$$Czz_{N}$) weakly tends to limit measure

$\#$

$i$

$\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}$

for Borel set EcR.

$s$

Moreover(6.2)$G_{\mu}(z)= \frac{1}{z-G_{C(\mu)}(z)}(\Leftrightarrow R_{\mu}^{\Delta}(z)=z)$.

The proofis the same like theorem 5.1 so we omit it.
Corollary 6.2&then the corresponding A-convolution is related
Ifwe take as ameasure
to the convolution discovered by Muraki-Lou and the central limit measure is the arcsinus low$d\varpi(x)=|x|\chi_{[-1,1]}$$\frac{1}{\pi}\frac{1}{\sqrt{2-x^{2}}}$&.

In the proof ofthe corollary use the fact that$\mu=\frac{1}{\pi\sqrt{2-x^{2}}}$$C( \mu)=\frac{1}{\pi}\sqrt{2-x^{2}}\chi_{[\dashv\overline{2},\sqrt{2}]}$&if&. Moreover
1$G_{\mu}(z)=$

1

$z-$

$\gamma_{2}$$z-$$\gamma_{2}$

$z-$

.

$z- \frac{\gamma_{2}}{z-}..$

and1$G_{\mathrm{C}(\mu)}(z)=$$\gamma_{2}$$z-$$\gamma_{2}$$z-$$\gamma_{2}$

$z-$

.

$z- \frac{\gamma_{2}}{z-}..$

so evidently the equation (6.2) is satisfied.
Problem 5

Characterize all central limit measures for all moment sequences$\mathrm{A}=(\delta_{n})$in the case of

A-convolution.
Remark 6.3

One can show that the classical Gauss measure$\frac{1}{\sqrt{2\pi}}e^{\frac{x^{2}}{2}}\$

measure for any A-convolution Hint: Use the equation (6.2)

is not the central limit

108
Remark 6.4

In our case A-convolution are commutative but as was shown by Muraki his monotonic

convolution is not comutative.
Remark 6.5

If we take$C(\mu)=\mu\square \delta_{s}=D_{s}(\mu)$i.e. that$\delta_{n}=m_{n}(a))=s^{n}$then we obtain aquite

interesting defomation of the ffae convolution. The case $s=0$ is again Boolean convolution.
The corresponding central limit measure for that convolution is the measure$\mu=\mu_{s}$,forwhich Cauchy transform satisfies the equation.
$G_{\mu}(z)= \frac{1}{z-\frac{1}{s}G_{\mu}(\frac{z}{s})}$If $0\leq s<1$ , then one can verify that our central limit measure$\mu_{l}$is discrete measure

and the orthogonal polynomials with respect that measure satisfy equation:,$xP_{n}(x)=P_{n+1}(x)+s^{2n-2}P_{n-1}(x)$$P_{0}(x)=1$,$P_{1}(x)=x$.

That last fact about orthogonal polynomials is equivalent (by Stieltjes theorem) that the
Cauchy transform ofthe measure$\mu_{s}$has the form:
1$G_{h}(z)=$

1

$z-$

$s^{2}$

$z-$
$z$

$-z$

$- \frac{s_{S^{6}}^{4}}{z-}..$.

We will finish our note with the following problem related to the last example ofconvolution.
Problem 6(i)Calculate the support ofthe measure(ii)

Consider also the case $s>1$ .

$\mu_{s}$for $s<1$ .

109Acknowledgement

The author wish to thank to Professors Nobuaki Obata, Akihito Hora and Kazuhiko
Aomoto for the collaboration in the Joint Research Project “Infinite Dimensional Harmonic
Analysis” under the auspices of JSPS and PAN in the years 2000-2001.
We also thanks professor N.Muraki for many stimulating conversations concerning the
subject ofthe paper and his monotonic independence.
Some parts this note was presented at RIMS workshop at Kyoto (2000) and at IIAS at
Kyoto during workshop on quantum probabiity.
We would like to thank professor T. Eda and professor N. Obata for nice working
atmosphere during their conferences at Kyoto and Nagoya.

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