Operator-Valued Moment Series of the Generating Operator of L(F_2) Over the Commutator Group von Neumann algebra L(K)

0.18 MB·English

by  Ilwoo Cho

OPERATOR-VALUED MOMENT SERIES OF THE GENERATING
OPERATOR OF L(F ) OVER THE COMMUTATOR GROUP VON
5 2
0 NEUMANN ALGEBRA L(K)
0
2
n ILWOOCHO
a
J

4
2
Abstract. In this paper, we will consider the generating operator x = a+
] b+a−1 +b−1 of the free group factor L(F2), where F2 =< a,b > is the
A free group with two generators a and b. Let K =< h > be the commutator
O group of a and b with one generator h = aba−1b−1. Then we can construct
the group von Neumann algebra L(K) and the conditional expectation E :
.
h
at L(F2)→L(K),definedbyE g∈F2αgg!=k∈Kαkk,forallg∈F2αgginL(F2).
m Then(L(F2),E)istheW∗-probaPbilityspacewPithamalgamatiPonoverL(K).In
this paper,wewillcompute the trivialoperator-valuedmomentseriesofthe
[ generatingoperatora+b+a−1+b−1 ofL(F2),overL(K).Thiscomputation

is the good example for studying the operator-valued distribution, since the
1
operator-valued moment series of operator-valued random variables contain
v
algebraic and combinatorial free probability information about the opeartor-
8
valueddistributions.
6
3
1 From mid 1980’s,Free Probability Theory has been developed. Here, the classi-
0 calconceptofIndependenceinProbabilitytheoryisreplacedbyanoncommutative
5 analogue called Freeness (See [9]). There are two approaches to study Free Proba-
0
bility Theory. One of them is the original analytic approachof Voiculescu and the
/
h other one is the combinatorialapproachof Speicher and Nica (See [1], [2] and [3]).
t
a Speicher defined the free cumulants which are the main objects in Combinatorial
m
approach of Free Probability Theory. The free cumulants of random variables are
:
v gottenfromthefreemomentsofrandomvariablesviatheMo¨biusinversion(orvice
i versa). But inthis paper,we willconcentrateonly oncomputing the free moments
X
ofcertainrandomvariables(See[3]). Also,Speicherconsideredtheoperator-valued
r
a free probability theory, which is also defined and observedoriginally by Voiculescu
(See [9]). In this paper, we will observe the important example of such operator-
valued free probability.
LetF beafreegroupwithN-generatorsandletL(F )bethefreegroupfactor
N N
defined by
w
L(F )=C[F ] .
N N
Key words and phrases. FreeGroupFactors,MomentSerieswithAmalgamation,Generating
OperatorsorRadialOperators,TheRecurrenceDiagramforN ∈N..
1
2 ILWOOCHO
Inthispaper,byusingso-calledtherecurrencediagramfoundin[13]and[14],we
willcompute the trivial operator-valuedmomentseries ofthe genaratingoperator
G of the free group factor L(F ), defined by
2
G=a+b+a−1+b−1 ∈L(F ),
2
over the group von Neumann algebra L(K), where F =< a,b > and K =<
2
aba−1b−1 > is the commutator group of a and b in F . i.e,
2
def w
L(K) = C[<aba−1b−1 >] .
Throughoutthis paper,wewillfixaandbasthegeneratorsofthe freegroupF
2
and we will also fix h =aba−1b−1 as the generator of the group K which is group
isomorphictotheintegersZ.NoticethatthecommutatorgroupK isasubgroupof
F andhence the groupvonNeumannalgebraL(K)isa W∗-subalgebraofthe free
2
group factor L(F ). Let x be an operator in L(F ). Then there exists the Fourier
2 2
expansion of x,
x= α u , with α ∈C, for all g ∈F .
g g g 2
g∈F2
P
We can regard all g ∈ F as unitaries u in L(F ). For the convenience, we will
2 g 2
denote these unitaries u just by g. With this notation, it is easy to check that
g
g∗ =u∗g =u−g1 =ug−1 =g−1 in L(F2),
where g−1 is the group inverse of g in F . We can define the conditional expec-
2
tation E :L(F )→L(K) by
2
E α g = α k.
g k
g∈F2 ! k∈K
P P
Then we have the W∗-probability space (L(F ),E) with amalgamation over
2
L(K). Let G be the generating operator, which is also said to be the radial op-
erator of F , a+b+a−1+b−1 in L(F ).
2 2
Now,letB ⊂AbevonNeumannalgebrasandassumethatthereisaconditional
expectationF :A→B.Thenthealgebraicpair(A,F)iscalledtheW∗-probability
space with amalgamation over B (in short, the W∗-probability space over B).
We say that each element a of (A,F) is a B-valued (or operator-valued) random
variable. Voiculescudefinedthe n-thB-valued(or operator-valued)momentofthe
B-valued random variable a by
F (b a...b a), for all n∈N,
1 n
OPERATOR-VALUED MOMENTS OF THE GENERATING OPERATOR OF L(F2) 3
where b ,...,b ∈ B are arbitrary. (Recall that, in [1], Speicher defined the n-
1 n
th moment of a ∈ (A,F) by F (ab a...b a), for arbitrary b ,...,b ∈ B) Different
2 n 2 n
from the scalar-valued case when B = C, since B and A are not cummutative,
in general, we have to consider the case when b ,...,b are not 1 = 1 . When
1 n B A
b = ... = b = 1 , we say that the n-th B-valued moment of a is the trivial n-th
1 n B
B-valued moment of a. However, if B and A are commutative (in other words, if
B =C (B)), then the trivialmomentF(an)contains the full free probability data
A
for the B-valued moments, since
F(b a...b a)=(b ···b )F(an), for all n∈N.
1 n 1 n
Remark that, in our case, L(K) and L(F ) are not commutative, because, for
2
instance,
ah=a2ba−1b−1 6=aba−1b−1a=ha.
Inthispaper,wewillconcentrateoncomputingthetrivialL(K)-valuedmoments
of the generating operator G = a+b+a−1 +b−1 of L(F ). As we mentioned in
2
the previous paragraph, since L(K) and L(F ) are not commutative, these trivial
2
operator-valued moments of G do not contain the complete free probability data
of G. But the computation, itself, provides the way how to compute the operator-
valued moments of G.
It is easy to see that the first, second and third trivial L(K)-moments of G
vanish. i.e,
E Gk =0 , for k =1,2,3,
L(K)
(cid:0) (cid:1)
since Gk does not contain the hn-term, for k = 1,2,3 and for n ∈ Z, where
h = aba−1b−1 and h−1 = bab−1a−1. However, fourth trivial L(K)-moment E(G4)
of G contains the h-term and the h−1-term. So, finding the trivial L(K)-moments
of G is to find the hk-terms of Gn, for all k ∈Z and n∈N.
Thefollowingrecurrencediagramwillplayakeyroletofindsuchtrivialoperator-
valued moment series of the generating operator G of L(F ) ;
2
4 ILWOOCHO
p2 =2N
0
↓
q3 =(2N −1)+2N
1
ււ ցց
p4 p4
2 0
ււ ց ւ
q5 q5
3 1
ււ ց ւ ցց
p6 p6 p6
4 2 0
ււ ց ւ ց ւ
q7 q7 q7
5 3 1
ււ ց ւ ց ւ ցց
p8 p8 p8 p8
6 4 2 0
. . . .
. . . .
. . . .
where
ււ : (2N −1)+[former term]
ց : (2N −1)·[former term]
ւ : ·+[former term]
and
ցց : (2N)·[former term].
In this paper, we obtain good applications about the above recurrence diagram.
Wewillre-computethe momentseriesofthegeneratingoperatorsofthe freegroup
factor L(F ), for all N ∈ N, by using the above recurrence diagram. This would
N
be the one application of this recurrence diagram (See Chapter 1). When N =
2, we can apply this recurrence diagram to compute the trivial operator-valued
moment series of the generating operator of L(F ) (See Chapter 2). Remark that
2
tostudy(scalar-valuedoroperator-valued)momentseriesofelementsinanoperator
algebrais to study (scalar-valuedor operator-valued)free distributions ofelements
in that operator algebra. So, the computations in this paper about generating
operators contain the free probability information about free distribution of those
generating operators. And the free probability information is determined by the
above recurrence diagram.
In Chapter 1, we will re-compute the (scalar-valued) moment series of the gen-
erating operator of L(F ), by using the recurence diagram found in [13] and [14].
N
The momentseriesofthe generatingoperatorofL(F )is alreadyknown,but here
N
we willcompute itagain,by using the aboverecurrencediagram. InChapter 2, by
using the reccurence diagram when N = 2, we will compute the trivial operator-
valued moment series of the generating operator G = a+b+a−1+b−1 of L(F ).
2
Remark that the moment series in Chapter 1 is a scalar-valued moment series
and the trivial operator-valued moment series in Chapter 2 is a operator-valued
(operator-valued) moment series.
OPERATOR-VALUED MOMENTS OF THE GENERATING OPERATOR OF L(F2) 5
1. Moment Series of the Generating Operator of L(F )
N
Let A be a von Neumann algebra and let τ :A →C be the normalized faithful
trace. Then we call the algebraic pair (A,τ), the W∗-probability space and we
call elements in (A,τ), random variables. Define the collection Θ , consists of
s
all formal series without the constant terms in noncommutative indeterminants
z ,...,z (s ∈ N). Then we can regard the moment series of random variables as
1 s
elements of Θ .
s
Definition 1.1. Let (A,τ) be a W∗-probability space with its normalized faithful
trace τ and let a ∈ (A,τ) be a random variale. The moment series of a is defined
by the formal series in Θ ,
1
M (z)= ∞ τ(an)zn .
a n=1
P
The coefficients τ(an) are called the n-th moments of a, for all n∈N.
Let H be a group and let L(H) be a group von Neumann algebra. i.e,
w
L(H)=C[H] .
Precisely, we can regardL(H) as a weak-closure of the group algebra generated
by H and hence
w
L(H)={ t g :g ∈H} .
g
g∈H
P
It is well known that L(H) is a factor if and only if the given group H is icc.
(Sincethe freegroupF withN-generatorsisicc,the vonNeumanngroupalgebra
N
L(F ) is a factor and it is called the free group factor.)
N
Now, define the canonical trace τ :L(H)→C by
τ t g =t , for all t g ∈L(H),
g eH g
g∈H ! g∈H
P P
where e is the identity of the group H. It is easy to check that the trace τ is
H
normalizedand faithful. So, the algebraicpair (L(H),τ) is a W∗-pobability space.
AssumethatthegroupH hasitsgenerators{g : j ∈I}.Wesaythattheoperator
j
G= g + g−1,
j j
j∈I j∈I
P P
6 ILWOOCHO
the generating operator of L(H). For instance, if we have a free group F =<
N
g ,...,g >, then the generating operator of the free group factor L(F ) is
1 N N
g +...+g +g−1+...+g−1.
1 N 1 N
Rest of this chapter, we will consider the moment series of the generating oper-
ator G of L(F ).
N
From now, fix n∈N. And we will denote free group factor L(F ) by A. i.e
N
w
A={ t g :t ∈C} .
g g
g∈FN
P
Recall that there is the canonical trace τ :A→C defined by
τ t g =t ,
g e
g∈FN !
P
where e ∈ F is the identity of F and hence e ∈ L(F ) is the unity 1 .
N N N L(FN)
The algebraic pair (L(F ),τ) is a W∗-probability space. Let G be the generating
N
operator of L(F ). i.e
N
G=g +...+g +g−1+...+g−1,
1 N 1 N
where F =< g ,...,g > . It is well-known that if we denote the sum of all
N 1 N
words with length n in {g , g−1 , ..., g , g−1} by
1 1 N N
X = w ∈A, for all n∈N,
n
|w|=n
P
then the following two recurrence relations (1.1) and (1.2) hold true, whenever
N ≥2 ;
(1.1) X X =X +2N ·e (n=1)
1 1 2
and
(1.2) X X =X +(2N −1)X (n≥2)
1 n n+1 n−1
(See [15]). In our case, we can regardour generating operator G as X in A, by
1
the very definition of G.
By using the relation (1.1) and (1.2), we can express Gn in terms of X ’s ; For
k
example, G=X ,
1
OPERATOR-VALUED MOMENTS OF THE GENERATING OPERATOR OF L(F2) 7
G2 =X X =X +2N ·e,
1 1 2
G3 =X ·X2 =X (X +(2N)e)=X X +(2N)X
1 1 1 2 1 2 1
=X +(2N −1)X +(2N)X =X +((2N −1)+2N)X ,
3 1 1 3 1
G4 =X +((2N −1)+(2N −1)+2N)X +(2N)((2N −1)+(2N))e,
4 2
G5 =X +((2N −1)+(2N −1)+(2N −1)+2N)X
5 3
+((2N −1)((2N −1)+(2N −1)+(2N))+(2N)((2N −1)+(2N)))X ,
1
G6 =X +((2N −1)+(2N −1)+(2N −1)+(2N −1)+2N)X
6 4
+{(2N −1)((2N −1)+(2N −1)+(2N −1)+(2N))
+(2N −1)((2N −1)+(2N −1)+(2N))
+(2N −1)((2N −1)+(2N))}X
2
+(2N)((2N −1)((2N −1)+(2N −1)+(2N)))+(2N)((2N −1)+(2N)))e,
etc.
So, we can find a recurrence relation to get Gn (n ∈ N) with respect to X ’s
k
(k ≤n). Inductively, G2k−1 andG2k have their representationsin terms of X ’s as
j
follows ;
G2k−1 =X2k−1 =X +q2k−1X +q2k−1X +...+q2k−1X +q2k−1X
1 2k−1 2k−3 2k−3 2k−5 2k−5 3 3 1 1
and
G2k =X2k =X +p2k X +p2k X +...+p2kX +p2ke,
1 2k 2k−2 2k−2 2k−4 2k−4 2 2 0
where k ≥2. Also, we have the following recurrence relation, by the straightfor-
ward computation ;
Proposition1.1. Let’s fix k ∈N\{1}.Let q2k−1 and p2k (i=1,3,5,...,2k−1,....
i j
and j =0,2,4,...,2k,...)be given as before. If p2 =2N and q3 =(2N−1)+(2N)2,
0 1
then we have the following recurrence relations ;
(1) Let
G2k−1 =X +q2k−1X +...+q2k−1X +q2k−1X .
2k−1 2k−3 2k−3 3 3 1 1
Then
G2k =X + (2N −1)+q2k−1 X + (2N −1)q2k−1+q2k−1 X
2k 2k−3 2k−2 2k−3 2k−5 2k−4
(cid:0) (cid:1) (cid:0) (cid:1)
+ (2N −1)q2k−1+q2k−1 X +
2k−5 2k−7 2k−6
(cid:0) (cid:1)
+...+ (2N −1)q2k−1+q2k−1 X +(2N)q2k−1e.
3 1 2 1
(cid:0) (cid:1)
8 ILWOOCHO
i.e,
p2k =(2N −1)+q2k−1,
2k−2 2k−3
p2k =(2N −1)q2k−1+q2k−1,
2k−4 2k−3 2k−5
........,
p2k =(2N −1)q2k−1+q2k−1
2 3 1
and
p2k =(2N)q2k−1.
0 1
(2) Let
G2k =X +p2k X +...+p2kX +p2ke.
2k 2k−2 2k−2 2 2 0
Then
G2k+1 =X + (2N −1)+p2k X + (2N −1)p2k +p2k X
2k+1 2k−2 2k−1 2k−2 2k−4 2k−3
(cid:0) (cid:1) (cid:0) (cid:1)
+ (2N −1)p2k +p2k X +
2k−4 2k−6 2k−5
(cid:0) (cid:1)
+...+ (2N −1)p2k+p2k X + (2N −1)p2k+p2k X .
4 2 3 2 0 1
i.e,
(cid:0) (cid:1) (cid:0) (cid:1)
q2k+1 =(2N −1)+p2k ,
2k−1 2k−2
q2k+1 =(2N −1)p2k +p2k ,
2k−3 2k−2 2k−4
......,
q2k+1 =(2N −1)p2k+p2k
3 4 2
and
q2k+1 =(2N −1)p2k+p2k.
1 2 0
(cid:3)
Example 1.1. Suppose that N = 2. and let p2 = 4 and q3 = 3+p2 = 3+4 = 7.
0 1 0
Put
G8 =X +p8X +p8X +p8X +p8e.
8 6 6 4 4 2 4 0
Then, by the previous proposition, we have that
p8 =3+q7, p8 =3q7+q7, p8 =3q7+q7 and p8 =4q7.
6 5 4 5 3 2 3 1 0 1
Similarly, by the previous proposition,
q7 =3+p6, q7 =3p6+p6 and q7 =3p6+p6,
5 4 3 4 2 1 2 0
p6 =3+q5, p6 =3q5+q5 and p6 =4q5,
4 3 2 3 1 0 1
q5 =3+p4 and q5 =3p4+p4,
3 2 1 2 2
OPERATOR-VALUED MOMENTS OF THE GENERATING OPERATOR OF L(F2) 9
p4 =3+q3 and p4 =4q3,
2 1 0 1
and
q3 =3+p2 =7.
1 0
Therefore, combining all information,
G8 =X +22X +202X +744X +1316e.
8 6 4 2
We have the following diagram with arrows which mean that
ււ : (2N −1)+[former term]
ց : (2N −1)·[former term]
ւ : ·+[former term]
and
ցց : (2N)·[former term].
p2 =2N
0
↓
q3 =(2N −1)+2N
1
ււ ցց
p4 p4
2 0
ււ ց ւ
q5 q5
3 1
ււ ց ւ ցց
p6 p6 p6
4 2 0
ււ ց ւ ց ւ
q7 q7 q7
5 3 1
ււ ց ւ ց ւ ցց
p8 p8 p8 p8
6 4 2 0
. . . .
. . . .
. . . .
Notation From now, we will call the above diagram the recur-
rence diagram for N. (cid:3)
For examplet, when N =2, we can compute p6, as follows ;
4
p2 =4, q3 =7,
0 1
p4 =3+7=10, p4 =28,
2 0
q5 =3+10=13, q5 =3·10+28=58.
3 1
10 ILWOOCHO
and hence p6 =3+13=16.
4
By the recurrence diagram for N, we can get that ;
Theorem 1.2. Let G∈(A,τ) be the generating operator. Then the moment series
of G is
0 if n is odd
τ(Gn)=

pn if n is even,
 0

for all n∈N.
Proof. Assume that n is odd. Then
Gn =X +qn X +...+qnX +qnX .
n n−2 n−2 3 3 1 1
So, Gn does not contain the e-terms. Therefore,
τ(Gn)=τ X +qn X +...+qnX +qnX =0.
n n−2 n−2 3 3 1 1
(cid:0) (cid:1)
Assume that n is even. Then
Gn =X +pn X +...+pnX +pne.
n n−2 n−2 2 2 0
So, we have that
τ(Gn)=τ X +pn X +...+pnX +pne =pn.
n n−2 n−2 2 2 0 0
(cid:0) (cid:1)
Remark that the n-th moments of the generating operator in (A,τ) is totally
depending on the recurrence diagram for N.
The moments ofthe generatingoperatorG of A=L(F ) is well-know. But the
N
abovetheoremprovidesthemethodhowtogetthemomentsofgeneratingoperator
G by using the recurrence diagram.
RecallthatNicaandSpeicherdefinedtheevenrandomvariableina∗-probability
space. Let(B,τ )bea∗-probabilityspace,whereτ :B →Cisalinearfunctional
0 0
satisfying that τ (b∗) = τ (b), for all b ∈ B, and let b ∈ (B,τ ) be a random
0 0 0
variable. We say that the random variable b ∈ (B,τ ) is even if it is self-adjoint
0
and it satisfies the following moment relation ;
τ (bn)=0, whenever n is odd.
0