Extensions of the notions of polynomial and rational hullbanach2019/pdf/Izzo.pdf · 2019. 7....
Transcript of Extensions of the notions of polynomial and rational hullbanach2019/pdf/Izzo.pdf · 2019. 7....
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Extensions of the notions of polynomial and rational hull
Alexander J. Izzo
Banach Algebras and Applications, July 2019, University of Manitoba
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Outline
I. Polynomial and Rational Hulls
II. Motivating Questions
III. The New Hulls
IV. Applications
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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= ∅.
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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= ∅.
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 .
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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.
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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.
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.
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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.
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).
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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}.
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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}.
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= ∅.
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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}.
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 .
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Analytic Structure in Polynomial Hulls
Observe: If X ⊂ Cn bounds an analytic variety V , then by the maximum
principle, V is contained in X̂.
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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.
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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.
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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.
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 .
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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.
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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)
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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)
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.
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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.
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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.
Question (Bercovici 2014): Does there exist a (nonsmooth) 1-dimensional
manifold with nontrivial polynomial hull without analytic discs?
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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.
Question (Bercovici 2014): Does there exist a (nonsmooth) 1-dimensional
manifold with nontrivial polynomial hull without analytic discs?
A similar question was raised by Wermer (1954) but for 1-dimensional
manifold in C2.
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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.
Question (Bercovici 2014): Does there exist a (nonsmooth) 1-dimensional
manifold with nontrivial polynomial hull without analytic discs?
A similar question was raised by Wermer (1954) but for 1-dimensional
manifold in C2.
Question: Which compact spaces can be embedded in some CN so as
to have nontrivial polynomial hull without analytic discs?
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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.
Question (Bercovici 2014): Does there exist a (nonsmooth) 1-dimensional
manifold with nontrivial polynomial hull without analytic discs?
A similar question was raised by Wermer (1954) but for 1-dimensional
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?
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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.
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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.
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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?
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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?
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.
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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 .
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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 .
How to define k-polynomial hull?
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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 .
How to define k-polynomial hull?
Note: If z ∈ h2r(X), then for V = {p = 0} (p a polynomial), z ∈
hr((X ∩ V )).
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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 .
How to define k-polynomial hull?
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 .
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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 .
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 .
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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 .
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.
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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”.
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Hulls without Analytic Discs
Fundamental Construction (ii)
(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.
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Hulls without Analytic Discs
Fundamental Construction (ii)
(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.
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:
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Hulls without Analytic Discs
Fundamental Construction (ii)
(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.
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:
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.
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Hulls without Analytic Discs
Fundamental Construction (ii)
(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.
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:
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.
What makes the proof of the lemma work is that ∂̂B2
= B.
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Hulls without Analytic Discs
Fundamental Construction (ii)
(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.
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:
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.
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.
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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).
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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).
Theorem: There exists a Cantor set in CN (N ≥ 2) whose (N − 1)-
rational hull contains an open set of CN .
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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).
Theorem: There exists a Cantor set in CN (N ≥ 2) whose (N − 1)-
rational hull contains an open set of CN .
Corollary: For N ≥ 2, there exists a totally disconnected, perfect set in
CN that intersects every analytic subvariety of CN of positive dimension.
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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).
Theorem: There exists a Cantor set in CN (N ≥ 2) whose (N − 1)-
rational hull contains an open set of CN .
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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).
Theorem: There exists a Cantor set in CN (N ≥ 2) whose (N − 1)-
rational hull contains an open set of CN .
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).
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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).
Theorem: There exists a Cantor set in CN (N ≥ 2) whose (N − 1)-
rational hull contains an open set of CN .
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).
Corollary: There exists a simple closed curve (and an arc) in C3 with
nontrivial polynomial hull containing no analytic discs.
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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).
Theorem: There exists a Cantor set in CN (N ≥ 2) whose (N − 1)-
rational hull contains an open set of CN .
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).
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.
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Open Question
Does there exist a Cantor set in C2 with a nontrivial polynomial hull
that contains no analytic discs?