On rationally convex hulls
Authors: Basener Richard F. 
Published: July 2010

For a compact set

X
⊆

C

n

X \subseteq {{\mathbf {C}}^n}

, let

h
r

(
X
)

{h_r}(X)

denote the rationally convex hull of X; let

Δ

{\mathbf {\Delta }}

denote the closed unit disk in C; and, following Wermer, for a compact set S such that

∂

Δ

⊆
S
⊆

Δ

\partial {\mathbf {\Delta }} \subseteq S \subseteq {\mathbf {\Delta }}

let

X
S

=
S
×
S
∩
∂

Δ

2

{X_S} = S \times S \cap \partial {{\mathbf {\Delta }}^2}

. It is shown that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h Subscript r Baseline left-parenthesis upper X Subscript upper S Baseline right-parenthesis equals StartSet left-parenthesis z comma w right-parenthesis element-of upper S times upper S vertical-bar u Subscript upper S Baseline left-parenthesis z right-parenthesis Superscript plus Baseline u Subscript upper S Baseline left-parenthesis w right-parenthesis less-than-or-equal-to 1 EndSet">

h
r

(

X
S

)
=
{
(
z
,
w
)
∈
S
×
S

|

u
S

(
z

)
+

u
S

(
w
)
≤
1
}

{h_r}({X_S}) = \{ (z,w) \in S \times S|{u_S}{(z)^ + }{u_S}(w) \leq 1\}

\]
where

u
S

{u_S}

is a function on S which, in the case when S is smoothly bounded, is specified by requiring
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u Subscript upper S Baseline StartAbsoluteValue equals 0 comma u Subscript upper S Baseline EndAbsoluteValue Subscript partial-differential bold upper Delta Baseline Subscript partial-differential upper S minus partial-differential bold upper Delta Baseline equals 1">

u
S

|

∂

Δ

=
0
,

u
S

|

∂
S
∖
∂

Δ

=
1

{u_S}{|_{\partial {\mathbf {\Delta }}}} = 0,{u_S}{|_{\partial S\backslash \partial {\mathbf {\Delta }}}} = 1

and

u
S

|

int
⁡
S

{u_S}{|_{\operatorname {int} S}}

harmonic. In particular this provides a precise description of

h
r

(
X
)

{h_r}(X)

for certain sets

X
⊆

C

2

X \subseteq {{\mathbf {C}}^2}

with the property that

h
r

(
X
)
≠
X

{h_r}(X) \ne X

, but

h
r

(
X
)

{h_r}(X)

does not contain analytic structure (as Wermer demonstrated, there are S for which

X
=

X
S

X = {X_S}

has these properties). Furthermore, it follows that whenever

h
r

(

X
S

)
≠

X
S

{h_r}({X_S}) \ne {X_S}

then there is a Gleason part of

h
r

(

X
S

)

{h_r}({X_S})

for the algebra

R
(

X
S

)

R({X_S})

with positive four-dimensional measure. In fact, the Gleason part of any point

(
z
,
w
)
∈

h
r

(

X
S

)
∩
int
⁡

Δ

2

(z,w) \in {h_r}({X_S}) \cap \operatorname {int} {{\mathbf {\Delta }}^2}

such that

u
S

(
z
)
+

u
S

(
w
)
>
1

{u_S}(z) + {u_S}(w) > 1

has positive four-dimensional measure. A similar idea is then used to construct a compact rationally convex set

Y
⊆

C

2

Y \subseteq {{\mathbf {C}}^2}

such that each point of Y is a peak point for

R
(
Y
)

R(Y)

even though

R
(
Y
)
≠
C
(
Y
)

R(Y) \ne C(Y)

; namely,
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y equals upper X overTilde Subscript upper T Baseline equals StartSet left-parenthesis z comma w right-parenthesis element-of bold upper C squared StartAbsoluteValue z element-of upper T comma EndAbsoluteValue w vertical-bar equals StartRoot 1 minus StartAbsoluteValue z EndAbsoluteValue squared EndRoot EndSet">

Y
=

X
~

T

=
{
(
z
,
w
)
∈

C

2

|

z
∈
T
,

|

w

|

=

1
−

|

z

|

2

}

Y = {\tilde X_T} = \{ (z,w) \in {{\mathbf {C}}^2}|z \in T,|w| = \sqrt {1 - |z{|^2}} \}

where T is any compact subset of

int
⁡

Δ

\operatorname {int} {\mathbf {\Delta }}

having the property that

R
(
T
)
≠
C
(
T
)

R(T) \ne C(T)

even though there are no nontrivial Jensen measures for

R
(
T
)

R(T)

. This example is more concrete than the original example of such a uniform algebra which was discovered by Cole. It is possible to show, for instance, that

R
(

X
~

T

)

R({\tilde X_T})

is not even in general locally dense in

C
(

X
~

T

)

C({\tilde X_T})

, a possibility which had been suggested by Stuart Sidney. Finally, smooth examples (3-spheres in

C

6

{{\mathbf {C}}^6}

) with the same pathological properties are obtained from

X
S

{X_S}

and

X
~

T

{\tilde X_T}

.