While writing arXiv:1510.05757v2, I found myself proving some basic facts about products of Banach algebra elements over an infinite totally ordered set. (Statements and proofs are in Appendix C.) The product over a totally ordered set $A$ is defined as you'd expect.
Definition. The product $\prod_{p \in A} x_p$ is the limit of the net that gives the product over each finite subset of $A$.
I prove things like the following.
Equivariance property. If $\prod_{p \in A} x_p$ converges, then $\phi\left(\prod_{p \in A} x_p\right) = \prod_{p \in A} \phi(x_p)$ for any continuous homomorphism $\phi$ from the algebra to itself.
Convergence condition. If the sum $C = \sum_{p \in A} \|x_p - 1\|$ converges, the product $\prod_{p \in A} x_p$ converges as well.
Estimates. When the sum $C$ converges, the norm of $\prod_{p \in A} x_p$ and its distance from $1$ are bounded in terms of $C$. (See the preprint for explicit bounds.)
For products over $\mathbb{N}$, these facts are well-established. The convergence condition, for example, is proven in Welstead's "Infinite products in a Banach algebra". My arguments for products over $A$ are small (though sometimes tricky) modifications of the arguments for products over $\mathbb{N}$.
Can I find these sorts of results in the literature instead of using my own proofs?
I don't need the results to be stated for general Banach algebras; I only use them for matrix algebras in the preprint. I'd be happy even to find a text that defines infinite ordered products like this, or to learn of a standard name for them.
