The dimension conjecture for polydisc algebras (Q1062250)

Let \(A(D^ d)\) be the algebra of continuous complex-valued functions on d-dimensional polydisc \(\bar D^ d\) that are analytic on \(D^ d\), equipped with sup-norm. The author shows that for \(m<n\) there is no linear embedding of the dual space \(A(D^ n)^*\) into \(A(D^ m)^*\). As an immediate corollary he gets that there is no linear isomorphism between \(A(D^ m)\) and \(A(D^ n)\) whenever \(m\neq n\). The corresponding invariant is infinite-dimensional and is connected with certain multi- indexed martingales. It is shown that the space \((AD^ 2)^*\) is not finitely representable in \(A(D)^*\). The same arguments show that given some strictly pseudoconvex bounded closed domains \(U_ i\subset {\mathbb{C}}^{m_ i}\) and \(V_ j\subset {\mathbb{C}}^{m'\!_ j}\), where \(1\leq i\leq d\), \(1\leq j\leq d'\), with \(C^ 2\)-smooth boundaries, such that \(U=U_ 1\times...\times U_ d\) and \(V=V_ 1\times...\times V_{d'}\) are domains of holomorphy in \({\mathbb{C}}^ m\) and \(C^{m'}\) respectively, \(m=m_ 1+...+m_ d\), \(m'=m'\!_ 1+...+m'\!_{d'}\), the corresponding algebras A(U) and A(V) are not linearly isomorphic whenever \(d\neq d'\). As a corollary it is shown that many pairs of algebras of analytic functions in the same number of complex variables are not isomorphic.