Exercise 18, § 4

Let A be a commutative unital Banach algebra.

a) For every finite family I of elements of A, let $S(I)\subset \mathbf{C}^I$ be the simultaneous spectrum of I. If $I'$ extends I, the canonical surjection pr$_{I,I'}$ of $\mathbf{C}^{I'}$ onto $\mathbf{C}^I$ is such that pr$_{I,I'}(S(I')) = S(I)$, whence an injective morphism $\varphi_{II'}$ of $\mathscr{O}(S(I))$ into $\mathscr{O}(S(I'))$. Let $\mathscr{O}(\mathsf{X}(A))$ be the inductive limit of the $\mathscr{O}(S(I))$ relative to the $\varphi_{II'}$ (one indexes the finite families of elements of A by the finite subsets of $A\times \mathbf{N}$). The canonical mappings

are injective.

We shall endow $\mathscr{O}(\mathsf{X}(A))$ with the inductive limit topology of those of the $\mathscr{O}(S(I))$. For every $a\in A$, the coordinate function in the neighbourhood of $S(a)\subset \mathbf{C}$ defines an element $z_a$ of $\mathscr{O}(\mathsf{X}(A))$.

b) The continuous surjection $\mathsf{X}(A)\rightarrow$ S(I) defines a continuous morphism $\mathscr{C}(S(I))\rightarrow \mathscr{C}(\mathsf{X}(A))$. The morphisms $\mathscr{O}(S(I))\rightarrow \mathscr{C}(S(I))\rightarrow \mathscr{C}(\mathsf{X}(A))$ define a continuous morphism:

This morphism is not injective in general.

c) There exists a unique continuous unital morphism $\psi$ of $\mathscr{O}(\mathsf{X}(A))$ into A such that $\psi (z_a) =a$ for every $a\in A$. (Use theorem 1 of I, p. 51.) Its composite with the Gelfand transform is $\varphi$.

d) Let E be a separated locally convex space, U a weakly open subset of E and $f: U\rightarrow \mathbf{C}$ a function. We say that $f$ is weakly holomorphic if, for every $u\in U$, there exists a weakly open neighbourhood $V_u$ of $u$, a finite codimension closed vector subspace $F_u$ of E, and a holomorphic function $g_u$ on $p_u(V_u)$, where $p_u$ denotes the canonical mapping of E onto $E/F_u$, such that $f|V_u=g_u\circ p_u$.

Show that if $f$ is weakly holomorphic, and if K is a weakly compact subset of U, there exists a weakly open neighbourhood $V_K$ of K in U, a finite codimension closed vector subspace $F_K$ of E, and a holomorphic function $g_K$ on $p_K(V_K)$ such that $f|V_K=g_K\circ p_K$, where $p_K$ is the canonical mapping of E onto $E/F_K$.

e) Show that $\mathscr{O}(\mathsf{X}(A))$ is the inductive limit of the $\mathscr{O}(S)$, where S runs through the set of weakly open neighbourhoods of $\mathsf{X}(A)$ in the dual of A, and where $\mathscr{O}(S)$ denotes the algebra of weakly holomorphic functions in S, endowed with the topology of compact convergence.