Title Deformed free probability of Voiculescu (Infinite DimensionalAnalysis and Quantum Probability Theory)

Author(s) Bozejko, Marek

Citation 数理解析研究所講究録 (2001), 1227: 96-113

Issue Date 2001-08

URL http://hdl.handle.net/2433/41406

Right

Type Departmental Bulletin Paper

Textversion publisher

Kyoto University

Deformed free probability ofVoiculescu

MAREK BO\.{Z}EJKO*

Institute Mathematics, Wroclaw UniversityPl. Grunwaldzki 2/4, 50384 Wroclaw, Poland

Abstract. We introduce r-ffee product of states on the ffoe product ofC*

-algebras and r-freeconvolution 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. $\mathrm{r}$-free product $(0\leq r\leq 1)$ of states.

$\mathrm{a}$ . $r=1$ -free product ofVoiculescu$\mathrm{b}$ . $r=0$ -Boolean product

3. $r$-Fock Space and $r$-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.

.Partially supported by KBN grant $2\mathrm{P}\mathrm{O}3\mathrm{A}05415$

数理解析研究所講究録 1227巻 2001年 96-113

96

1. Introduction

As each discrete group $G$ with $N$ generators is ahomorphism image ofthe ffee group $\mathrm{F}_{\mathrm{N}}$ in

the same manner we would like to say that $\mathrm{e}\mathrm{a}\mathrm{c}\mathrm{h}" \mathrm{n}\mathrm{a}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{a}\mathrm{l}$ ”probability is adeformation of

the free probability Voiculescu. In the papers [BS] [BKS] we considered defomed

classical probability and we get so called $q$-deformed Fock space, $q$-second quantization

and $q$-Gaussian processes. In this note we propose some versions of deformation ofthe

ffee probability Voiculescu using our technique coming ffom the conditional ffoe

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

for $r=1$ the ffee probability and for $r=0$ the Boolean probability.

One ofthe main result this paper is the construction on the ffee product ofnon-unital$\mathrm{C}^{*}$-algebras4with states $\varphi_{i}$ : $4arrow \mathbb{C}$ , (we recall that by astate on anon-unital algebraswe mean positive ffinctional norm 1), anew examples of states $\varphi:*4arrow \mathbb{C}$ such that

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

$\mathrm{i}\mathrm{i}$ . (Voiculescuproperty If $\varphi(a_{l})=0$ for $i=1$ , $\ldots$ , $n$, and$a_{j}\in 4_{j}$ , $i_{1}\neq i_{2}\neq\ldots$ ,

then $\varphi(a_{1}a_{2}\ldots a_{n})=0$

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

Using the construction $\mathrm{o}\mathrm{f}r$-ffee product of states $(0\leq r\leq 1)$ we can form the r-ffeeconvolution ofprobability measures on R. Then we introduce the analogue of $\mathrm{R}\{\mathrm{j}\mathrm{j})-\mathrm{R}-$

transform. The main ideas comes ffom the our paper $[\mathrm{B}\mathrm{L}\mathrm{S},\mathrm{B}\mathrm{W}2]$ .

As an example of application $\mathrm{o}\mathrm{f}\mathrm{R}$-transfom we obtain central limit theorem for r-convolution. Our central limit measure $\mu$, is the “symmetrization” ofthe MarcenkO-Pasturmeasure (the ffee Poisson measure) which Cauchy transform is ofthe $\mathrm{f}\mathrm{o}\mathrm{m}$ :

1$G_{h}(z)=$ 1

$z-$$r$

$z-$ 1$z-$

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

97

which is 2-periodic continued $\mathrm{f}\mathrm{f}\mathrm{f}\mathrm{a}\alpha \mathrm{i}\mathrm{o}\mathrm{n}$ and the measure $\mu_{r}$ is supported on two intervals if05 $r<1$ .

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 wehave 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. $r$-Free Product of StatesLet 4be anon-unital $\mathrm{C}^{*}$-algebra4with states $\varphi_{i}$ : $4arrow \mathbb{C}$ . Let $\tilde{4}$ be the unitalization of4(i.e. $\tilde{4}=4$ $+\mathbb{C}1$ ) and we define the extension of $\varphi_{t}$ as $\overline{\varphi_{i}}(1)=1,\tilde{\varphi_{i}}|=\varphi_{i}4^{\cdot}$ Moreover let

define anew state $\psi_{i}=r\varphi_{\mathrm{i}}+(1-r)\delta_{1}$ where $\delta_{1}$ is the $\mathrm{f}\mathrm{f}\mathrm{i}\mathrm{n}\alpha \mathrm{i}\mathrm{o}\mathrm{f}\mathrm{f}\mathrm{i}$ defined as

$\delta_{1}(x)=\{$

0if $x$ $\neq\lambda 1$

$\lambda$ if $x$ $=\lambda 1$

then $\psi_{i}$ is also astate on unital algebra 4and we can fom the conditional ffee product state$\tilde{\varphi}$ on the ffee product $\mathrm{C}^{\cdot}$-algebra A$\sim=*\tilde{4.}=\overline{*4}$ :

$\tilde{\varphi}^{=}*(\tilde{\varphi_{i}},\psi_{i})$ .

By [BLS] we knew that $\tilde{\varphi}$ is astate on $\mathrm{C}^{\cdot}$-algebra A$\sim$

. Hence also we get state $\varphi=\tilde{\varphi}|_{A}$ o$\mathrm{n}$

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

(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:$\vee$ $\vee$

(iii)$\varphi(a_{1}a_{2}\ldots a_{n})=r\sum_{J}\varphi(a_{j})\varphi(a_{1}\ldots a_{j}\ldots a_{n})-r^{2}\sum_{i<j}\varphi(a_{i})\varphi(a_{j}w(a_{1}\ldots a_{i}\ldots a_{j}\ldots a_{n})$

$+\ldots+[(-1)^{n+1}r^{n}+(1-r)^{n}]\varphi(\mathrm{q})\ldots\varphi(a_{n})$ .

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

The most natural state on the group algebra ofthe ffae group $\mathrm{F}_{\mathrm{N}}$ with the ffee generators$x_{1}$ , $x_{2},\ldots$ , $x_{N}$ is the Haagerup stat

$H_{q}(g)=q^{l(g)}$ , $g\in \mathrm{F}_{\mathrm{N}}$

98

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

$x\ovalbox{\tt\small REJECT}_{X_{\ovalbox{\tt\small REJECT}}}^{\ovalbox{\tt\small REJECT}};:$; \ldots $x_{::}$ , 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 ofC’algebras and$j_{\ovalbox{\tt\small REJECT}}\mathit{1}$

$H_{q}=P_{q\mathrm{o}}*\ldots*{}_{0}P_{q}$

is the Boolean ffee product, where $P_{q}(n)$ $=d]$ , $(n\in \mathbb{Z})$ is the classical Fourier transform ofthePoisson kernel.One can see that in the case $r=1$ our construction give Voiculescu ffae product of states inthe case when the algebras 4are unital.

Remark 2.1. If $(A,\varphi)=*,(4,\varphi_{i})$ is the rfree $pr\ovalbox{\tt\small REJECT} ct$ as defined above then $\mathrm{i}\mathrm{f}a_{i}$ are indifferent algebras $A_{i}$, then $\varphi(\mathrm{q}a_{2}\ldots a_{n})=\varphi(\mathrm{q})\varphi(a_{2})\ldots\varphi(a_{n})$ .

Moreover if 4, $a_{2}\in 4$ , $b\in A_{j}$ , $i\neq j$ , 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 $r$-ffoe product is notassociative $\mathrm{i}.\mathrm{e}$ . if $(\varphi_{1r}*\varphi_{2})_{r}*\varphi_{3}=\varphi$ $*r(\varphi_{2r}*\varphi_{3})$ then $r=0$ or $r=1$ . $[]$

Remark 2.3. From the formula (iii) we see that the $r$-ffoe product of states $(\varphi^{(r)}=*,\varphi_{i})$ hasVoiculescuproperty:

If $\varphi(a_{j})=0$ for allj and $a_{j}\in 4_{j}$ , $\mathrm{t}$ $\neq i_{2}\neq\ldots$ , then $\varphi(a_{1}a_{2}\ldots a_{n})=0$

Also for $r\neq 1\varphi^{(\prime)}$ is different ffom the ffee product ofVoiculescu.Problem 1. Find other examples of states $F$ on $*(4, \varphi_{i})$ such that:

(i) $F|_{A}=\varphi_{i}$

(ii) $\mathrm{F}$ satisfies Voiculescu property

Problem 2. $\mathrm{I}\mathrm{f}r$-free product of states is again astate for r $>1$ ?

2. $r$-Fock space and $\mathrm{r}$-Gaussian random variables

Let $H$ be areal Hilbert space and $H_{\mathrm{C}}\mathrm{w}\mathrm{i}\mathrm{U}$ be its complexification. We define the ffee Fock

space $\mathrm{F}(\mathrm{H}\mathrm{C})$ $\mathbb{C}\mathrm{O}\mathrm{O}\bigoplus_{n=1}^{\infty}H_{\mathrm{c}}^{\otimes n}$ . Now we make deformation ofthe scalar product as follows:

For $x_{n},y_{n}\in H_{\mathbb{C}}^{\otimes n}$ we put$\langle x_{n},y_{n}\rangle,$ $=r^{k}\langle x_{n},y_{n}\rangle$ ifn $=2k$ or $n=2k+1$ , $k=0,1,2,3$, $\ldots$ .Moreover $\langle \mathrm{q}\Omega\rangle_{r}=\langle \mathrm{q}\Omega\rangle=1$ .

We can see ( $\mathrm{x},$$x\rangle_{r}=\langle x,x\rangle$ for $x\in H$. The completion $\mathrm{o}\mathrm{f}F(H_{\mathrm{c}})$ with respect the scalar

product $\langle,\rangle_{r}$ we called $r$-Fock space and will be denoted $F(H, r)$. Moreover $\mathrm{f}\mathrm{o}\mathrm{r}f\in H$ wedefine the $r$-creation operation $A^{+}\zeta f$)$x_{1}\otimes\ldots\otimes x_{n}=f\otimes\triangleleft\otimes\ldots\otimes x_{n}$ and the r-annihilationoperator $A(f)$ such that $\mathrm{A}(\mathrm{f})\mathrm{Q}$ and $A(f)\eta\otimes\ldots\otimes x_{n}=\lambda_{n}\langle f,x_{1}\rangle_{h}\otimes\ldots\otimes\chi_{n}$ ,

99

,.$\ovalbox{\tt\small REJECT}$ $\mathrm{C}$

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

if $n\ovalbox{\tt\small REJECT}$ 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 $\mathrm{X}(\mathrm{N})$ @ $\ldots$$x_{n}=\lambda_{n}x_{1}\otimes\ldots x_{n}$ .

(iv) $\mathrm{I}\mathrm{f}P$ is the orthogonal projection ofF(H, $r$) onto ,$\bigoplus_{\triangleright 0}^{\infty}H^{\emptyset 2n}$ , then

$\mathrm{X}(\mathrm{N})=rP+(I-P)=I+(r-1)P$ .

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

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

$=| \sum a_{l}a_{j}.|||\lambda(N)[$ .

Since $\lambda(N)$ is the diagonal operator, therefore $[ \lambda(N)||=\max(1,r)$ .

Now we define $\mathrm{r}- Msg\dot{|a}l$ random variables. For $\mathrm{f}\in H$ $G(f)=A(f)+A^{+}(f)$ and forbounded operator $T$ on $\mathrm{f}(H, r)$ we define the vacuum state $\epsilon(T)=\{T\mathrm{q}\Omega\rangle$ .

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, $\ldots$ ,

$2\mathrm{n}\}$ ,$e(V)=\#$ {$B_{j}\in V$ : $d_{v}(B_{j})$ is even number}. Here $d\langle B_{j}$) is the depth ofthe block $B_{j}$ in thepartition $V$ as was defined in $[]$ .

Theorem 3.3. $\mathrm{I}\mathrm{f}f_{j}\in H$ then(3.1) $\epsilon(G(f_{1})G\mathrm{C}f_{2})\ldots G(f_{2n}))=\sum_{V\in NC_{2}(1\ldots,2n)}\langle f_{\mathrm{i}},f_{j_{1}}\rangle\ldots\langle f_{j_{*}},f_{j_{*}}\rangle r.(V)$.

The proof of the formula (3.1) folows ffom general result which was proven by us in

the paper with Accardi [AB]

100

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 $\mathrm{i}\mathrm{f}||f||=1$ , we see that the distribution ofthe Boolean Gaussian random variables $G\eta$

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 $r$-free Gaussian random variables.Moreover in the Boolean case we have much more that corollary 3.2.

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:

$(\begin{array}{ll}0 T\Gamma 0\end{array})(\begin{array}{ll}0 TT^{*} 0\end{array})=(\begin{array}{ll}\Pi^{*} 00 T^{*}T\end{array})$

and $T= \sum a_{i}\otimes G_{i}=\{\begin{array}{llll}0 \mathrm{q} .\cdot a_{n}\dot{\mathrm{q}} \vdots 0 a_{\dot{n}} \end{array}\}$

0 $a_{1}$...

$a_{n}$

$a\mathrm{i}$

$.\cdot$

.$a_{\dot{n}}$

0

Problem 3Let VNO(N) $=\mathrm{I}_{r}(G_{1}, \ldots, G_{n})$ will be the von Neumann algebra generated by$G_{1},\ldots,G_{n}$ in the $r$-Fock space $F(H, r)$ .

$\mathrm{I}\mathrm{f}r=0$ , then VNO(N) $=M_{N}(\mathbb{C})$ .If $\mathrm{r}=1$ , then $\mathrm{V}\mathrm{N}\mathrm{i}(\mathrm{N})$ is the ffoe group $\mathrm{f}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{o}\mathrm{r}-\mathrm{V}\mathrm{N}(\mathrm{F}_{\mathrm{N}})$.Try to verify if $\mathrm{V}\mathrm{N}\mathrm{r}(\mathrm{N})$ is also afactorial von Neumann algebra for $0<r<1$ .When does exist a $\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$ on $mXN$)?

3. $r$-Free Convolution ofProbability Measures on R.

In this section we will work mainly with probability measures $\mu$ on $\mathbb{R}$ with compact support$(\mu\in \mathcal{F}^{\mathrm{p}})$ . Let

$m_{k}( \mu)=\int_{\mathrm{R}}x^{k}d\mu(x)$ , $\mathrm{k}=0,1,2,\ldots$

101

and we treat the measure $\mu$ as astate on the algebra ofpolynomials $\mathbb{C}\langle X\rangle:\mu[X^{k}]=\mu_{k}(\mu)$ .

Ifwe take two probability measures $l4,\mu_{2}\in P$ we define their$r$-ffee convolution $(n^{[eggR]}\mu_{2})$ as follows:

(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 $r$-free product of states on the algebra ofnon-commutative polynomials$\mathbb{C}\langle X_{1},X_{2}\rangle$ .

On the other hand using our conditionally ffee product ofpairs probability measure as wasdone in [BLS]. The $r$ convolution measure $\mathrm{A}$ , $\mu_{2}$ is the measure $\mu$ denoted as $n^{[eggR]}\mu_{2}$

can be obtained in the following way:$(\mathrm{A}, V_{r}(\mathrm{A}))\mathrm{f}\mathrm{f}1(\mu_{2},V,(\mu_{2}))=(\mu,v)$ , where $\nu$ is the Voiculescu ffee product$V,(/\eta)\mathrm{f}\mathrm{f}1V_{r}(\mu_{2})=\nu$ .

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 $R$-transform $R_{\mu}(k)=R(\mu,V_{r}(\mu)\mathrm{X}k)$ we have the

following formula for calculation ofmoments for $\mu\in P$ :

(4.2) $\int x^{n}d\mu(x)=\sum_{V\in \mathfrak{X}(n)}R_{l’}(V)r^{*(\psi)}$,

where $R_{\mu}(V)= \prod_{B\in V}R_{\mu}(\# B)$ and $e$ is asuitable function on the set ofnon crossing partitions

$NC(\mathit{2}n)$ .The important property ofthe function $e$ 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]

(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 $r$-ffoe convolution ofprobability measure is associative.Moreover if $\delta_{x}$ is Dirac measure at point $x\in \mathbb{R}$ , then $\delta_{x}\copyright\delta_{y}=\delta_{(x+y)}$ .

Problem 4

From theorem (3.3) we know that for 2-n0n-cr0ssing partition $V$,$\mathrm{e}(\mathrm{V})$ {$B\in V:d_{V}(B)$ is even}. Find description ofthe ffinction for $\underline{\mathrm{a}\mathrm{u}}$ non-crossingpartitions.

After this consideration we can now formulate our result:Proposition 4.1

If $\mu\in P$ and for $z\in \mathbb{C}^{+}=\{z\in \mathbb{C}:\mathrm{h}\mathrm{n}(z)>0\}$ then

(4.4) $\frac{1}{G_{\mu}(z)}=z-R_{\mu}(ae_{\mu}(z)+(1-r)\frac{1}{z})$, where

102

$G.(z)\ovalbox{\tt\small REJECT}$

$\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}$, $R_{\ovalbox{\tt\small REJECT}}\ovalbox{\tt\small REJECT}^{)}(zi^{\ovalbox{\tt\small REJECT}}$ $R.(z)\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT} R.(k)z$’

z-x $1\ovalbox{\tt\small REJECT} 1$

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

$\frac{1}{G_{\mu}(z)}=z-R_{\mu}^{(0)}(\frac{1}{z})$

4. Central Limit Theorem

This section is devoted to the main result ofthis paper.Theorem 5.1

Let $X_{j}=X_{i}^{\cdot}\in(A,\varphi)$ , where A is a $C^{*}$ algebra with astate $\varphi$ and $X_{1}$ , $X_{2},\ldots$ are freerandom variables in the probabiistic system $(A,\varphi)$ . That means that $A=*r4$ , $\varphi^{=}*\varphi_{j}r$ and$X_{i}=X_{i}^{\cdot}\in(4,\varphi_{i})$ . Assume that:

(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 $\mu_{r}$ has the following continued ffaction $\mathrm{f}\mathrm{o}\mathrm{m}$:

1$G_{\mu}(z)=$ 1

$z-$$r$

$z-$ 1$z-$

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

103

Proof. The limit measure $\mu$, is such that(5.1) $R_{\hslash}^{(r)}(z)=z$ .

The argument is almost the same as in the proofofthe ffee probabiity central limit theoremso we omit it (see [VDN]).

Hence the Cauchy transform $G(z)=G_{h}(z)$ measure $\mu_{r}$ satisfies the equation:

(5.2) $\frac{1}{G(z)}=z-(rG(z)+(1-r)\frac{1}{z})$ .

1Now 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

$\eta \mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$ :(5.3) $zrH^{2}+(1-z^{2}-r)H+z=0$ .

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 $\mu_{r}$ . For this let us observe thatthe folowing $\hslash\alpha$ holds:

(5.4) $m_{2n}( \mu,)=\frac{1}{r}m_{n}(p,)$ ,

where $n>0$ and $p_{r}$ is the ffee Poisson measure with intensivity $r$.To show (5.4) let us recall that

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

For the proof that fact see $[\mathrm{S}\mathrm{p}1,\mathrm{B}\mathrm{L}\mathrm{S}]$ . On the other side let us calculate the moments of ourlimit measure $\mu_{r}$ and the ffae Poisson measure $p_{r}$ using our theorem ffom [AB]:

$m_{2}(\mu,)=1$ $n\}(p_{r})=r$

$m_{4}(\mu_{r})=1\dagger r$ $m_{2}(p_{r})=r(1+r)$

$m_{6}(\mu,)=1+3r\dagger r^{2}$ $m_{6}(p_{r})=r(1+3r+r^{2})$

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

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Lemma 5.2 Let $f\in L^{1}(\mathbb{R})$ , and $\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}ff\succ^{-}\mathrm{I}\subset \mathbb{R}^{+}$ and

$\tilde{f}(x)=\frac{1}{2}(f(x)\chi_{l}(x)+f(-x)\chi_{(-l)}(x))$

then the Cauchy transform of $\tilde{f}$ is ofthe $\mathrm{f}\mathrm{o}\mathrm{m}$ :

(5.7) $G_{\tilde{f}}(z)=zG_{F}(z^{2})$ , where $F(x)= \frac{f(\sqrt{x})}{2\sqrt{x}}$ .

Since $P,$ $=(1-r)\delta_{0}+F_{r}(x)\$ , where $0\leq r\leq 1$ .This implies that:

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

Therefore by lemma 5.2 we get that

(5.9) $\mu_{r}=\tilde{f}_{r}(x)\ = \frac{1}{2}(f(x)\chi_{I}(x)+f(-x)\chi_{(-I)}(x))$ &.

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 $/\mathrm{r}(\mathrm{x})$ $2xF_{r}(x^{2})= \frac{1}{\pi x}\sqrt{4r-(x^{2}-(1+r))^{2}}$

and $\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}f_{r}=I_{r}$, where $I_{r}$ is interval ofthe form $I_{r}=[1-\sqrt{r},1+\sqrt{r}]$ , therefore this completesthe proofof theorem 5.1.

Remark 5.3 Ifr $=1$ , we have $f_{1}(x)$ $= \frac{1}{\pi}\sqrt{4-x^{2}}$ .

Therefore $\mu_{1}=\frac{1}{2\pi}\sqrt{4-x^{2}}\chi_{[-2,2]}$ &-so this is semicircle low ofWigner (free Gaussian

random variables).

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

5. Remarks on Muraki-Lou convolution and A-convolution.

In this chapter we present some generalization $\mathrm{o}\mathrm{f}r$-ffee convolution ofprobability measures.For this let $C$ :Prob(E) Prob(E) will be some map and Prob(E) is the set of allprobability measures on the real line. We will define $C$-ffee convolution ofmeasures asfollows:

(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 ofmeasure is conditionallyfiee convolution ([BLS]).

The formula (6.1) defines $C$-ffoe convolution of $\mu_{1}[eggc]\mu_{2}=\mu$ .

105

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 methodwe obtain again $r$-free convolution. Here $\square$ denote the multiplicative convolution ofprobability measures on real line.Another example ofdeformed ffee convolution was presented by Wysoczanski and myself(see $[\mathrm{B}\mathrm{W}1,\mathrm{B}\mathrm{W}2]$). This corresponds to C-convolution, where $C=U_{t}(t \geq 0)$ is defined by the

$\eta \mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$

$\frac{1}{G_{\mu(t)}(z)}=\frac{t}{G_{\mu}(z)}+(1-t)z$ , where $\mu(t)=U_{t}(\mu)\mathrm{N}\mathrm{C}(\mathrm{n})$ . In that example the central

limit measure $K_{t}$ is the Kesten measure which is the spectral measure for the random walks on

the ffoe group $\mathrm{F}_{N}$ and the parameter $t=1- \frac{1}{2N}$ . The Cauchy transfom ofthe measure $K_{t}$ has

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

-convolution which corresponds to the map $C$ Prob(R) $arrow$ Prob(R) done by themultiplicative convolution $\square$ on the real he by the suitable measure

$a),\cdot$ so we define $C(\mu)=\mu$ $[]$ $\varpi$ or in another words if $\delta_{n}=\int x^{n}d\varpi(x)$ , then$m_{n}(C(\mu))=\delta_{n}m_{n}(\mu)$ , $n=0,1$ , $\ldots$ . In that case our A-convolution is associative, since wehave $R-\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{s}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}-R_{\mu}^{\Delta}=R^{\Delta}(\mu)$ which make linearization of our $C$-convolution. That exactly

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:

$\int_{\mathrm{B}}x^{n}d\mu(x)=\sum_{\nu^{\mathrm{r}}\epsilon \mathfrak{X}(n)}R_{\mu}^{\Delta}(V)t(V,\mathrm{A})$

for proper $\mathrm{f}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{c}\dot{\mathrm{u}}\mathrm{o}\mathrm{n}$ $t(\cdot,\mathrm{A})$ on non-crossing partition set $\mathrm{N}\mathrm{C}(\mathrm{n})$ Now we can present ageneralization of central limit theorem for $\mathrm{A}-\omega \mathrm{n}\mathrm{v}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$ .

We recall that dilatation $D_{s}$ ofthe measure $\mu$ is defined as $D_{s}( \mu\cross E)=\mu(\frac{1}{s}E)$ for Borel set

$E\subset \mathbb{R}$ and $s\neq 0$ .Theorem 6.1

Let $\mu_{l}\in \mathrm{P}\mathrm{r}\mathrm{o}\mathrm{b}(\mathrm{R})$ and all moments ofmeasures $\mu_{i}$ are finite. Assume that

(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 $\mathrm{a}\mathrm{U}\mathrm{j}$ ,

106

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

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

)$Czz_{N}$ ) weakly tends to limit measure $\#$

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

1$D_{s}(\ovalbox{\tt\small REJECT} \mathrm{u}\mathrm{X}\mathrm{E})\ovalbox{\tt\small REJECT}$ $\mathrm{p}(,\mathrm{A}\ovalbox{\tt\small REJECT})$ 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

Ifwe take as ameasure $d\varpi(x)=|x|\chi_{[-1,1]}$ &then the corresponding A-convolution is relatedto the convolution discovered by Muraki-Lou and the central limit measure is the arcsinus low

$\frac{1}{\pi}\frac{1}{\sqrt{2-x^{2}}}$ &.

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

$\mu=\frac{1}{\pi\sqrt{2-x^{2}}}$ &. Moreover

1$G_{\mu}(z)=$

1$z-$

$\gamma_{2}$

$z-$

$z-$$\gamma_{2}$

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

.and

1$G_{\mathrm{C}(\mu)}(z)=$

$\gamma_{2}$

$z-$$\gamma_{2}$

$z-$

$z-$$\gamma_{2}$

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

.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}}\$ is not the central limit

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

107

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 $\mu=\mu_{s}$ , for

which 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 of

convolution.

Problem 6

(i) Calculate the support ofthe measure $\mu_{s}$ for $s<1$ .

(ii) Consider also the case $s>1$ .

108

Acknowledgement

The author wish to thank to Professors Nobuaki Obata, Akihito Hora and KazuhikoAomoto for the collaboration in the Joint Research Project “Infinite Dimensional HarmonicAnalysis” under the auspices ofJSPS and PAN in the years 2000-2001.

We also thanks professor N.Muraki for many stimulating conversations concerning thesubject ofthe paper and his monotonic independence.

Some parts this note was presented at RIMS workshop at Kyoto (2000) and at IIAS atKyoto during workshop on quantum probabiity.

We would like to thank professor T. Eda and professor N. Obata for nice workingatmosphere during their conferences at Kyoto and Nagoya.

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