Theorem 5

Let A be a unital Banach algebra, not necessarily commutative. Let $a$ be an element of A and $z$ the germ of the identity function of $\mathbf{C}$ in a neighbourhood of Sp$_A(a)$. There exists a unique continuous unital morphism $\varphi_a$ of $\mathscr{O}$(Sp$_A(a)$) into A such that $\varphi_a(z) =a$.

The image of $\varphi_a$ is contained in the full closed subalgebra of A generated by $a$. In particular, it is contained in the bicommutant of $a$.

Let us prove the existence of the morphism $\varphi_a$. Let B be the full closed subalgebra of A generated by $a$. It is commutative, and we have Sp$_B(a) =$ Sp$_A(a)$ (I, p. 5, No$^o5$). The mapping $\Theta_a$ of the holomorphic functional calculus on B is a continuous unital morphism of $\mathscr{O}$(Sp$_B(a)$) into B such that $\Theta_a(z) =a$ (Theorem 1 of I, p. 51). The morphism composed of $\Theta_a$ and the canonical injection of B into A is a continuous unital morphism $\varphi_a$ of $\mathscr{O}$(Sp$_A(a)$) into A such that the image of $z$ is $a$.

Let us prove uniqueness. Let $\varphi '_a$ be a continuous unital morphism from $\mathscr{O}$(Sp$_A(a)$) into A such that $\varphi '_a(z) =a$. Then $\varphi_a$ and $\varphi '_a$ coincide on the set of germs of polynomials in a neighbourhood of Sp$_A(a)$, hence on the set of germs of rational fractions holomorphic in a neighbourhood of Sp$_A(a)$. But these germs are dense in $\mathscr{O}$(Sp$_A(a)$) (I, p. 69, th. 3). This implies that $\varphi_a=\varphi '_a$.

The construction of $\varphi_a$ proves that its image is contained in the commutative subalgebra B, which is contained in the bicommutant of $a($I, p. 6).

If the radical of the algebra A is zero, the uniqueness of the morphism $\varphi_a$ is valid without requiring that it be continuous (cf. prop. 9 of I, p. 40). This is not so in general, cf. G. R. Allan, Embedding the algebra of formal power series in a Banach algebra, Proc. London Math. Soc. (3) 25 (1972), 329–340.

For every Banach algebra A, every element $a$ of A and every germ $f\in \mathscr{O}$(Sp$_A(a)$), one denotes by $f(a)$ the element $\varphi_a(f)$ of theorem 5. If A is a commutative Banach algebra, this element $f(a)$ coincides with the element $f(a)$ furnished by the holomorphic functional calculus on a commutative Banach algebra (theorem 1 of I, p. 51).

Let B be the closed full subalgebra of A generated by $a$, so that Sp$_A(a) =$ Sp$_B(a)$. The element $f(a)$ of A belongs to B, and coincides with the element $f(a)$ calculated relative to the algebra B.