In discussion with a colleague recently (Jan 2017), $\newcommand{\AD}{A({\bf D})}\newcommand{\CT}{C({\bf T})}$ I was reminded that if $A(D)$ denotes the disc algebra and $\iota: \AD\to \CT$ is the standard isometric embedding, then $$\iota\otimes\iota : \AD\hat\otimes\AD \to \CT\hat\otimes\CT$$ has closed range. Since $\AD$ is not complemented as a Banach space in $\CT$ this is far from obvious, and to my knowledge the only proofs go via a version for $\AD$ of Grothendieck's theorem/inequality, established originally by Bourgain in the 1980s.
Many years ago I was interested in what was known for other uniform algebras. My recollection is that if one replaces $\AD$ with the polydisc algebra or the ball algebra in $n$ dimensions, $n\geq 2$, then it was unknown if the corresponding map $\iota\otimes\iota$ into the appropriate $C(K)\hat\otimes C(K)$ has closed range.
Q1. Is it known by now if the corresponding result for polydisc algebras or ball algebras is false?
Q2. Are there any other examples known where the analogous result is true? (Obviously one could do trivial modifications of the disc algebra case by taking a uniform algebra that contains a finite-codimension closed copy of $\AD$; I mean genuinely new examples such as $R(K)$ for other domains.)
