Definition 2

The image of the mapping $\mathsf{X}(h)$ is called the simultaneous spectrum of $x$, and denoted by Sp$^{\Lambda}_A(x)$ or Sp$^{\Lambda}(x)$.

The simultaneous spectrum of $x$ is a compact subset of $\mathbf{C}^{\Lambda}$. An element $c= (c_{\lambda})\in \mathbf{C}^{\Lambda}$ belong to Sp$^{\Lambda}_A(x)$ if and only if the elements $x_{\lambda}-c_{\lambda}$ belong to one and the same maximal ideal of A, in other words if the family $(x_{\lambda}-c_{\lambda})_{\lambda\in\Lambda}$ does not generate the algebra A.

If Λ contains only one element, so that the family $x$ reduces to a single element $x\in A$, we have Sp$^{\Lambda}_A(x) =$ Sp$_A(x)$ (I, p. 37, prop. 6, b)). If $\Lambda '\subset \Lambda$, then Sp$^{\Lambda'}_A((x_{\lambda})_{\lambda\in\Lambda'})$ is the image of Sp$^{\Lambda}_A((x_{\lambda})_{\lambda\in\Lambda})$ under the canonical projection mapping of $\mathbf{C}^{\Lambda}$ onto $\mathbf{C}^{\Lambda'}$. In particular, we have

Sp$^{\Lambda}_A(x)\subset \prod_{\lambda\in\Lambda}$ Sp$_A(x_{\lambda})$.

Let $z_{\lambda}$, for $\lambda \in \Lambda$, denote the coordinate functions on $\mathbf{C}^{\Lambda}$. If $\chi \in$ $\mathsf{X}(A)$, the value at $\chi$ of $z_{\lambda}\circ \mathsf{X}(h)$ is $\chi (x_{\lambda})$, hence $z_{\lambda}\circ \mathsf{X}(h) =\mathscr{G}(x_{\lambda})$.

Let A and B be commutative unital Banach algebras, $\varphi$ a unital morphism from A into B, and $x= (x_{\lambda})_{\lambda\in\Lambda}$ a family of elements of A. Let $\varphi (x)$ denote the family $(\varphi (x_{\lambda}))_{\lambda\in\Lambda}$ of elements of B. We have, for every $\chi \in \mathsf{X}(B)$, and every $\lambda \in \Lambda$

hence Sp$^{\Lambda}_B(\varphi (x))\subset$ Sp$^{\Lambda}_A(x)$. The diagram

Sp$_B(\varphi (x))\rightarrow$ Sp$_A(x)$

where $i$ denotes the inclusion, and where the vertical arrows denote the mappings defined by the families $\varphi (x)$ and $x$, is therefore commutative.