Weakly strongly regular uniform algebras

Given a uniform algebra



A
A



on a compact Hausdorff space



X
X



and a point



x
x



in



X
X



, denote by




M
x

M_x



the ideal of functions in



A
A



that vanish at



x
x



and by




J
x

J_x



the ideal of functions in



A
A



that vanish on a neighborhood of



x
x



. It is shown that for each integer




m

≥


2

m\geq 2



, there exists a compact plane set



K
K



containing the origin such that in




R
(
K
)

R(K)



we have






J
x


¯




⊃



M
x


\overline {J_x}\supset M_x



for every




x

∈


K

∖


{
0
}

x\in K\setminus \{0\}



and






J
0


¯




⊃



M
0
m


\overline {J_0}\supset M_0^m



but






J
0


¯



⊅

M
0

m

−


1



\overline {J_0} \not \supset M_0^{m-1}



. This result establishes a recent conjecture of Alexander Izzo [Trans. Amer. Math. Soc. 378 (2025), 8967–8988]. For the proof we introduce a construction that could be described as taking square roots of Swiss cheeses.