Operator-valued free Fisher information of random matrices

0.18 MB·

by  Bin Meng

OPERATOR-VALUED FREE FISHER INFORMATION OF
RANDOM MATRICES
6
0
0
2 BINMENG

n
a Abstract. We study the operator-valued freeFisher information of random
J matricesinanoperator-valuednoncommutative probabilityspace. Weobtain
2 aformulaforΦ∗M2(B)(A,A∗,M2(B),η), whereA∈M2(B)isa2×2operator
2 matrixonB,andη islinearoperators on M2(B). Thenweconsider aspecial
setting: A is an operator-valued semicircular matrix with conditional expec-
] tation covariance, and find that Φ∗B(c,c∗ : B,id) = 2Index(E), where E is a
A conditionalexpectationofBontoDandcisacircularvariablewithcovariance
O E.
.
h
t
a
m
1. Introduction and preliminaries
[

1 Free probability theory is a noncommutative probability theory where the clas-
v sical concept of independence is replaced by the notion of ”freeness”. This theory,
7 due to D.Voiculescu, has very important applications on operator algebras (see
2 [6, 21])
5
Originally, a noncommutative probability space is a pair ( ,τ), where is a
1 A A
C∗ or von Neumann algebra and τ is a state on . Free independence is defined
0
− A
6 in terms of reduced free product relation given by τ. This notion was generalized
0 byD.Voiculescuandotherstoanalgebravaluednoncommutativeprobabilityspace
h/ whereτ is replacedby a conditionalexpectationEB ontoa subalgebra of , and
B A
t freeness is replaced by freeness with amalgamation.
a
m
Definition 1. [20] Let be a unital algebra over C, and let be a subalgebra of
A B
: , 1 . E : is a conditional expectation, i.e. a linear map such that
v B
A ∈ B A −→ B
i EB(b1ab2)=b1EB(a)b2, EB(b)=b, for any b,b1,b2 , a . We call ( ,EB, )
X ∈B ∈A A B
an operator-valued (or valued) noncommutative probability space and elements
r in are called ranBdo−m variables.
a A B−
The algebra freely generated by and an indeterminate X will be denoted by
B
[X]. The distribution ofa is a conditional expectation µ : [X] , µ =
a a
B ∈A B −→B
E τ ,whereτ : [X] istheuniquehomomorphism suchthatτ (b)=b,for
B a a a
◦ B −→A
any b , τ (X)=a. Let ,(i I) be subalgebras. The family ( )
a i i i∈I
∈B B ⊂A ⊂A ∈ A
will be called free with amalgamation over (or free), if E (a a a ) = 0
B 1 2 n
B B− ···
whenever a with i =i = =i and E (a )=0,1 j n. A sequence
j ∈Aij 1 6 2 6 ···6 n B j ≤ ≤
a willbecalledfreewithamalgamation over ifthefamilyofsubalgebras
i i∈I
{ } ⊆A B
generated by ( a ) is free.
i i∈I
B { } B−
S
2000Mathematics Subject Classification. Primary46L54, 42C15.
Keywords andphrases. Conjugatevariable,freeFisherinformation,randommatrix,modular
frame.
1
2 BINMENG
Inthispaperweonlyconsider valuedW∗ noncommutativeprobabilityspace,
B− −
which means , are vonNeumann (sub)algebrasand E is a faithful conditional
B
A B
expectation (i.e. projection with norm 1) of onto .
A B
Classical Fisher information is derived from the statistical estimation theory
which is defined by R.A.Fisher (for some details we refer to [2]). By analogy
with the classical case D.Voiculescu introduced the free Fisher information of self-
adjoint random variables in a tracial W∗ noncommutative probability space (see
−
[22, 23]). Then A.Nica, D.Shlykhteko and R.Speicher investigated it in view of
cumulant function (see [11]) and study the free Fisher information of random ma-
trices (see [12]). They found the following equality holds: Φ∗( a ,a∗ :
{ ij ij}1≤i,j≤d
) = d3Φ∗( A,A∗ : M ( )), where Φ∗ denotes the free Fisher information,
d
B { } B
A = [a ]d M ( ) := M (C) is an operator matrix with entries in ,
ij i,j=1 ∈ d A A⊗ d A
and is a subalgebra of . D.Shlykhteko even introduced free Fisher infor-
B ⊆ A A
mation with respect to a completely positive map (see [15]). All of these works
are done in the tracial von Neumann algebra framework. In the case of states,
the non-tracial framework was worked out by D. Shlyakhtenko in [17]. In [9, 10],
we generalized the notion of free Fisher information to the operator-valuedsetting
and this work is done in the general von Neumann algebra framework and found
that the free Fisher information of an operator-valued semicircular variable with
conditional expectation covariance is closely related to modular frames.
In the present paper, based on the notion of operator-valued free Fisher infor-
mation with respect to a map which introduced in [10], we study the free Fisher
informationofrandommatricesandobtainaformulaofoperator-valuedfreeFisher
information of random matrices, which generalized Theorem 1.2, Proposition 4.1
in [12], and the method we prove it is quite different from that of Theorem 1.2 in
[12], since the conditional expectations are not ”tracial” in general. Furthermore,
we find that the free Fisher information of circular variables is closely related to
modular frames and the index of conditional expectations.
Our main method to prove the results is R.Speicher’s cumulant function tech-
nique (see [14]), just as in [9, 10]. We use k := (k(n)) and E := (E(n))
B B n≥1 B B
to denote the cumulant and moment functions induced by E respectively where
B
E(n)(a a ):=E (a a ) and k(n) ics determined by the focllowing recur-
B 1⊗···⊗ n B 1··· n B
rence formula (where E(0) =1 formally):
B
E(n)(a a )=
B 1⊗···⊗ n
n−1
= k(r+1)(a E(i(1)−2)(a
B 1 B 1⊗
Xr=01<i(1)<i(X2)···<i(r)≤n
a ) a E(i(2)−i(1)−1)(a a )
···⊗ i(1)−1 ⊗ i(1) B i(1)+1⊗···⊗ i(2)−1
a a E(n−i(r))(a a ))
⊗ i(2)···⊗ i(r) B i(r)+1⊗···⊗ n
for alln N anda , ,a . It is easyto see E and k determine eachother
1 n B B
∈ ··· ∈A
uniquely by the above recurrence formula.
Another tools we use is the frame theory. Thecframectheory in Hilbert space
is derived from the wavelets theory. D.Larson and D.Han etc. studied frames by
the operator theory (see [7]). Furthermore, M.Frank and D.Larsongeneralized the
frame notion to the Hilbert C∗ module setting called modular frames (see [4, 5]).
−
For an unital C∗ algebra , a Hilbert module is a linear space and algebraic
− D D−
OPERATOR-VALUED FREE FISHER INFORMATION OF RANDOM MATRICES 3
module together with a valued inner product .,. and complete with
D
D− M D− h i
respecttotheHilbertC∗ modulenorm: .,. D 12. Wereferto[8]formoredetails
− kh i k
on Hilbert C∗ module theory. A sequence f :j J is said to be a frame
j
− { ∈ }⊆M
if there are real constants C,D >0 such that
C x,x x,f f ,x D x,x , for all x .
D i D i D D
h i ≤ h i h i ≤ h i ∈M
i
X
If one can choose C = D = 1 the frame will be called normalized tight. Note
that for a frame the sum in the middle only converges weakly, in general. If the
sum converges in norm then the frame will be called standard (see [4]). M. Frank
and D. Larson also proved that every algebraically finitely or countably generated
Hilbert C∗ module possesses a standard normalized tight frame ([4]). In this
−
paper, when we say ”frames”, it means ”standard frames”.
The following lemma will be used frequently.
Lemma 2. [9] Let ( ,E , ) be a noncommutative probability space, E :
B
A B B → D
be a conditional expectation, and let X ( ,E , ) be a semicircular variable with
B
∈ A B
covarianceE. DenotetheHilbertC∗ modulegeneratedby by 2( )whoseinner
− B LD B
product induced by E and let ∗( )= . Let f be a frame in 2( ). Then
LD B B { i}⊆B LD B
(1) Φ∗(X : )= f f∗ =Index(E)
B B i i
i
X
and X is free from with amalgamation over .
A D
In the abovelemma, Index(E) denotes the index of the conditional expectation
E. We refer to [1, 3, 18] for more details on the index of conditional expectations.
Thepaperisorganizedasfollows: Insection2,wereviewthedefinitionandsome
results ofthe operator-valuedfree Fisher informationwith respectto a linear map.
Wegetaexplicitformulafortheoperator-valuedfreeFisherinformationofrandom
matriceswithrespecttoacertainkindoflinearmaps. Insection3,weconsiderthe
semicircular random matrices’ free Fisher information. We can compute the free
Fisher information of circular variables with covarianceE by modular frames, and
furthermore, we point out that it is just equal to the double of index of E.
2. The free Fisher information of random matrices
In[10],weintroducedtheoperator-valuedfreeFisherinformationofonevariable
with respect to a linear map which is a generalization of D.Shlyakhtenko’s notion
in[15]. Sucha notioncanbe generalizedto severalrandomvariables’settingeasily.
Definition3. Let( ,E , )bea valuedW∗ noncommutativeprobabilityspace.
B
A B B− −
Suppose that is a von Neumann subalgebra and X n is
B ⊆ C ⊆ A { i}i=1 ⊆ A
algebraically free from over . Denote by 2( [X ,X , ,X ]) the Hilbert
C B LB C 1 2 ··· n
modulegeneratedby [X ,X , ,X ],whoseinnerproductisdefinedby x,y :=
1 2 n
B− C ··· h i
E (x∗y), for any x,y [X ,X , ,X ]. Let η n be a sequence of linear
B ∈ C 1 2 ··· n { i}i=1
maps on . ξ n 2( [X ,X , ,X ]) will be called the conjugate system of
C { i}i=1 ⊆LB C 1 2 ··· n
X n with respect to , η n , if it satisfying:
{ i}i=1 C { i}i=1
(2) E (ξ c X c X X c )
B i 0 i1 1 i2··· im m
m
= δ E (η E (c X X c )) E (c X X c )
i,ij B i C 0 i1··· ij−1 j−1 · B j ij+1··· im m
j=1
X
4 BINMENG
for any m 0, c , ,c and i , ,i 1,2, ,m .
0 m 1 m
≥ ··· ∈C ··· ∈{ ··· }
If such a ξ n exists, then we define Φ∗(X , ,X : ,η , ,η )–the free
{ i}i=1 B 1 ··· n C 1 ··· n
Fisher information of X n with respect to , η n by ξ ξ∗, that is,
{ i}i=1 C { i}i=1 i i
i
n P
Φ∗(X , ,X : ,η , ,η )= ξ ξ∗
B 1 ··· n C 1 ··· n i i
i=1
X
From[10,11],weknowthatthe conjugatesystemcanbeexpressedbycumulant
function.
Proposition 4. With the above notations, let E be a conditional expectation of
C
onto such that E =E E and let (k(n)) be the cumulant function induced
A C B B C C n≥1
by E . Then ξ n 2( [X ,X , ,X ]) is the conjugate system of X n
C { i}i=1 ⊆ LB C 1 2 ··· n { i}i=1
with respect to , η n if and only if the following equations hold
C { i}i=1
k(1)(ξ c)=0, i 1,2, ,n ,
C i ∀ ∈{ ··· }
kC(2)(ξi⊗ca)=δa,XiEB(ηi(c)), ∀i∈{1,2,··· ,n},
k(m+1)(ξ c a c a )=0, i 1,2, ,n ,m 2
C i⊗ 1 1⊗···⊗ m m ∀ ∈{ ··· } ≥
where c,c1, ,cm , a,a1, ,am X1, ,Xn .
··· ∈C ··· ∈{ ··· }∪C
Weintroducesomenotationsaboutrandommatrices. M ( ):= M (C),M ( ):=
d d d
A A⊗ B
M (C) are the sets of all the d d matrices on , respectively. If E is
d B
B ⊗ × A B
a conditional expectation of onto , then define the conditional expectation
A B
E I : M (C) M (C)by(E I )([a ]d ):=[E (a )]d ,forall
B⊗ d A⊗ d →B⊗ d B⊗ d ij i,j=1 B ij i,j=1
A := [a ]d M ( ). Obviously, (M ( ),E I ,M ( )) is a M ( ) valued
ij i,j=1 ∈ d A d A B ⊗ d d B d B −
noncommutative probability and the elements in it are called operator-valuedran-
dom matrices in general.
Since there is no essential difference between 2 2 and n n matrices, for
× ×
convenience, we only consider 2 2 matrices below.
×
The following theorem describe the free Fisher information of random matrices
in terms of its entries’ free Fisher information.
Theorem 5. Let ( ,E , ) be a valued W∗ noncommutative probability space
B
A B B− −
and let be a subalgebra. η ,ξ 2 is a sequence of linear map on .
B ⊆C ⊆A { ij ij}i,j=1 C
c c η (c )+η (c ) 0
Defineη :M ( ) M ( )byη 11 12 = 11 11 21 22
2 C → 2 C c21 c22 0 η12(c11)+η22(c22)
(cid:18) (cid:19) (cid:18) (cid:19)
c c ξ (c )+ξ (c ) 0
andξ :M ( ) M ( )byξ 11 12 = 11 11 12 22 .
2 C → 2 C c21 c22 0 ξ21(c11)+ξ22(c22)
(cid:18) (cid:19) (cid:18) (cid:19)
A:=[a ]2 M ( ). Let x ,y 2 betheconjugatesystemof a ,a∗ 2
ij i,j=1 ∈ 2 A { ij ij}i,j=1 { ij ij}i,j=1
with respect to η ,ξ 2 . Then
{ ij ij}i,j=1
A A
Φ∗ (A,A∗ :M ( ),η,ξ)= 11 12
M2(B) 2 C A21 A22
(cid:18) (cid:19)
where
A =Φ∗(a ,a∗ : ,η ,ξ )+E (x x∗ )+E (y y∗ )
11 B 11 11 C 11 11 B 21 21 B 12 12
A =E (x x∗ +x x∗ +y y∗ +y y∗ )
12 B 11 12 21 22 11 21 12 22
A =E (x x∗ +x x∗ +y y∗ +y y∗ )
21 B 12 11 22 21 21 11 22 12
A =Φ∗(a ,a∗ : ,η ,ξ )+E (x x∗ )+E (y y∗ )
22 B 22 22 C 22 22 B 12 12 B 21 21
OPERATOR-VALUED FREE FISHER INFORMATION OF RANDOM MATRICES 5
x x y y
Proof. Let X := 11 21 , X := 11 12 . Below we show X ,X
1 x12 x22 2 y21 y22 { 1 2}
(cid:18) (cid:19) (cid:18) (cid:19)
is the conjugate system of A,A∗ with respect to η,ξ using Proposition 4.
{ } { }
Denote by (K(n)) the cumulantfunction inducedby(E I ). Thenforany
n≥1 C 2
⊗
C :=[c(k)]2 , we get the following equalities.
k ij i,j=1
x x c(1) c(1)
K(1)(X C )=(E I ) 11 21 11 12
1 1 C ⊗ 2 (cid:18) x12 x22 (cid:19) c(211) c(212) !
x c(1)+x c(1) x c(1)+x c(1)
=(E I ) 11 11 21 21 11 12 21 22
C ⊗ 2 x12c(111)+x22c(211) x12c(112)+x22c(212) !
=0;
By the same way, we can prove K(1)(X C )=0.
2 1
K(2)(X C A)
1 1
⊗
=(E I )(2)(X C A) (E I )(1)(X (E I )(C A))
C 2 1 1 C 2 1 C 2 1
⊗ ⊗ − ⊗ ⊗
=(E I )(X C A)
C 2 1 1
⊗
2 2
x c(1) a x c(1) a
i11 i1,i2 i21 i11 i1i2 i22
=(EC ⊗I2) i1,Pi22=1x c(1) a i1,Pi22=1x c(1) a 
 i12 i1i2 i21 i12 i1i2 i22 
 i1,i2=1 i1,i2=1 
 P P 
E (η (c(1))+η (c(1))) 0
= B 11 11 21 22
0 EB(η12(c(111))+η22(c(212))) !
=(E I )(η(C ))
B 2 1
⊗
Similarly,wegetK(2)(X C A∗)=E (ξ(C )); K(2)(X C A∗)=0; K(2)(X
2 1 B 1 1 1 2
⊗ ⊗ ⊗
C A)=0.
1
To prove
(3) K(m+1)(X C A C A )=0, m 2,
1 1 1 m m
⊗ ⊗···⊗ ∀ ≥
where A A,A∗,M ( ) we need to induce on m.
i 2
∈{ C }
When m=2,
K(3)(X C A C A )
1 1 1 2 2
⊗ ⊗
=(E I )(X C A C A ) K(2)(X C A (E I )(C A ))
C 2 1 1 1 2 2 1 1 1 C 2 2 2
⊗ · · − ⊗ ⊗
K(2)(X (E I )(C A )C A )
1 C 2 1 1 2 2
− ⊗ ⊗
=(E I )(X C A C A ) δ (E I )(η(C ))(E I )(C A )
C ⊗ 2 1· 1 1· 2 2 − A1,A C ⊗ 2 1 C ⊗ 2 2 2
δ (E I )η((E I )(C A )C )
− A2A C ⊗ 2 C ⊗ 2 1 1 2
6 BINMENG
On the other hand,
(E I )(X C A C A )
C 2 1 1 1 2 2
⊗ · ·
x x c(1) c(1) a(1) a(1) c(2) c(2) a(2) a(2)
=(E I ) 11 21 11 12 11 12 11 12 11 12
C ⊗ 2 "(cid:18) x12 x22 (cid:19) c(211) c(212) ! a(211) a(212) ! c(211) c(222) ! a(221) a(222) !#
2
2
= E(3)(x (c(1) a(1) ) (c(2) a(2) ))
 C i3i1 ⊗ i3i4 i4i5 ⊗ i5i6 i6i2 
i3,i4X,i5,i6=1 i1,i2=1
 
2
= k(1)(x E (c(1) a(1) c(2) a(2) ))
" i3i1 C i3i4 i4i5 i5i6 i6i2
i3,i4X,i5,i6=1
+k(2)(x c(1) a(1) E (c(2) a(2) ))
i3i1 ⊗ i3i4 i4i5 C i5i6 i6i2
2
+k(2)(x E (c(1) a(1) ) c(2) a(2) )+k(3)(x c(1) a(1) c(2) a(2) )
i3i1 C i3i4 i4i5 ⊗ i5i6 i6i2 i3i1 ⊗ i2i3 i3i4 ⊗ i4i5 i5i1 #
i1,i2=1
=δ (E I )(η(C ))(E I )(C A )+δ (E I )η((E I )(C A )C ),
A1A B⊗ 2 1 C ⊗ 2 2 2 A2A B⊗ 2 C ⊗ 2 1 1 2
so, K(3)(X C A C A )=0
1 1 1 2 2
⊗ ⊗
OPERATOR-VALUED FREE FISHER INFORMATION OF RANDOM MATRICES 7
Now assume that the Eq.(3) holds for the numbers m 1, then
≤ −
K(m+1)(X C A C A )
1 1 1 m m
⊗ ⊗···⊗
=(E I )(X C A C A )
C 2 1 1 1 m m
⊗ · ···
m+1
K(2)(X (E I )(j−1)(C A C A )
1 C 2 1 1 j−1 j−1
− ⊗ ⊗···⊗
j=1
X
C A (E I )(m−j)(C A C A ))
j j C 2 j+1 j+1 m m
⊗ ⊗ ⊗···⊗
=(E I )(X C A C A )
C 2 1 1 1 m m
⊗ · ···
m+1
δ (E I )(η(E I )(C A C A )C )
− AAj B⊗ 2 C ⊗ 2 1 1··· j−1 j−1 j
j=1
X
(E I )(m−j)(C A C A ))
C 2 j+1 j+1 m m
⊗ ⊗···⊗
2 2
= E (x (c(1) a(1) ) (c(m) a(m) ))
" C i3i1 i3i4 i4i5 ··· i2(m+1)−1 i2(m+1)i2 #
i3,i4,···X,i2(m+1)=1 i1,i2=1
m+1
δ (E I )η((E I )(C A C A )C )
− AAj B⊗ 2 C ⊗ 2 1 1··· j−1 j−1 j
j=1
X
(E I )(m−j)(C A C A ))
C 2 j+1 j+1 m m
⊗ ⊗···⊗
2 m
= k(2)(x E (c a(1) c(j) a(j) )
" i3i1 C i3i4 i4i5··· i2(j+1)−1i2(j+1) i2(j+1)i2(j+1)+1
i3,i4,··X·,i2(m+1)Xj=0
2
c(j+1) a(j+1) E (c(j+2) a c(m) a(m) ))
⊗ i2(j+2)−1i2(j+2) i2(j+2)i2(j+2) C i2(j+2)+1i2(j+2)+2 i2(j+2)+2i2(j+2)+3··· i2m+1i2m+2 i2m+2i2 #
i1,i2=1
m+1
δ (E I )η((E I )(C A C A )C )
− AAj B⊗ 2 C ⊗ 2 1 1··· j−1 j−1 j
j=1
X
(E I )(m−j)(C A C A ))
C 2 j+1 j+1 m m
⊗ ⊗···⊗
2
= δ E η(E (c a(1) c(j) a(j) )
"i3,···,iX2(m+1)=1 ai3i1,ai(2j+(j1+)2)i2(j+2) B C i3i4 i4i5··· i2(j+1)−1i2(j+1) i2(j+1)i2(j+1)+1
2
c(j+1) )E (c(j+2) a c(m) a(m) ))
i2(j+2)−1i2(j+2) C i2(j+2)+1i2(j+2)+2 i2(j+2)+2i2(j+2)+3··· i2m+1i2m+2 i2m+2i2 #
i1,i2=1
m+1
δ (E I )η((E I )(C A C A )C )
− AAj B⊗ 2 C ⊗ 2 1 1··· j−1 j−1 j
j=1
X
(E I )(m−j)(C A C A ))
C 2 j+1 j+1 m m
⊗ ⊗···⊗
=0
8 BINMENG
ThusweclaimthatX ,X istheconjugatesystemof(A,A∗)withrespectto(η,ξ).
1 2
So,
Φ∗(A,A∗ :M ( ),η,ξ)
B 2 C
x x x∗ x∗ y y y∗ y∗
= (E I ) 11 21 11 12 + 11 12 11 21
B⊗ 2 (cid:20)(cid:18) x12 x22 (cid:19)(cid:18) x∗21 x∗22 (cid:19) (cid:18) y21 y22 (cid:19)(cid:18) y1∗2 y2∗2 (cid:19)(cid:21)
A A
= 11 12
A A
21 22
(cid:18) (cid:19)
(cid:3)
The following corollary is a analogue of Theorem 1.2 in [12].
b 0 b b
Corollary 6. Let (2) := b and (E tr ) 11 12 :=
BT 0 b | ∈B B ⊗ T b21 b22
(cid:26)(cid:18) (cid:19) (cid:27) (cid:18) (cid:19)
b +b
E 11 22 0
B
 (cid:18) 2 (cid:19) . Then in the (2) valued noncommuta-
0 E b11+b22 BT −
B
 2 
 (cid:18) (cid:19) 
tive probability space (M ( ),E tr, (2)), for a random matrix A = [a ]2
2 A B ⊗ T BT ij i,j=1
we have
2
Φ∗(a ,a∗ : ,id) 0
1 B ij ij C
Φ∗B(2)(A,A∗ :M2(C),id)= 8 i,Pj=1 2 
T 0 Φ∗(a ,a∗ : ,id)
 B ij ij C 
 i,j=1 
 P 
Proof. WiththenotationsasinTheoremanditsproof,letη =ξ =1, i,j =1,2.
ij ij
Then from the proof of Theorem, we know
E (c +c )
B 11 22
1
K(2)( X CA) = 2
2 1⊗  EB(c11+c22) 
 2 
= (E tr )(C) 
B T
⊗
and
E (c +c )
B 11 22
1
K(2)( X CA∗) = 2
2 2⊗  EB(c11+c22) 
 2 
= (E tr )(C). 
B T
⊗
Hence 1X ,1X is the conjugate system of A,A∗ with respect to (2),id and
{2 1 2 2} BT
Φ∗ (A,A∗ :M ( ),id)
B(2) 2 C
T
1 1
= E tr (X X∗)+ E tr (X X∗)
4 B⊗ T 1 1 4 B⊗ T 2 2
2
Φ∗(a ,a∗ : ,id) 0
1 B ij ij C
=  i,j=1 
8 P 2
0 Φ∗(a ,a∗ : ,id)
 B ij ij C 
 i,j=1 
 P  (cid:3)
OPERATOR-VALUED FREE FISHER INFORMATION OF RANDOM MATRICES 9
3. The free Fisher information of random matrices with semicircular
and circular entries
RecallthatX ( ,E , )willbe calledasemicircularvariablewith covariance
B
∈ A B
η(orη semicircularvariable,whereηisalinearmapon ),ifitsatisfies: k(1)(X)=
− B B
0, k(2)(X bX)=η(b), k(m+1)(X b X b X)=0, m 2,where(k(n))
B ⊗ B ⊗ 1 ⊗···⊗ m ∀ ≥ B n≥1
is the cumulant function induced by E . Let X,Y be two free η semicircular
B
X+iY −
variables, then C := will be called a circular variable in ( ,E , ) with
B
√2 A B
covariance η (or η circular).
−
f c
In the classical free probability theory, it is well known that 1
c∗ f2 ∈
(cid:18) (cid:19)
(M ( ),τ tr,C) is semicircular, where f ,f , c,c∗ is a free family in ( ,τ),
2 1 2
A ⊗ { } A
f ,f are semicircular and c is circular(see [19]). In fact we can get more.
1 2
Lemma 7. [16] Let ( ,E , ) be a valued noncommutative probability space,
B
A B B−
X,Y ( ,E , ) be η semicircular, and let C ( ,E , ) be η circular. If
B B
∈ A B − ∈ A B −
X C
X,Y, C,C∗ is a freefamily, then (M ( ),E I ,M ( ))is a
{ } B− C∗ Y ∈ 2 A B⊗ 2 2 B
(cid:18) (cid:19)
b b η(b +b ) 0
semicircularvariablewithcovarianceη+,whereη+ 11 12 := 11 22
b b 0 η(b +b )
21 22 11 22
(cid:18) (cid:19) (cid:18) (cid:19)
Theorem 8. Let c ( ,E , ) be E circular, where E : is a conditional
B
∈ A B − B →D
expectation. Then
(4) Φ∗(c,c∗ : ,E)=2
B B
Proof. Let f be anormalizedtightframeinthe Hilbert C∗ module 2( )
{ i}⊆B − LD B
which induced by E. Let s ,s ( ,E , ) be two E semicircular variables such
1 2 B
∈ A B −
that s ,s , c,c∗ is a free family with amalgamationover , and let a,x,y,b be
1 2
{ } B { }
the conjugate system of s ,c,c∗,s with respect to E Then we claim E (ax) =
1 2 B
{ }
E (ay)=E (xb)=E (yb)=0 since a, x,y ,b is a free family. By Theorem 5,
B B B
{ }
s c
(5) Φ∗ 1 :M ( ),E+
M2(B) c∗ s2 2 B
(cid:18)(cid:18) (cid:19) (cid:19)
Φ∗(s ; ,E)+E (yy∗) 0
= B 1 B B
0 Φ∗(s ; ,E) E (xx∗)
(cid:18) B 2 B B (cid:19)
and from [10], we know Φ∗(s ; ,E)=Φ∗(s ; ,E)=1.
B 1 B B 2 B
s c
On the other hand, 1 (M ( ),E I ,M ( )) is E+semicircular.
c∗ s2 ∈ 2 A B ⊗ 2 2 B
(cid:18) (cid:19)
d 0 d 0
NotethatE+ isnotaconditionalexpectation,sinceE+ = in
0 d 6 0 d
(cid:18) (cid:19) (cid:18) (cid:19)
f 0 0 f 0 0 0 0
general. Butfor i i ,
0 0 0 0 f 0 0 f
i i i i i i
we claim that Yn:=(cid:18) fi(cid:19)o0 Sn(cid:18)s1 c (cid:19)Eo+Sn(cid:18)fi∗ 0 (cid:19)o Sn(cid:18) (cid:19)o
0 0 c∗ s 0 0
i (cid:18) (cid:19)(cid:18) 2 (cid:19) (cid:18)(cid:18) (cid:19)(cid:19)
0 0 sP c 0 0 s c
+ 1 E+ istheconjugatevariableof 1
0 f c∗ s 0 f∗ c∗ s
i (cid:18) i (cid:19)(cid:18) 2 (cid:19) (cid:18)(cid:18) i (cid:19)(cid:19) (cid:18) 2 (cid:19)
P
10 BINMENG
with respect to M ( ),E+ in (M ( ),E I ,M ( )). In fact,
2 2 B 2 2
B A ⊗ B
b b s c
K(2) Y 11 12 1
⊗ b21 b22 c∗ s2
(cid:18) (cid:18) (cid:19)(cid:18) (cid:19)(cid:19)
f 0 f∗ 0 b b
= i E+ i E+ 11 12
0 0 0 0 b b
21 22
i (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
X
0 f 0 0 b b
+ i E+ E+ 11 12
0 0 f∗ 0 b b
i (cid:18) (cid:19) (cid:18) i (cid:19) (cid:18) 21 22 (cid:19)
X
0 0 0 f∗ b b
+ E+ i E+ 11 12
f 0 0 0 b b
i 21 22
i (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
X
0 0 0 0 b b
+ E+ E+ 11 12
0 f 0 f∗ b b
i (cid:18) i (cid:19) (cid:18) i (cid:19) (cid:18) 21 22 (cid:19)
X
E(b +b ) 0 b b
= 11 22 =E+ 11 12 ,
0 E(b +b ) b b
11 22 21 22
(cid:18) (cid:19) (cid:18) (cid:19)
and
s c s c
K(m+1) Y B 1 B 1 =0,
⊗ 1 c∗ s2 ⊗···⊗ m c∗ s2
(cid:18) (cid:18) (cid:19) (cid:18) (cid:19)(cid:19)
s c
for all B , ,B M ( ), m=1, since 1 is semicircular.
1 ··· m ∈ 2 B 6 c∗ c∗
(cid:18) (cid:19)