A function algebraic characterization of Riemann domains (Q1204069)

The author proves the following result on the characterization of Riemann domains: Theorem: i) Let \(A\) be an \(F\)-algebra with defining sequence of seminorms \(\{p_ k\}\) and let \(M(A)\) be the set of nonzero continuous linear functionals on \(A\). Denote by \(\hat A\) the algebra of all Gelfand transforms \(\hat f\), \(f\in A\). Suppose there exist \(f_ 1,\dots,f_ n\in A\) and derivations \(D_ 1,\dots,D_ n\) on \(A\) such that \(\alpha)\) the map \(\hat F\): \(M(A)\to\mathbb{C}^ n\), \(\varphi\to(\hat f_ 1(\varphi),\dots,\hat f_ n(\varphi))\), is locally injective, i.e. for each \(\varphi\in M(A)\) there exists an open neighbourhood \(U\) of \(\varphi\) on which \(F\) is injective; \(\beta)\) the matrix \((\varphi(D_ j(f_ i)))_{i,j}\) has rank \(n\) for each \(\varphi\in M(A)\); \(\gamma)\) there is a sequence \(\{r_ i\}\) of positive numbers such that \[ p_ k(D^ v(f))\leq v!\;p_{k+1}(f)(r_ k)^{-v}\quad\text{for all } v\in\mathbb{N}^ n_ 0,\;f\in A,\;k\in\mathbb{N}. \] Then \(M(A)\) can be given the structure of a holomorphically and algebraically convex Riemann domain spread over \(\mathbb{C}^ n\) such that \(A={\mathcal O}(M(A))\). ii) if \(A\) is moreover semisimple, then \(A\) is topologically and algebraically isomorphic to \({\mathcal O}(M(A))\).