Definition 5

The direct limit E of the direct system $(E_J, f_{J'J})$ is called the tensor product of the family of A-algebras $(E_i)_{i \in I}$.

If I is finite, E is identified with $\bigotimes_{i \in I} E_i$. By an abuse of notation, E is also denoted by $\bigotimes_{i \in I} E_i$ even if I is infinite.

For every finite subset J of I, let $f_J$ denote the canonical homomorphism $\bigotimes_{i \in J} E_i \to \bigotimes_{i \in I} E_i$ (writing $f_i$ instead of $f_{\{i\}}$); if e is the unit element of $\bigotimes_{i \in I} E_i$, then $f_J(e_J) = e$ for all $J \in \mathscr{F}(I)$. It is immediate that if all the algebras E_i are commutative, so is $\bigotimes_{i \in I} E_i$.