Download Where is pointwise multiplication on CK

Where is pointwise multiplication on CK -spaces
locally open?
Ehrhard Behrends
Freie Universität Berlin
Andalucia, September 2015
... a workshop in honour of R. Payá ...
A survey
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Open linear mappings
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Open multilinear mappings: some first observations
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The Lodz connection
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“Walk the dog”
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Multiplication from (CR [ 0, 1 ])n to CR [ 0, 1 ]
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The variant for CK -spaces
I
Complex scalars
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Open problems
Open linear mappings
Everybody here knows that linear continuous surjective mappings
between Banach spaces are open. And it easy to find examples
where the map is not open in the nonlinear case.
But what happens with mappings that are “mildly” nonlinear, e.g.,
with multilinear mappings?
Open multilinear mappings: some first observations
Let us concentrate on the seemingly innocent case of the map
M : (f , g ) 7→ f · g (pointwise multiplication) from
CR [ 0, 1 ] × CR [ 0, 1 ] to CR [ 0, 1 ]. This map is not open!
Open multilinear mappings: some first observations
Let us concentrate on the seemingly innocent case of the map
M : (f , g ) 7→ f · g (pointwise multiplication) from
CR [ 0, 1 ] × CR [ 0, 1 ] to CR [ 0, 1 ]. This map is not open!
Suppose it were. Let O be the open set
{(f , g ) | ||f − f0 ||, ||g − g0 || < 1/2},
where f0 = g0 is the map t 7→ t − 0.5. If M were open there would
exist δ > 0 such that M(O) contains the δ-ball B = B(f02 , δ)
around M(f0 , g0 ) = f02
B contains a strictly positive function, but – due to the
intermediate value theorem – every function in B(O) vanishes
somewhere in [ 0, 1 ]. This contradiction shows that M is not open.
The following definition will be important: A map φ : A 7→ B
between metric spaces A, B will be called locally open at x0 ∈ A, if
for every ε > 0 there is a positive δ such that the image of the
ε-ball around x0 contains the δ-ball around φ(x0 ).
Clearly φ is open iff it is locally open at every x0 .
The Lodz connection
Some years ago I came in touch with these problems by a paper of
Balcerzak and his colleagues from the university of Lodz, Poland.
(This lovely city is pronounced “wu:dsch”.)
The Lodz connection
Some years ago I came in touch with these problems by a paper of
Balcerzak and his colleagues from the university of Lodz, Poland.
(This lovely city is pronounced “wu:dsch”.)
Here are some typical results that were found by the Balcerzak
group and by collaboration of this group with myself:
I
If 1/p + 1/q = 1, then the map (x, y ) 7→ xy from Lp × Lq to
L1 is open (pointwise multiplication).
The Lodz connection
Some years ago I came in touch with these problems by a paper of
Balcerzak and his colleagues from the university of Lodz, Poland.
(This lovely city is pronounced “wu:dsch”.)
Here are some typical results that were found by the Balcerzak
group and by collaboration of this group with myself:
I
If 1/p + 1/q = 1, then the map (x, y ) 7→ xy from Lp × Lq to
L1 is open (pointwise multiplication).
I
Let X1 , . . . , Xn be normed spaces over the scalar field K and
φ : X1 × · · · × Xn → K a nontrivial multilinear map. Then φ is
open.
More generally, things are well understood in the case of natural
multiplication mappings between spaces of measurable functions
and in the case where the range space is the scalar field.
In the present talk we will mainly concentrate on spaces of
continuous functions.
“Walk the dog”
In view of the above counterexample the following question is
natural:
Let f , g ∈ CR [ 0, 1 ]) be given. Can one decide whether
pointwise multiplication is locally open at (f , g )?
To state it otherwise: Is it true or not that for every ε > 0 one can
choose a positive δ such that for h with ||h − fg || ≤ δ one finds
f˜, g̃ with f˜g̃ = h and ||f − f˜||, ||g − g̃ || ≤ ε ?
A characterization was given in my paper “Walk the dog”
(Functiones et Approximatio 44, 2011).
The title was chosen since the problem can be translated into the
possibility of certain walks of a dog “close” to his master.
Given f , g it will be important to consider the walk
γ : t 7→ f (t), g (t)
in R2 .
Definition: We say that γ has a positive saddle point crossing if
there is an interval [ a, b ] ⊂ [ 0, 1 ] such that
γ(t) ∈ Q −− ∪ Q ++ = {(x, y ) | x, y ≤ 0} ∪ {(x, y ) | x, y ≥ 0}
for t ∈ [ a, b ] and γ(t) ∈ (Q −− )o for some t as well as
γ(t) ∈ (Q ++ )o for some t.
A negative saddle point crossing is defined similarly, there the
quadrants Q −+ and Q +− come into play.
Here is our characterization:
Theorem: Pointwise multiplication is locally open iff γ admits no
positive and no negative saddle point crossings.
One direction of the proof is simple: If there is a positive or a
negative saddle point crossing then multiplication will not be
locally open at (f , g ). One only has to adapt the proof of the
standard counterexample. The proof of the other direction is rather
involved. A crucial observation is the following lemma:
Multiplication will be locally open at (f , g ) iff for every ε there is a
δ such that B(f , ε) · B(g , ε) contains fg ± δ.
Multiplication from (CR [ 0, 1 ])n to CR [ 0, 1 ]
In my paper “Products of n open sets in the space of continuous
functions on [ 0, 1 ]” (Studia Math. 204, 2011) I generalized the
preceding characterization to the multiplication operator from
n
C [ 0, 1 ] to C [ 0, 1 ].
There one deals with “walks” in Rn , and one of the main
difficulties was to generalize the notion of positive/negative saddle
point crossings to this new situation.
This paper also contains a first discussion of the case of complex
valued functions on [ 0, 1 ] (see below).
The variant for CK -spaces
The most recent results concern multiplication on general real
CK -spaces. (The corresponding paper is submitted.) We fix a
compact Hausdorff space K and we want to characterize the pairs
(f , g ) in CR K where pointwise multiplication is locally open.
As in the case K = [ 0, 1 ] one can show that this is the case iff for
every ε there is a δ such that fg ± δ lie in B(f , ε)B(g , ε).
Fix positive ε and η. The following subsets of R2 will play an
important role in our characterization:
Figure 1: Mη,1 , Mη,ε,1 and Mη,ε,2 ; Mη,2 = Mη,ε,1 ∪ Mη,ε,2 .
Figure 2: M−η,1 , M−η,ε,1 and M−η,ε,2 ; M−η,2 = M−η,ε,1 ∪ M−η,ε,2 .
Let us concentrate on the first three sets:
The main result: For every ε there is a δ with
fg − δ ∈ B(f , ε)B(g , ε) iff the set {k | −η ≤ f (k)g (k) ≤ η}
“splits” appropriately. More precisely:
The relevant condition is the following: For every ε > 0 there is an
η > 0 such that {k | −η ≤ f (k)g (k) ≤ η} can
be written as the
disjoint union K1 ∪ K2 such that f (k), g (k) ∈ Mη,ε,1 for k ∈ K1
and f (k), g (k) ∈ Mη,ε,2 for k ∈ K2 .
Complex scalars
What happens if one replaces CR K by CC K ? Here we only have
the following result:
The multiplication map from CC [ 0, 1 ] × CC [ 0, 1 ] to CC [ 0, 1 ] is
open.
In the proof it is crucial that the boundary of the unit ball in C is –
in contrast to the real case – connected.
Open problems
The following problems should be investigated next:
1. Given K , characterize the
n n-tupels (f1 , . . . , fn ) where
multiplication (from CR K to CR K is locally open. The main
difficultity is that there is no analogue to the case of two factors
where it suffices to verify that fg ± δ are in B(f , ε)B(g , ε).
Open problems
The following problems should be investigated next:
1. Given K , characterize the
n n-tupels (f1 , . . . , fn ) where
multiplication (from CR K to CR K is locally open. The main
difficultity is that there is no analogue to the case of two factors
where it suffices to verify that fg ± δ are in B(f , ε)B(g , ε).
2. Multiplication from CC [ 0, 1 ] × CC [ 0, 1 ] to CC [ 0, 1 ] is open,
what about multiplication
n
I from CC [ 0, 1 ]
to CC [ 0, 1 ] ?
Open problems
The following problems should be investigated next:
1. Given K , characterize the
n n-tupels (f1 , . . . , fn ) where
multiplication (from CR K to CR K is locally open. The main
difficultity is that there is no analogue to the case of two factors
where it suffices to verify that fg ± δ are in B(f , ε)B(g , ε).
2. Multiplication from CC [ 0, 1 ] × CC [ 0, 1 ] to CC [ 0, 1 ] is open,
what about multiplication
n
I from CC [ 0, 1 ]
to CC [ 0, 1 ] ?
I from CC K × CC K to CC K for general K ?
Open problems
The following problems should be investigated next:
1. Given K , characterize the
n n-tupels (f1 , . . . , fn ) where
multiplication (from CR K to CR K is locally open. The main
difficultity is that there is no analogue to the case of two factors
where it suffices to verify that fg ± δ are in B(f , ε)B(g , ε).
2. Multiplication from CC [ 0, 1 ] × CC [ 0, 1 ] to CC [ 0, 1 ] is open,
what about multiplication
n
I from CC [ 0, 1 ]
to CC [ 0, 1 ] ?
I from CC K × CC K to CC K for general K ?
3. The investigations presented so far are special cases of the
following situation:
Let A be a Banach algebra. Characterize the pairs (x0 , y0 )
where multiplication (from A × A to A) is locally open.
Even for seemingly simple situations (like finite-dimensional matrix
algebras) very few results are known.
Thank you for your attention!
... a workshop in honour of R. Payá ...