Abstract
It is shown that form≠n the polydisc algebrasA(D m) andA(D n) are not isomorphic as Banach spaces. More precisely, there is no linear embedding of the dual spaceA(D n)* intoA(D m)* form<n. The invariant is infinite dimensional and is based on certain, multi-indexed martingales related to those considered by Davis et al. [10]. In the one-dimensional case, i.e. for the spaceA(D)*, a finite inequality is proved, implying thatA(D 2)* is not finitely representable inA(D)*. Extensions to algebras on products of strictly pseudoconvex domains are outlined. They imply in particular the non-isomorphism of certain algebras in the same number of variables, for instance A (D4) ≠ A (B2xB2).
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Bourgain, J. The dimension conjecture for polydisc algebras. Israel J. Math. 48, 289–304 (1984). https://doi.org/10.1007/BF02760630
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DOI: https://doi.org/10.1007/BF02760630