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.

`Axiom 1`

using ctrl+F and reading the text above it. $\endgroup$ – Ella Sharakanski Sep 9 '19 at 13:55theoremthat the bounded functions in $D(\mathcal{E})$, that is $\mathcal{A} := D(\mathcal{E}) \cap L^\infty(M)$, do in fact form an algebra. $\endgroup$ – Nate Eldredge Sep 11 '19 at 17:34Dirichlet Forms and Symmetric Markov Processes. I am pretty sure this theorem can be found there. $\endgroup$ – Nate Eldredge Sep 11 '19 at 18:02