Proposition 9

Let $(E_i)_{i \in I}$ be a family of $A$*-algebras and for each* $i \in I$ let $B_i$ be a basis of $E_i$ such that the unit element $e_i$ belongs to $B_i$. Let $B$ be the set of elements of the form $\bigotimes_{i \in I} x_i$, where $(x_i)$ runs through the set of elements of $\prod_{i \in I} B_i$ such that the family $(x_i - e_i)$ has finite support. Then $B$ is a basis of the algebra $\bigotimes_{i \in I} E_i$ and this basis contains the unit element $e$.

For every finite subset $J$ of $I$, let $B_J$ be the basis of $E_J = \bigotimes_{i \in J} E_i$ the tensor product of the bases $B_i$ for $i \in J$ (II, § 3, no. 9). It follows immediately from the definitions that $B$ is the union of the $f_J(B_J)$ when $J$ runs through $\mathscr{F}(I)$ and that $f_{J',J}(B_J) \subset B_{J'}$ when $J \subset J'$; hence $(B_J)$ is a direct system of subsets of the $E_J$ and $B = \lim \rightarrow B_J$; the conclusion then follows from II, § 6, no. 2, Corollary to Proposition 5.

The basis $B$ is also called the tensor product of the bases $B_i$ for $i \in I$; when the conditions of Proposition 9 are fulfilled, the canonical homomorphisms $f_J : E_J \to E = \bigotimes_{i \in I} E_i$ are injective for every subset $J$ of $I$, for if $B_J$ is the basis of $E_J$ the tensor product of the $B_i$ for $i \in J$, it is immediately verified that the restriction of $f_J$ to $B_J$ is injective and maps $B_J$ onto a subset of $B$.