Corollary

Let $(E_i)_{i \in I},\ (E'_i)_{i \in I}$ be two families of $A$*-algebras with the same indexing set and, for all* $i \in I$, let $u_i : E_i \to E'_i$ be an algebra homomorphism. Then there exists one and only one $A$*-algebra homomorphism* $u : \bigotimes_{i \in I} E_i \to \bigotimes_{i \in I} E'_i$ such that, for all $i \in I$, the diagram

is commutative, $f_i$ and $f'_i$ denoting the canonical homomorphisms.

It suffices to apply Proposition 8 to the homomorphism $f'_i \circ u_i$.

The homomorphism $u$ defined in the Corollary to Proposition 8 is denoted by $\bigotimes_{i \in I} u_i$. If $J$ is any subset of $I$, Proposition 8 can be applied to the family $(f_i)_{i \in J}$ of canonical homomorphisms $f_i : E_i \to \bigotimes_{i \in I} E_i = E$; a canonical homomorphism $E_J \to E$ is derived which is also denoted by $f_J$ and which, when $J$ is finite, coincides with the homomorphism denoted thus above.

Now let $(x_i)_{i \in I}$ be an element of $\prod_{i \in I} E_i$ such that the family $(x_i - e_i)_{i \in I}$ has finite support $H$. It is immediate that, if $J$ and $J'$ are two finite subsets of $I$ containing $H$, then

The common value of the $f_J((x_i)_{i \in J})$ for the finite subsets $J \supset H$ of $I$ is denoted by $\bigotimes_{i \in I} x_i$.