On the approximation of functions of several complex variables on fat compact subsets of $\mathbf C^n$ by polynomials

For a compact set $J\subset\mathbf C^n$, we denote by $P(J)$ the algebra of all functions on $J$ which can be approximated uniformly (on $J$) by polynomials in $n$ complex variables, and by $A(J)$ the algebra of all continuous functions on $J$ which are analytic at the interior points of $J$. We shall say that $J$ is fat if it is the closure of an open set.
In this paper, we consider the problem of approximating functions of several complex variables on fat compact sets with connected interior by polynomials. We prove the following theorems.
Theorem 1. There exists a fat polynomially convex $($holomorphically$)$ contractible compact subset $J$ of $\mathbf C^2$ whose interior is homeomorphic to the four-dimensional open ball and such that $P(J)\ne A(J)$.
Theorem 2. There exists a fat polynomially convex contractible compact subset $J$ of $\mathbf C^3$ whose interior is homeomorphic to the six-dimensional open ball and such that $P(J)\ne A(J)$, although the minimal boundaries of the algebras $P(J)$ and $A(J)$ coincide.
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