A Survey of Topological Free Entropy Dimensions
Joint with: Qihui Li, Weihua Li, Junhao Shen
1
Let’s take a moment
(M; ) von Neumann algebra with a faithful trace
M = W ( x1 ; : : : ; x n ) ; ~
x = ( x1 ; : : : ; x n )
m = m (t1; : : : ; tn) -monomial in n free variables
(m (x1; : : : ; xn)) = mth moment of ~
x.
Suppose (N ; ) and N = W (y1; : : : ; yn).
THM (m (~
x)) = (m (~
y )) for all -monomials m
, 9 normal -isomorphism
: M ! N such that
= .
If N 2 N and " > 0; we say that (~
x; ) and (~
y ; ) are
(N; ")-close if
j (m (~
x))
(m (~
y ))j < "
for all -monomials m with deg m
N.
Voiculescu’s Free Entropy Dimension
Given M = W (x1; : : : ; xn) and faithful trace .
Let R > max fkx1k ; : : : ; kxnkg = k~
xk
M k (C) = k
k complex matrices, k = k1 T r
For " > 0; N 2 N and k 2 N de…ne
x; N; "; k)
R (~
n
o
~ 2 M k (C)n : A
~ < R by:
A
~ 2 R (~
~ k
A
x; N; "; k) , (~
x; ) ; A;
are (N; ") -close
We want to "measure" the sets R (~
x; N; "; k).
COVERING NUMBERS
(X; d) is a totally bounded metric space,
!>0
d (X; ! )
= minimal number of ! -balls needed to cover
X.
Ex: (1=! )m
m
kk (ballR ; ! )
(3=! )m
A C*-algebra dimC A = d < 1 =)
2
(1=! )d
:
kk (U (A) ; ! )
2
(9 e=! )d
BOX DIMENSION
The Box Dimension (Minkowski Dimension) is de…ned
as
dimbox (X ) = lim sup
!!0+
If X
then
log d (X; ! )
:
log !
Rn, X bounded with positive Lebesgue measure,
dimbox (X ) = n
Voiculescu’s free entropy dimension
De…ne for ~
x = ( x1 ; : : : ; x n )
0 ( x1 ; : : : ; x n )
=
log (v ( R (~
x; N; "; ; k); ! )
log !
Voiculescu’s free entropy dimension
De…ne for ~
x = ( x1 ; : : : ; x n )
0 ( x1 ; : : : ; x n )
=
log (v ( R (~
x; N; "; ; k); ! )
k2 log !
Voiculescu’s free entropy dimension
De…ne for ~
x = ( x1 ; : : : ; x n )
0 ( x1 ; : : : ; x n )
=
log (v ( R (~
x; N; "; ; k); ! )
k2 log !
k!1
lim sup
Voiculescu’s free entropy dimension
De…ne for ~
x = ( x1 ; : : : ; x n )
0 ( x1 ; : : : ; x n )
=
log (v ( R (~
x; N; "; ; k); ! )
N 2N;">0 k!1
k2 log !
inf
lim sup
Voiculescu’s free entropy dimension
De…ne for ~
x = ( x1 ; : : : ; x n )
0 ( x1 ; : : : ; x n )
=
log (v ( R (~
x; N; "; ; k); ! )
k2 log !
!!0+ N 2N;">0 k!1
lim sup
inf
lim sup
This is independent of R > k~
xk :
Szarek: 0 unchanged if we use the covering numbers
with respect to kk2.
APPLICATIONS
Many results of the form: If M = W (x1; : : : ; xn) has
some particular property, then 0 (~
x) 1.
conclusion: If 0 (~
x) > 1 (e.g., LFn ; n
does not have the property.
2), then M
Voiculescu: Cartan subalgebra
Ge: Prime Factors, Simple Masas
Ge, Shen: Deep results absorbing all previous results
(generalized and simpli…ed by Dostal and H.)
FREE ORBIT DIMENSION Shen, H: New invariant K2
involving covering by unitary orbits of ! -balls that gives
elementary proofs of extended Ge-Shen Dostal-Hadwin
results.
1. K2 = 0 is generator-independent.
2. 0
1 + K2
3. M hyper…nite =) K2 (M) = 0
4. K2 (N ) = 0; A; B 2 N normal, p (A) \ p (B ) =
?; AT = T B =) K2 (W (N [ fT g)) = 0
6. K2 (N1) = K2 (N2) = 0 and N1 \ N2 di¤use =)
K2 (W (N1 [ N2)) 0.
5. K2 = 0 is preserved under (…nitely generated) direct
limits.
Def of Free orbit Dimension:
! -orbit-ball = unitary orbit of an ! -ball
O2 (E; ! ) = covering number withh kk2-norm ! -orbit
balls
K 2 ( x1 ; : : : ; x n ) =
log O2 ( R (x1; : : : ; xn; N; k; ") ; w)
sup inf lim sup
k2 log !
0<!<1 N;"
k!1
Weihua Li and H. de…ne a generator-indepedent invariant K3 (M) and has most of the properties of K2; and
is lower semicontinuous with respect to arbitrary direct
limits.
Moreover, K3 is de…ned for nonseparable (e.g., ultraproducts, uncountable group von Neumann algebras) algebras tracially embeddable in an ultrapower of the hyper…nite II1 factor R.
K3 (R ) = 0 but maybe K3
Y
k2N
Junsheng Fang and and H.:R and
isomorphic.
Mk (C) 6= 0.
Y
k2N
Mk (C) not
Many other exciting recent results for free entropy dimension: Jung, Popa, Dykema,
Shlyakhtenko, etc.
Thm (H 1998: Shanghai Theorem) Suppose M =
W (x1; : : : ; xn) is hyper…nite with a faithful normal
trace and suppose ! > 0 and R > max1 j n xj .
(R not needed).Then there are " > 0 and N 2 N such
~ B
~ 2 Nn
that, for every …nite factor (N; ) and every A;
~
~
~
with A
R; if A;
and B;
are both (N; ")close to (~
x; ) ; then there is a unitary U 2 N such that
n
X
j=1
Bj
U Aj U
2
< !:
~ then any B
~ is almost unitarily
Note: If you …nd a nice A
equivalent to A
COR. W (x1; : : : ; xn) hyper…nite =) 0(x1; : : : ; xn)
1:
COR If A a unital separable C*-algebra, R hyper…nite
II1 factor with trace , and ; : A ! R unital homomorphusms, then
=
() 9 fUng
U (R) s.t. kUn (a) Un
(a)k2 ! 0 for all a 2 A:
The proof is based on tracial ultraproducts.
Sakai A tracial ultraproduct of …nite factors is a factor
von Neumann algebra
Weihua Li & H: A tracial ultraproduct of C*-algebras is
a von Neumann algebra (simple proof)
C*-algebra Version
A = C ( x1 ; : : : ; x n ) ; B = C ( y 1 ; : : : ; y n )
fp1 (t1; : : : ; tn) ; p2 (t1; : : : ; tn) ; p2 (t1; : : : ; tn) ; : : :g= all
-polynomials over Q + iQ.
THM There is a unital -isomorphism
ing each xj to yj i¤
: A ! B send-
kpm (~
x)k = kpm (~
y )k
for all m
1.
~
x; ~
y are topologically (N; ")-close , top-(N; ")-close
,
pj (~
x)
for 1
j
N.
pj (~
y)
<"
Voiculescu’s Topological Free Entropy Dimension
Given A = C (x1; : : : ; xn).
For " > 0; N 2 N and k 2 N de…ne
top (~
x; N; "; k)
Mk (C)n by:
~ are top- (N; ") -close
~ 2 top (~
A
x; N; "; k) , ~
x; A
top(x1 ; : : : ; xn)
=
log (v ( R (~
x; N; "; ; k); ! )
k2 log !
!!0+ N 2N;">0 k!1
lim sup
inf
lim sup
Szarek: top unchanged if we use the covering numbers
with respect to kk2.
Junhao Shen & H: A =C (~
x) commutative or …nitedimensional =)
1
top(x1 ; : : : ; xn) = 1
dim A
Shen & H: Formula for direct sums, Change of variables
formula
Shen, Qihui Li & H: top additive for full free products
Shen, Qihui Li & H:
A nuclear =) top(x1; : : : ; xn)
1:
Domain of
0
) de…ned , 8 " > 0; N 2 N 9k 2 N
(~
x; N; "; k) 6= ?
0 ( x1 ; : : : ; x n ;
such that
()
W (x1; : : : ; xn) embeddable in a tracial ultrapower of
the hyper…nite II1 factor (Connes’embedding problem)
()
W (x1; : : : ; xn) embeddable in a tracial ultraproduct
Q
(M k (C) ; k )
k2N
Domain of
top
de…ned , 8" > 0; N 2 N 9k 2 N
(~
x; N; "; k) 6= ?.
top(x1 ; : : : ; xn)
such that
,
C (~
x) is an MF algebra (Blackadar-Kirchberg), i.e.,
C (~
x) can be embedded in
Y
1 s<1
M ks (C) =
X
1 s<1
M ks (C)
for some increasing sequence fksg of positive integers
()
C (~
x) can be embedded in a C*-ultraproduct
Q
k2N
Mk (C).
C (x1; : : : ; xn) is an M F -algebra if there are sequences
~ s = (A1s; : : : ; Ans) 2
fksg of positive integers and A
Mks (C)n such that, for every -polynomial p,
lim kp (A1s; : : : ; Ans)k = kp (x1; : : : ; xn)k .
s!1
We call this convergence in topological distribution,
and write
~ s top
A
!~
x:
In the tracial von Neumann algebra case, we write
~ s; k
A
s
~s
if lims!1 ks m A
m.
=
dist
! (~
x; )
(m (~
x)) for every -monomial
2
MF-traces
De…nition 1 Suppose A = C (x1; : : : ; xn) is an MF
C*algebra. A tracial state on A is an M F -trace if
~ s in Mk (C)nsuch
there are sequences fksg in N and A
s
that, for every -polynomial p,
1. limk!1 kp (A1k ; : : : ; Ank )k = kp (x1; : : : ; xn)k, and
2. limk!1 mk (p (A1k ; : : : ; Ank )) =
(p (x1; : : : ; xn)).
T (A) = the set of all tracial states on A
TM F (A) = the set of all MF-traces on A.
Notation:
a tracial state,
GNS representation for ; ^ :
(T e; e) faithful normal trace
is …nite-dimensional if dim
(a) = ( (a) e; e) =
(A)00 ! C, ^ (T ) =
(A ) < 1 .
Proposition 2 If A = C (x1; : : : ; xn) is M F; then
1. TM F (A) is a nonempty weak*-compact convex set.
2. Every …nite-dimensional tracial state on A is in TM F (A),
3. If is a unital -homomorphism on A and
M F -algebra, then
f'
4.
: ' 2 TM F ( (A))g
(A) is an
TM F (A ) :
2 TM F (A) () there is a free ultra…lter on N; and
Q
Mk (C) such that for every
embedding
: A !
a 2 A;
k2N
(a) = lim
k!
where
(a) = fAk g .
k (Ak ) ;
3
MF-nuclear C*-algebras
A = C (x1; : : : ; xn) is nuclear if, for every representation : A ! B (H ) we have (A)00 is hyper…nite.
A is MF-nuclear if A is MF and, for every
we have
(A)00 is hyper…nite.
2 T M F (A ) ;
The class of MF-nuclear C*-algebras is closed under:
direct sums, nice tensor products, direct limits, and
MF-representations.
THM IfA = C (x1; : : : ; xn) is MF and
fx1; : : : ; xng =
and C
m
[
Ej ;
j=1
Ej is MF-nuclear for 1
top(x1 ; : : : ; xn)
j
m:
m, then
Lower Bounds
If A = (A1; : : : ; An) 2 Mn
k (C) ; we de…ne the unitary
~ by
orbit of A
~ = f(U A1U; U A2U ; : : : ; U AnU ) : U 2 Uk g :
U A
THM (Dostal, H.) If W (x1; : : : ; xn) is hyper…nite and
~ s in Mk (C)nsuch
if there are sequences
fk
g
in
N
and
A
s
s
o
n
~ s is bounded, then
~ s~
that A
x and A
x)
0 (~
= lim suplim sup
!!0+ s!1
~ s ; !)
log (v (U A
k2s log !
. THM (Li,Li,Shen,H.) If A = C (~
x) is MF-nuclear,
then, for every 2 TM F
top(x1 ; : : : ; xn)
0 (x1 : : : ; xn;
):
Theorem 3 Suppose A = C (x1; : : : ; xn) is a unital
M F -algebra and
2 TM F (A). Suppose b = b 2
(A)00. Then
top (x1 ; : : : ; xn)
Voiculescu: x = x =)
0 ( x; ) = 1
X
2 p (x)
0 (b;
):
f g ( x)
so 0 (x; ) = 1 , x has no eigenvalues.
2
:
A unitary element U n a C*-algebra with a faihful trace
is a Haar unitary if and only if
(U n) = 0 when n 2 Zn f0g :
We de…ne Hk 2 Mk (C) as the diagonal matrix whose
eigenvalues are e2 m=k (0 m < k).
We call Hk the k
k Haar unitary matrix.
Lemma 4 Suppose M = W (x1; : : : ; xn) has a faithful normal tracial state and M is tracially embeddable
in an ultrapower of the hyper…nite II1 factor. The following are equivalent:
1. M is di¤use,
2. M contains a Haar unitary element
3. M has no normal …nite-dimensional representation
4.M has no nonzero central projection P such that P MP
is …nite-dimensional.
5. M has no minimal nonzero projection.
6. Not every normal element in M has an eigenvalue
7. For every > 0 there is a noncommutative polynomial p (t1; : : : ; tn), a positive integer N0 such, for every
k
N0, for every N
N0; for every 0 < " < 1=N0
~ = (A1; : : : ; An) 2 (~
and every A
x; N; "; k) there is a
unitary U 2 Uk such that
~
p A
U Hk U
2
< :
We call a unital C*-algebra C*-di¤use or NF if it has
no …nite-dimensional representations
Lemma 5 Suppose A = C (x1; : : : ; xn) is an MF algebra. The following are equivalent:
1. A is NFD.
2. For every MF trace
Neumann algebra.
on A,
(A)00 is a di¤use von
3. For every > 0 there is a positive integer N0 such, for
every k
N0, for every N
N0; for every 0 < " <
~ = (A1; : : : ; An) 2 top (~
1=N0 and every A
x; N; "; k)
there is a p 2 PN t1;; : : : ; tn a unitary U 2 Uk such
that
~
p A
k
2
U
U diag e2 i=k ; e2 i=k ; : : : ; e2 i=k
<
2
.
Theorem 6 Suppose A = C (x1; : : : ; xn) is an M F algebra and either
1. A is NF
2. A has in…nitely many non-unitarily equivalent …nite-dimensional
representations.
Then top (x1; : : : ; xn)
1.
Corollary 7 If A is a unital residually …nite-dimensional
C*-algebra, then, for any generating set fx1; : : : ; xng of
A, we have
1
:
1
top (x1 ; : : : ; xn)
dim A
If, in addition, A is MF-nuclear, then equality holds.
Corollary 8 Suppose A is a unital …nitely generated M F
C*-algebra and G is a …nitely generated in…nite abelian
group and : G ! Aut (A) is a group homomorphism.
If Ao G is M F , then, for every set fx1; : : : ; xng of
generators for Ao G, we have
top (x1 ; : : : ; xn)
1:
If, in addition, A is MF-nuclear, then
top (x1 ; : : : ; xn)
= 1:
Corollary 9 If A is a simple, MF-nuclear C*algebra, then,
for any generating set fx1; : : : ; xng of A, we have
top (x1 ; : : : ; xn)
=1
1
:
dim A
The MF-Ideal
IM F (A) = fa 2 A :
( a a ) = 08 2 T M F ( A ) g
Theorem 10 Suppose A = C (x1; : : : ; xn) is an M F algebra and dim A=JM F (A) = d < 1. Then
top (x1 ; : : : ; xn) = 1
1
:
d
Theorem 11 If A = C (x1; : : : ; xn) is an MF-nuclear
and A=JM F (A) is RFD algebra, then
top (x1 ; : : : ; xn) = 1
1
:
dim A=IM F
Example 12 Suppose B is a unital separable M F -C*algebra that is not nuclear, e:g:; B = Cr (F2),.and let
D = B K `2 . Then D is singly generated, and every
tracial state vanishes on D. Let D+ be the C*-algebra
obtained by adjoining the identity to D and suppose N
is a …nitely generated nuclear MF C*-algebra. Then A =
N D+ is …nitely generated and MF, but not nuclear.
However, 1 D+ IM F (A), so
I M F (A ) = I M F (N )
D:
Thus A=IM F (A) is isomorphic to N =IM F (N ), which
is nuclear. Hence, for every set fx1; : : : ; xng of generators of A, we have
top (x1 ; : : : ; xn)
1:
We now consider the class S of separable M F C*-algebras
for which every trace is an MF-trace, i.e., T S (A) =
TM (A).
Lemma 13 The following are true.
1. If A; B 2 S , then A
2. If A 2 S , then A
B 2 S.
Mn (C) 2 S for every n
1.
3. S is closed under direct (inductive) limits.
4. Every separable commutative C*-algebra is in S .
5. Every AH C*-algebra is in S .
6. A 2 S , every factor tracial state on A is in TM F (A)
7. If A; B 2 S , then A
min B 2 S .
8. Connes’ embedding counterexample =) There is MF
A2
=S
Theorem 14 Suppose A =C (x1; : : : ; xn) 2 S . Then
either
1. There is a 2 TM F (A) and an A 2
0 (A) = 1, or
(A)00 such that
2. A=IM F (A) is RFD.
Therefore, either top (x1; : : : ; xn)
1.
1 or dim A=IM F (A) <
Corollary 15 If A = C (x1; : : : ; xn) is MF-nuclear and
A 2 S , then
top (x1 ; : : : ; xn)
=1
1
dim A
We now want to focus on the class W of all separable
M F C*-algebras A such that IM F (A) = f0g. The
main reason is the following.
Proposition 16 Suppose A = C (x1; : : : ; xn) is MFnuclear and A 2 S \ W . Then
top (x1 ; : : : ; xn) = 1
1
:
dim A
Theorem 17 The following are true.
1. If fAi : i 2 Ig W , and A is a separable unital subalY
gebra of the C*-direct product i2I Ai, then A 2 W .
2. If A,B 2 W , then A
min B 2 W .
3. Every separable unital simple MF C*-algebra is in W .
4. Every separable unital RFD C*-algebra is in W .
5. W is not closed under direct limits.
NEW STUFF:
H., Qihui Li, Weihual Li, Junhao Shen.
top
De…ned K2
top
and K3
and proved analogues for top.
top
1 + K2
1. top
top
2. K3 is generator independent and de…ned for all MFalgebras
top
3 . If A is the direct limit of fAigi2I , then K3 (A)
top
lim inf K3 (Ai)
top
top
4. If K3 (A1) = K3 (A2) = 0 and A1 \ A2 is NF,
top
then K3 (C (A1 [ A2)) = 0,
top
5. If K3 (A) = 0 and D
A is NF and U is unitary
and U DU
A, then K3 (C (A[ fU g)) = 0.
The central sequence algebra of a separable C*-algebra A
is obtained by …rst embedding A (via constant sequences)
Y1
P1
in n=1 A= n=1 A and taking the commutant.
(We can do the same in any ultrapower.)
THM (Akemann pedersen) The central sequence algebra
is trivial (representable by sequences in Z (A)) i¤ A is a
continuous trace C*-algebra.
THM (H.,Li, Li, Shen) If the central sequence algebra of
an MF algebra A is NF, then K3 (A) = 0
Other Ideas:
There is a vN algbebra M such that every …nite factor
with a separable predual can be tracially embeddd in an
ultrapower of M
Choose a metric on the (ballM)n
Consider
log O2 ( 1 (x1; : : : ; xn; N; "; M) ; gw)
lim sup
:
log
!
+
!!0
Still 0 when W (x1; : : : ; xn) is hyper…nite..
For 0 <
2 de…ne Kt (x1; : : : ; xn) =
log O2 ( R (x1; : : : ; xn; N; k; ") ; w)
sup inf lim sup
kt log !
0<!<1 N;"
k!1
Kt has may properties of K2,and can perhaps distinguish
between LF2 LF2 and LF2 LF2 LF2 .