The authors study the following problem for \(U\subset X\), \(V\subset Y\), convex balanced open subsets of Banach spaces \(X\) and \(Y\). Suppose that two algebras of holomorphic functions \({F}(U)\) and \({F}(V)\) are topologically algebra-isomorphic, then are \(X\) and \(Y\) (or \(X'\) and \(Y'\)) isomorphic? The problem is studied mainly for two types of spaces of holomorphic functions: \({F}={H}_{wu}\), uniformly weakly continuous holomorphic functions; and \({F}={H}_b\), holomorphic functions of bounded type. Among the many interesting results are: If \(X'\) or \(Y'\) has the approximation property and \({H}_{wu}(U)\) is topologically algebra-isomorphic to \({H}_{wu}(V)\), then \(X'\) is isomorphic to \(Y'\). If every polynomial on \(X''\) or \(Y''\) is approximable (in norm, by finite-type polynomials) and \({H}_b(U)\) is topologically algebra-isomorphic to \({H}_b(V)\), then \(X'\) is isomorphic to \(Y'\). Both results are preceded by characterizations of continuous algebra homomorphisms as composition operators on certain hull-like sets obtained from \(U\) and \(V\). This in turn requires extensive consideration of the spectra of these algebras, in the course of which several interesting examples and results are presented. Finally, the authors show that, in the absence of approximability, an analogous study of this problem, and of composition operators, leads to pathological -- and surprising -- results. For example: Let \(X\) be a symmetrically regular Banach space with unconditional finite-dimensional Schauder decomposition, and suppose that there exists a continuous \(k\)-homogeneous polynomial which is not weakly-sequentially continuous. Then there is a continuous \((k+1)\)-homogeneous polynomial \(P:X\rightarrow X\) such that the mapping \(\phi \mapsto \phi \circ P^t\) on the spectrum of \({H}_b(X)\) is not continuous.