Extensions of the notions of polynomial and rational hullbanach2019/pdf/Izzo.pdf · 2019. 7....

Extensions of the notions of polynomial and rational hull

Alexander J. Izzo

Banach Algebras and Applications, July 2019, University of Manitoba

Outline

I. Polynomial and Rational Hulls

II. Motivating Questions

III. The New Hulls

IV. Applications

Polynomial and Rational Convexity

X ⊂ Cn compact

Definition: The polynomial hull of X ⊂ Cn is the set

X̂ = {z ∈ Cn : |p(z)| ≤ maxx∈X

|p(x)| for every polynomial p}.

X is said to be polynomially convex if X̂ = X .

X̂ is said to be nontrivial if X̂ \ X 6= ∅.


X ⊂ Cn compact


X̂ = {z ∈ Cn : |p(z)| ≤ maxx∈X


X is said to be polynomially convex if X̂ = X .

X̂ is said to be nontrivial if X̂ \ X 6= ∅.

P (X)= uniform closure of polynomials in z1, . . . , zn on X

X̂ is the maximal ideal space of P (X).

In particular, P (X) = C(X) =⇒ X̂ = X .

Examples

In the plane, a compact set is polynomially convex if and only its com-

plement is connected. The polynomial hull of a compact set in the plane

is obtained by filling in the holes.

Examples




The situation is vastly more complicated in CN for N ≥ 2. Consider

K1 = {(eiθ, 0) : 0 ≤ θ ≤ 2π} K2 = {(e

iθ, e−iθ) : 0 ≤ θ ≤ 2π}

Both are circles, but K̂1 is the disc {(z, 0) : |z| ≤ 1}, while K2 is poly-

nomially convex.

Examples




The situation is vastly more complicated in CN for N ≥ 2. Consider

K1 = {(eiθ, 0) : 0 ≤ θ ≤ 2π} K2 = {(e

iθ, e−iθ) : 0 ≤ θ ≤ 2π}

Both are circles, but K̂1 is the disc {(z, 0) : |z| ≤ 1}, while K2 is poly-

nomially convex.

There exist non-polynomially convex arcs (Wermer 1955) and Cantor

sets (Rudin 1956).


X ⊂ Cn compact


X̂ = {z ∈ Cn : |p(z)| ≤ maxx∈X



X ⊂ Cn compact


X̂ = {z ∈ Cn : |p(z)| ≤ maxx∈X


Definition: The rational hull of X ⊂ Cn is the set

hr(X) = {z ∈ CN : p(z) ∈ p(X) for all polynomials p}.

X is said to be rationally convex if hr(X) = X .

hr(X) is said to be nontrivial if hr(X) \ X 6= ∅.


X ⊂ Cn compact


X̂ = {z ∈ Cn : |p(z)| ≤ maxx∈X


Definition: The rational hull of X ⊂ Cn is the set

hr(X) = {z ∈ CN : p(z) ∈ p(X) for all polynomials p}.

X is said to be rationally convex if hr(X) = X .

hr(X) is said to be nontrivial if hr(X) \ X 6= ∅.

R(X)= uniform closure of rational functions holomorphic on X

hr(X) is the maximal ideal space of R(X).

In particular, R(X) = C(X) =⇒ hr(X) = X .

Analytic Structure in Polynomial Hulls

Observe: If X ⊂ Cn bounds an analytic variety V , then by the maximum

principle, V is contained in X̂.




“Conjecture”: Every nontrivial polynomial hull contains an analytic disc.





Wermer (1958) prove this in the case of real-analytic 1-manifolds.





Wermer (1958) prove this in the case of real-analytic 1-manifolds.

Theorem (Alexander 1971): If J is a rectifiable arc in Cn, then J is

polynomially convex (and P (J) = C(J)).

Theorem (Alexander 1971): If J is a rectifiable simple closed curve in

Cn, then either J is polynomially convex (and P (J) = C(J)) or else

Ĵ \ J a one-dimensional complex analytic subvariety of Cn \ J .

Polynomial Hulls without Analytic Discs

Theorem (Stolzenberg 1963): There exists a compact set X in C2 such

that X̂ \ X 6= ∅ but X̂ contains no analytic discs.




Many later examples:

Wermer (1970, 1982)

Duval and Levenberg (1997)

Alexander (1998)




Many later examples:

Wermer (1970, 1982)

Duval and Levenberg (1997)

Alexander (1998)

Polynomial Hulls with Dense Invertibles

Theorem (Dales and Feinstein 2008): There exists a compact set X in

C2 with X̂ \ X 6= ∅ such that P (X) has dense invertibles.

Topology of Sets with Hull without Analytic Discs

Theorem (I., Samuelsson Kalm, Wold, 2016; I., Stout 2018): Every

smooth compact manifold of real dimension ≥ 2 smoothly embeds in

CN for some N so as to have nontrivial polynomial hull without analytic

discs.





discs.

Question (Bercovici 2014): Does there exist a (nonsmooth) 1-dimensional

manifold with nontrivial polynomial hull without analytic discs?





discs.



A similar question was raised by Wermer (1954) but for 1-dimensional

manifold in C2.





discs.




manifold in C2.

Question: Which compact spaces can be embedded in some CN so as

to have nontrivial polynomial hull without analytic discs?





discs.




manifold in C2.

Question: Which compact spaces can be embedded in some CN so as

to have nontrivial polynomial hull without analytic discs?

Fundamental Question: Does there exist a Cantor set with nontrivial

polynomial hull without analytic discs?

Cantor Sets with Hull with Interior

Theorem (Vitushkin 1973): There exists a Cantor set in C2 whose

polynomial hull contains an open set of C2.




Theorem (Henkin 2006): There exists a Cantor set in C2 whose rational

hull contains an open set of C2.






Question: Can Henkin’s theorem be generalized to CN for N > 2?






Question: Can Henkin’s theorem be generalized to CN for N > 2?

Answer: Yes, but a direct generalization is not so interesting, because

while in C2 to say z ∈ hr(X) means every analytic variety through z

intersects X , in contrast, in CN , N > 2, hr(X) concerns only codimen-

sion 1 analytic varieties.

k-Hulls

Definition: For 1 ≤ k ≤ N , the k-rational hull hkr (X) of X is the set

hkr (X) = {z ∈ CN : every analytic subvariety of CN of pure

codimension ≤ k that passes through z intersects X}.

We say that X is k-rationally convex if hkr (X) = X .

k-Hulls





How to define k-polynomial hull?

k-Hulls






Note: If z ∈ h2r(X), then for V = {p = 0} (p a polynomial), z ∈

hr((X ∩ V )).

k-Hulls






Note: If z ∈ h2r(X), then for V = {p = 0} (p a polynomial), z ∈

hr((X ∩ V )).

To define 2-polynomial hull, replace z ∈ hr((X ∩ V )) by z ∈ X̂ ∩ V .

k-Hulls





Definition: For 2 ≤ k ≤ N , the k-polynomial hull X̂k of X is the set

X̂k = {z ∈ CN : z ∈ hk−1r (X) and z ∈ X̂ ∩ V for every analytic subvariety V

of CN of pure codimension ≤ k − 1 that passes through z}.

We say that X is k-polynomially convex if X̂k = X .

k-Hulls





Definition: For 2 ≤ k ≤ N , the k-polynomial hull X̂k of X is the set

X̂k = {z ∈ CN : z ∈ hk−1r (X) and z ∈ X̂ ∩ V for every analytic subvariety V

of CN of pure codimension ≤ k − 1 that passes through z}.

We say that X is k-polynomially convex if X̂k = X .

With these definitions

X̂ = X̂1 ⊃ hr(X) = h1

r(X) ⊃ X̂2 ⊃ h2r(X) ⊃ · · · ⊃ X̂

n ⊃ hnr (X) = X.

Hulls without Analytic Discs

Three Fundamental Constructions

(i) Stolzenberg 1963: Take a limit of boundaries of analytic varieties whose

hulls are such that their projections to the coordinate planes miss points

of a dense set.

(ii) Wermer 1970: Successively remove sets from the boundary of a domain

in CN in such a way that what is left in the limit has hull without analytic

discs.

(iii) Wermer 1982 (based on Cole 1968): Take a limit of graphs of multivalued

analytic functions involving square roots to get a “Riemann surface with

an infinite number of branch points”.


Fundamental Construction (ii)



discs.





discs.

The key to (ii) is to cut out subsets with the property that a point of

the domain lies in the rational hull of the set that remains if it does not

lie in the polynomial hull of the set removed. One approach uses:





discs.




Lemma: Let p be a polynomial on CN and X = {ℜp ≤ 0} ∩ ∂B. (B =

unit ball in CN ) Then X̂ = hr(X) = {ℜp ≤ 0} ∩ B.





discs.






What makes the proof of the lemma work is that ∂̂B2

= B.





discs.






What makes the proof of the lemma work is that ∂̂B2

= B.

With this observation one can generalize the lemma to get a very flexible

method of constructing hulls without analytic discs.

Applications

Theorem: Any compact set X ⊂ CN with nontrivial k-polynomial hull

(k ≥ 2) contains a subset Y with nontrivial (k − 1)-rational hull such

that Ŷ contains no analytic discs (and P (Y ) has dense invertibles).

Applications




Theorem: There exists a Cantor set in CN (N ≥ 2) whose (N − 1)-

rational hull contains an open set of CN .

Applications






Corollary: For N ≥ 2, there exists a totally disconnected, perfect set in

CN that intersects every analytic subvariety of CN of positive dimension.

Applications






Applications






Theorem: For N ≥ 3, there exists a Cantor set K in CN with nontrivial

(N−2)-rational hull and whose polynomial hull contains no analytic discs

(and such that P (K) has dense invertibles).

Applications









Corollary: There exists a simple closed curve (and an arc) in C3 with

nontrivial polynomial hull containing no analytic discs.

Applications









Corollary: There exists a simple closed curve (and an arc) in C3 with

nontrivial polynomial hull containing no analytic discs.

Corollary: Every uncountable, compact, metrizable space of finite topo-

logical dimension can be embedded in some CN so as to have nontrivial

polynomial hull without analytic discs.

Open Question

Does there exist a Cantor set in C2 with a nontrivial polynomial hull

that contains no analytic discs?

Extensions of the notions of polynomial and rational hullbanach2019/pdf/Izzo.pdf · 2019. 7....

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