Let X be a reduced complex space, with countable topology. Consider the Fréchet algebra A of all global holomorphic functions on X. The continuous spectrum \(S_ c(A)\) is the set of all continuous \({\mathbb{C}}\)- algebra homomorphisms \(A\to {\mathbb{C}}\) with the Gelfand topology. There is a canonical continuous map \(\chi\) : \(X\to S_ c(A)\) given by \(x\mapsto \hat x\), where \(\hat x(f)\):\(=f(x).\)
In this survey article the author gives a summary of the results on the use of \(S_ c(A)\) and \(\chi\) for examination of global properties of X, for representation of the envelope of holomorphy of X and on topological properties of \(S_ c(A)\). Some of the results have been obtained by the author herself and together with J. P. Vigué.