Download Ultraproducts of Banach Spaces

Ultraproducts of Banach Spaces
This talk is based on the wonderful 1980 survey article “Ultraproducts in Banach Space Theory”
by Stefan Heinrich.
Definition. We say U is an ultrafilter on a set I if U ⊆ P(I) is a collection of subsets of I satisfying
1. I ∈ U, ∅ ∈
/ U.
2. If A, B ∈ U, then A ∩ B ∈ U.
3. If A ∈ U, then A ⊆ B implies B ∈ U.
4. For all A ⊆ I, either A ∈ U or I \ A ∈ U.
If F satisfies (1), (2), and (3), it is called a filter - an ultrafilter is just a maximal filter. The
standard example on a filter is the cofinite sets.
One can think of elements of an ultrafilter U to be “big” sets in I, and the complements of such
sets as “small” sets: We get a finitely-additive measure by
(
1 if A ∈ U
m(A) =
0 if I \ A ∈ U
Definition. An ultrafilter U is called principal, or trivial, if it is determined by a single element
i0 ∈ I, i.e. A ∈ U if and only if i0 ∈ A. An ultrafilter U is free or non-trivial if and only if
\
A=∅
A∈U
A non-principal ultrafilter is called free, and generally we will be interested in free ultrafilters (which
will exist if we assume the axiom of choice). Free ultrafilters necessarily contain the cofinite sets.
Definition. An ultrafilter U is countably incomplete if there is a sequence of elements Ik of U
I1 ⊇ I2 ⊇ I3 ⊇ . . .
∞
\
Ik = ∅
k=1
For example each free ultrafilter on the set of natural numbers if countably incomplete.
Definition. Let Ai be an arbitrary collection of (non-empty)
sets indexed by I, and let U be an
Q
ultrafilter
on
I.
Consider
the
cartesian
product
A
.
Define
an equivalence relation ≡U on
i
i∈I
Q
A
by
i∈I i
(ai ) ≡U (bi ) if and only if {i ∈ I | ai = bi } ∈ U
We define
family (Ai )i∈IQwith respect to the ultrafilter U
Q the the set-theoretic ultraproduct of the Q
to be
A
/
≡
,
which
is
sometimes
denoted
U
i∈I i
i∈I Ai /U or
U Ai . If all the Ai are the same,
U
we call this the set-theoretic ultrapower and denote it by A .
A (set-theoretic) ultrapower of Banach spaces is a field under pointwise addition and multiplication (need to show this is well-defined, i.e. independent of the choice of representations), but
it is not a Banach space or even a metric space. For example, if U is a free ultrafilter on N, RU
is the hyperreals, and it contains infinitessimals and infinite elements. How can we modify our
construction to get a Banach space?
Definition. Given an ultrafilter U on I, we can define the ultralimit with respect to U on a metric
space X by limU xn = x if and only if for all ε > 0, {i : ||xi − x|| < ε} ∈ U.
Comment: this is a natural extension of the standard definition of a limit:
lim xn = x iff ∀ε > 0
n→∞
∀ε > 0
∃N ∈ N
n ≥ N ⇒ ||x − xn || < ε
{n : ||x − xn || < ε} is cofinite
In particular, if limn→∞ xn = x, then limU xn = x as well.
Proposition. Let K be a compact metric space. Then for each sequence (xi )i∈I in
limit
lim xi = x
Q
i∈I
K the
U
exists in K.
Proof. Simple compactness and Hausdorffness argument.
Real-valued ultralimits exend our normal definition of limit and have (almost) all the standard
properties of normal limits we know and love: they are linear, multiplicative, and respect order.
They are not shift invariant: limU xn 6= limU xn+1 in general (since either the even numbers or the
odd numbers are a big set).
Cute aside: know every convergent sequence of real numbers is bounded. Conversely, every
bounded sequence of real numbers has an ultralimit.
This notion of a Banach space ultraproduct was introduced by Dacunha-Castelle and Krivine
in 1972.
Definition. Let (Ei )i∈I be a family of Banach spaces. Consider the set
Y
`∞ (I, Ei ) = {(xi ) ∈
Ei : ||(xi )|| = sup ||xi ||Ei < ∞}
i∈I
i∈I
Then `∞ (I, Ei ) is a Banach space under componentwise addition and scalar multiplication. For U
an ultrafilter on I, let
NU = {(xi ) ∈ `∞ (I, Ei ) : lim ||xi ||Ei = 0}
U
In other words, this is the set of sequences (xi ) such that for all ε > 0,
{i ∈ I : ||(xi )||Ei < ε} ∈ U
It is easy to check NU is a closed linear subspace of `∞ (I, Ei ). Finally, we can define the Banach
space ultraproduct of the (Ei )i∈I with respect to U an ultrafilter on I by
(Ei )U = `∞ (I, Ei )/NU
2
We will denote the equivalence
Q class of an element (xi ) ∈ `∞ (I, Ei ) in the ultraproduct as (xi )U .
We equip the ultraproduct U Ei with the canonial quotient norm:
||(xi )U || =
inf
(ai )∈NU
||(xi − ai )||∞ =
sup ||xi − ai ||Ei
inf
(ai )∈NU i∈I
In fact, the norm can be computed as
||(xi )U || =
inf
sup ||(xi − ai )||Ei = lim ||xi ||
U
(ai )∈NU i∈I
Proof. The limit on the right hand side always exists, since if we consider the representative (xi )i∈I ,
supi∈I ||xi ||Ei = M < ∞, so each ||xi ||Ei ∈ [0, M ], a compact Hausdorff space. Let L = limU ||xi ||.
Now note that for any ε > 0,
Iε = {i ∈ I | ||x − L|| < ε} ∈ U
Define the sequence αi by
(
xi
αi =
0
if i ∈
/ Iε , i.e. ||xi − limU xi || ≥ ε
if i ∈ Iε
Then note
{i ∈ I | ai = 0} ⊇ Iε ∈ U
In particular,
lim αi = 0
U
Thus (αi ) ∈ NU , and for all i ∈ I,
(
0
if i ∈
/ Iε
||xi − αi || =
||xi || if i ∈ Iε
<L+ε
Thus
||(xi )U || =
inf
(ai )∈NU
||(xi − ai )|| ≤ ||(xi − αi )|| = sup ||xi − αi || ≤ L + ε
i
Conversely, if (ai ) ∈ NU , then
I(ai ),ε = {i ∈ I : ||ai || < 2ε } ∩ {i ∈ I : ||xi || − L < 2ε } ∈ U
Then for all i ∈ I(ai ),ε 6= ∅,
||xi − ai || ≥ ||xi || − ||ai || ≥ L −
ε
2
−
ε
2
=L−ε
Thus for any (ai ) ∈ NU ,
sup ||xi − ai || ≥ L − ε
i∈I
So
||(xi )U || =
inf
sup ||xi − ai || ≥ L − ε
(ai )∈NU i∈I
Thus we have shown for all ε > 0,
lim ||xi || − ε ≤ ||(xi )U || =
U
inf
(ai )∈NU
||(xi − ai )|| =
inf
And we conclude that
||(xi )U || = lim ||xi ||
U
as claimed.
3
sup ||xi − ai || ≤ lim ||xi || + ε
(ai )∈NU i∈I
U
Note that if U = Ui0 is the trivial or principal ultrafilter generated by i0 ∈ I then
Q
U
Ei ∼
= Ei0 .
If all the spaces Ei = E then we speak of the ultrapower E I /U or E U . There is a canonical
isometric embedding J of E into its ultrapower (E)U which is defined by J(x) = (xi )U where xi = x
for all i ∈ I.
Interestingly, the Banach space ultrapower contruction is only interesting in the infinite-dimensional
case. If E is finite-dimensional, then E U ∼
= E: the diagonal embedding is actually an isomorphism. This is because each element (xi )U ∈ E U has ||xi || ≤ M for some M , and the ball
{y ∈ E : ||y|| ≤ M } is compact in E since E is finite-dimensional. Thus limU xi = x exists in E,
and thus (xi )U = (x)U .
Recall the following cute fact: an ultraproduct of (non-empty) subsets Ak of the natural numbers with respect to a free ultrafilter U is either finite or uncountable. The following theorem is
the analogue of this fact for Banach spaces:
Theorem. An ultraproduct of Banach spaces with respect to a free ultrafilter U is either finitedimensional or non-separable.
Now that we have introduced the ultraproduct of Banach spaces, we can define the ultraproduct
of operators. If (Ei )i∈I and (Fi )i∈I are families of Banach spaces indexed by the same set I, and
for each i ∈ I, Ti ∈ B(Ei , Fi ) is a bounded linear map from Ei to Fi , such that
sup ||Ti || < ∞
i∈I
The ultraproduct of the family of operators (Ti )i∈I with respect to the ultrafilter U on I is (Ti )U
defined by
(xi ) 7→ (Ti xi )U
This map is well-defined, and ||(Ti )U || = limU ||Ti ||.
Proposition.
1. Banach algebras are stable under ultraproducts (under component-wise multiplication).
2. C ∗ -algebras are stable under ultraproducts (under component-wise adjoints).
Theorem.
1. Let Ki , i ∈ I be compact
Q Hausdorff spaces. Then there is a compact Hausdorff space K such
that the ultraproduct U C(Ki ) is linearly isometric to C(K). The isometry preserves the
multiplicative and lattice structure.
Q
2. Let 1 ≤ p < ∞, and let µi (i ∈ I) be arbitrary σ-additive measures. Then U Lp (µi ) is
order-isometric to Lp (ν) for a certain measure ν.
Q
Proof. Note that U C(Ki ) is a commutative C ∗ -algebra with identity. Thus by the Gelfand representation theorem, it is isomorphic to continuous functions on the character space (multiplicative
linear functionals/maximal ideal space) of this C ∗ -algebra, which is a compact Hausdorff space in
the weak∗ -topology.
4
Next, we would like to study the structure of the space K that satisfies
Y
C(Ki ) ∼
= C(K)
U
It is natural to ask if K is related to the the set-theoretic ultraproduct of the spaces Ki (topologized
in a natural way) and what topological properties of the Ki will be shared with K.
Given a familar of topological compact
Q Hausdorff spaces Ki (i ∈ I) and an ultrafilter U on I,
we form the set-theoretric ultraproduct U Ki and consider the following family of subsets
B = {(Ui )U ⊆ (Ki )U : Ui is an open subset of Ki for all i ∈ I}
This family forms a basis of a topology on (Ki )U : certainly it covers (Ki )U as the whole space is in
this family, and if (Ui )U , (Vi )U ∈ B, then
(Ui )U ∩ (Vi )U = (Ui ∩ Vi )U ∈ B
We will equip (Ki )U with the topology generated by this basis.
Q
Theorem. Let Ki (i ∈ I) be compact Hausdorff spaces and let C(K) ∼
= U C(Ki ) topologized as
above. Then
Q
1. U Ki is canonically homeomorphic to a dense subset of K.
2. If the spaces Ki are totally disconnected (only the empty set and singleton sets are connected),
then so is K.
Recall a topological space is extremally disconnected if the closure of every open set is itself
open. An extremally disconnected compact Hausdorff space is a Stonean space. For example, the
Stone-Čech compactification of a discrete space is Stonean, in particular, β(N) is a Stonean space.
It is a fact that if K is Stonean (compact, Hausdorff, and extremally disconnected) then C(K) is
an injective Banach space, and these are all the injective Banach spaces.
Recall (in the category of Banach spaces with contractive linear maps) a Banach space I is injective if for all Banach spaces A and B with A ⊆ B a subspace and φ : A → I a contractive linear
map there is an extension of φ to Φ : B → I a contractive linear map. For example, C is injective
by the Hahn-Banach theorem, and `∞ (N) is injective by using Hahn-Banach componentwise (the
direct sum of injective spaces is injective). Since every Banach space embeds isometrically into
∗ ), there are sufficiently many injectives, and being injective is
an injective space, B ,→ `∞ (B≤1
equivalent to being contractively complemented in any super Banach algebra, which is equivalent
to being contractively complemented in a super Banach algebra.
5
Namely, the Stone-Čech compactification of the natural numbers βN is Stonean, and C(βN) =
`∞ (N) are injective by the Hahn-Banach theorem, but [`∞ (N)]U is not injective: it embeds naturally
into `∞ (NU ) by sending
(fi )U ∈ [`∞ (N)]U 7→ Φ ∈ `∞ (NU ),
Φ( (xi )U ) := lim fi (xi )
U
for (xi )U ∈ NU
and there is no bounded projection from `∞ (NU ) onto `∞ (N)U , otherwise it would imply the existence of a bounded projection from `∞ (N) to c0 (N) which is known to be false.
The Local Structure of Ultraproducts
Finite Representability and Ultrapowers
This section is devoted to the study of finite-dimensional subspaces of ultraproducts. In the
case of ultrapowers, this leads to the connection with the notion of finite representability. We
discuss the subject of local properties.
Our first result show that the finite dimensional subspaces of an ultraproduct (Ei )U are arbitrarily close to suitable subspaces of the spaces Ei .
Recall that an operator T : E → F is a (1 + ε)-isomorphism if ||T ||, ||T −1 || < 1 + ε.
Q
Proposition. Let (Ei )i∈I be a family of Banach spaces, M a finite-dimensional subspace of U Ei ,
and ε > 0 be given. There there is a set I0 ∈ U and subspaces Mi ⊆ Ei for all i ∈ I0 which are
(1 + ε)-isomorphic to M .
There is a partial converse to this proposition as well :
Proposition. Let X be a Banach space and let B be a family of Banach spaces such that for each
ε > 0, and each finite-dimensional subspace M of X there is a space E = EM,ε ∈ B such that M
is (1 + ε)-isomorphic to a subspace of E. There there is an ultrafilter U onQan index set I and a
map from I into B sending i 7→ Ei ∈ B, so X is isometric to a subspace of U Ei .
Proof. Let I be the collection of all pairs (M, ε) where M is a finite dimensional subspace of X
and ε > 0. There is a canonical partial ordering on I given by
(M1 , ε1 ) ≺ (M2 , ε2 ) if and only if M1 ⊆ M2 and ε1 ≥ ε2
We get an order filter generated by all the “upsets” of any (M0 , ε0 ), i.e. all indices greater than or
equal to a particular fixed index:
↑ {(M0 , ε0 )} = {(M, ε) : (M0 , ε0 ) ≺ (M, ε)}
The set of subsets of I which contain an upset is a filter: upsets are non-empty, and I clearly
contains upsets, containing a set that contains an upset means you contain an upset, and the
intersection of two upsets is an upset:
↑ {(M1 , ε1 )}∩ ↑ {(M2 , ε2 )} = {(M, ε) : M1 , M2 ⊆ M, ε1 , ε2 ≤ ε} =↑ {(M1 + M2 , max(ε1 , ε2 ))}
Now, let U be an ultrafilter dominating the order filter, denote an element i ∈ I by (Mi , εi ).
6
By assumption, for each i ∈ I there
Q is a space Ei ∈ B and a (1 + εi )-isomorphism Ti : Mi →
Ni ⊆ Ei . We define a map J : X → U Ei for x ∈ X by
(
Ti (x) if x ∈ Mi
J(x) = (yi )U
yi =
0
otherwise
Then J is clearly linear, and for all ε0 > 0, the set
I0 =↑ {span(x), ε0 } = {(M, ε) : x ∈ M, ε ≤ ε0 } ∈ U
Then since yi = 0 or yi = Ti (x), we have
||yi || = ||Ti (x)|| ≤ ||Ti || ||x|| ≤ ε0 ||x||
and
||x|| = ||Ti−1 (yi )|| ≤ ||Ti−1 || ||yi ||
so
Thus
||x||
||x||
≤
||yi || ≤ ε0 ||x||
1 + ε0
||Ti−1 ||
||x||
≤ ||Jx|| = lim ||yi || ≤ (1 + ε0 )||x||
U
1 + ε0
Since this holds for all ε0 > 0, we can conclude J is an isometry of X into
Q
U
Ei as claimed.
This type of approach is typical: the finite substructures of a given object are used to build
up an ultrafilter which then allows one to reproduce the whole object as a substructure of a corresponding ultraproduct.
We are ready to establish the connection between finite representability and ultrapowers: Recall
that a Banach space F is finitely representable in a Banach space E if for each finite-dimensional
subspace M ⊆ F and each ε > 0 this is a (1 + ε)-isomorphism subspace N ⊆ E.
Theorem. F is finitely representable in E if and only if there is an ultrafilter U such that F is
isometric to a subspace of E U .
Moreover, if F is separable and finitely representable in E, then F embeds isometrically into
each ultrapower E U where U is a countable incomplete ultrafilter.
Definition. If P is a Banach space property, recall that a Banach space E is said to be super-P
if and only if each Banach space F which is finite representable in E is P.
A precise knowledge about the structure of ultrapowers allow one to localize infinite dimensional
results - to transform connections between infinite dimensional properties to connections between
the local properties. Here is an example:
Proposition. A Banach space E is super-reflexive if and only if each ultrapower E U is reflexive.
Proposition. Let F be a subspace of E. If F and E/F are super-reflexive, then E is superreflexive.
7
Proof. Recall the classical result that if F and E/F are reflexive, then E is reflexive. We want to
localize this result. Note that (E/F )U is canonically isometric to (E)U /(F )U , since if
(ēi ) ∈ (E/F )U ,
||(ēi )|| = lim ||ēi || = lim inf ||ei − fi ||
U
U fi ∈F
On the other hand,
(ei ) ∈ (E)U /(F )U ,
||(ei )|| =
inf
(fi )∈(F )U
||(ei ) − (fi )|| =
inf
lim ||ei − fi ||
(fi )∈(F )U U
Which is the same as the above. By assumption, F and E/F are super-reflexive, so (F )U and
(E/F )U ∼
= (E)U /(F )U are reflexive, so E U is reflexive, and thus, E is super-reflexive.
This gives us a framework for defining a local variant of a Banach space property P by saying
E has local P if and only if every ultrapower of E has P . By our above work, local P is determined
by behavior on finite-dimensional subspaces.
Let us express a relation between a Banach space and its second dual using ultrapowers and
the principle of local reflexivity. Recall any Banach space E embeds canonically into its double
dual by the map ι which sends an element to evaluation at that element.
Theorem (The Principal of Local Reflexivity). If E is a Banach space, then for all M a finite
dimensional subspace of the double dual E ∗∗ , all finite subsets {f1 , . . . , fn } in the dual E ∗ , and
each ε > 0, there is an operator T : M → E which is a (1 + ε)-isomorphism onto its image, which
satisfies
T |M ∩ι(E) (ι(x)) = x
hT (u)|fk i = hu, fk i for all u ∈ M, 1 ≤ k ≤ n
An immediate corollary of the Principal of Local Reflexivitiy is that E ∗∗ is finitely representable
in E, and in particular, it follows from our previous work that E ∗∗ embeds isometrically into the
ultrapower (E)U . In fact, more is true:
Theorem. For each Banach space E there exists an ultrafilter U and an isometric embedding
J : E ∗∗ → E U . The restriction of J to ι(E) is the canonical diagonal embedding of E into its
ultrapower. Furthermore, there is a projection of norm 1 π : E U J(E ∗∗ ).
The proof is similar to our previous construction.
Q
0
Theorem.
Let
E
be
a
family
of
Banach
spaces
and
U
is
countably
incomplete.
Then
(
i
U Ei ) =
Q
0
U Ei if and only if (E)i is reflexive.
Corollary. If U is countably complete, then (E U )0 = (E 0 )U if and only if E is super-reflexive.
References
[1] Stefan Heinrich. Ultraproducts in Banach space theory. J. Reine Angew. Math., 313:72–104,
1980.
[2] Brailey Sims. “Ultra”-techniques in Banach space theory, volume 60 of Queen’s Papers in Pure
and Applied Mathematics. Queen’s University, Kingston, ON, 1982.
[3] Jacques Stern. Ultrapowers and local properties of Banach spaces. Trans. Amer. Math. Soc.,
240:231–252, 1978.
8