Why do we assume that $\mathcal{A}$ is an algebra in this 2003 paper of Bobkov and Tetali?

In the following paper (extended version here), at the beginning of section 3, the authors give two axioms about $$\mathcal{A}$$. Axiom 1 is about $$\mathcal{A}$$ being an algebra. I do not see where this axiom is being used. Any ideas?

Conference version: Bobkov, Sergey; Tetali, Prasad, Modified log-Sobolev inequalities, mixing and hypercontractivity, Proceedings of the thirty-fifth annual ACM symposium on theory of computing (STOC 2003), San Diego, CA, USA,. New York, NY: ACM Press (ISBN 1-58113-674-9). 287-296, electronic only (2003). ZBL1192.60020.

Journal version: Bobkov, Sergey G.; Tetali, Prasad, Modified logarithmic Sobolev inequalities in discrete settings, J. Theor. Probab. 19, No. 2, 289-336 (2006). ZBL1113.60072.

• Where in the cited document does A appear? What page, paragraph? – Ken W. Smith Sep 8 at 18:56
• @KenW.Smith: p290, Axiom 1. So I haven’t managed to digest the paper yet, but is it important that $f^2\in \mathcal A$? (I notice that $f^2$ appears in various places; and a linear space of functions is closed under squaring if and only if it’s closed under taking products). – Anthony Quas Sep 8 at 19:33
• @KenW.Smith $\mathcal{A}$ appears in the first paragraph of section 3 ("POINCARE AND LOG-SOBOLEV IN ABSTRACT SETTINGS"). You can easily find it by searching for Axiom 1 using ctrl+F and reading the text above it. – Ella Sharakanski Sep 9 at 13:55
• As a general tip, when you mention a paper, I suggest giving a full citation instead of just a link. It helps people find the paper more reliably. (In particular, your link to the extended version could only be used by people whose institutions participate in OpenAthens.) Also, I feel it is a courtesy to the authors to mention their names and the other info about their paper. There is a "cite" button in the question editor that will give you a full citation with DOI and Zentralblatt link. I've added citations to your post - perhaps you could double check that they are correct. – Nate Eldredge Sep 11 at 17:24
• I also don't immediately see this assumption used explicitly. However, I would note that they are using a very simplified definition of "Dirichlet form". In the more usual development, a Dirichlet form $\mathcal{E}$ is defined on a dense subspace $D(\mathcal{E})$ of $L^2(M,\pi)$, which should be thought of as a space of "finite energy" functions having one "derivative" in $L^2$, like the Sobolev space $W^{1,2}_0$. Then it is a theorem that the bounded functions in $D(\mathcal{E})$, that is $\mathcal{A} := D(\mathcal{E}) \cap L^\infty(M)$, do in fact form an algebra. – Nate Eldredge Sep 11 at 17:34