From 42433c40caf51d9d4ca7af306fa7af3d5b3805fe Mon Sep 17 00:00:00 2001
From: Bokuan Li <bokuan.li@mail.mcgill.ca>
Date: Tue, 2 Jun 2026 20:26:07 -0400
Subject: [PATCH] Upgraded prose.

---
 src/op/banach/multiplicative.tex | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/src/op/banach/multiplicative.tex b/src/op/banach/multiplicative.tex
index 350fa62..25e94fa 100644
--- a/src/op/banach/multiplicative.tex
+++ b/src/op/banach/multiplicative.tex
@@ -97,12 +97,12 @@
         &= 2[x(yxy) + y(xyx)] \in \ker \phi
     \end{align*}
 
-    and $(xy - yx)^2 \in \ker\phi$ as well, and
+    and $(xy - yx)^2 \in \ker\phi$. Since
     \[
     (\phi(xy - yx))^2 = \phi((xy - yx)^2) = 0
     \]
 
-    Therefore $2xy = (xy + yx) - (xy - yx) \in \ker \phi$, $\ker \phi$ is an ideal, and $\phi$ is a homomorphism. 
+    The commutator $xy - yx \in \ker \phi$ as well. Therefore $2xy = (xy + yx) - (xy - yx) \in \ker \phi$, $\ker \phi$ is an ideal, and $\phi$ is a homomorphism. 
 \end{proof}