From 4a56953006f365c70ffe514c4c230ac8cdb03576 Mon Sep 17 00:00:00 2001
From: Michael Orlitzky <michael@orlitzky.com>
Date: Sun, 20 Apr 2025 11:08:30 -0400
Subject: [PATCH] examples,mjo-eja: add the Jordan-algebraic trace \tr{}

---
 examples.tex |  3 ++-
 mjo-eja.tex  | 16 ++++++++++++++++
 2 files changed, 18 insertions(+), 1 deletion(-)

diff --git a/examples.tex b/examples.tex
index accec5d..f8165b6 100644
--- a/examples.tex
+++ b/examples.tex
@@ -170,7 +170,8 @@
     $\JAut{V}$. Two popular operators in an EJA are its quadratic
     representation and ``left multiplication by'' operator. For a
     given $x$, they are, respectively, $\quadrepr{x}$ and
-    $\leftmult{x}$.
+    $\leftmult{x}$. The Jordan-algebraic trace is available either as
+    a function $\tr{x}$, or in its operator form $\tr{}$.
 
     The one EJA that fits better here than anywhere else is the Jordan
     spin algebra, $\spineja$. Its ambient space is $\Rn$, but having a
diff --git a/mjo-eja.tex b/mjo-eja.tex
index b9e072e..1d245fa 100644
--- a/mjo-eja.tex
+++ b/mjo-eja.tex
@@ -22,6 +22,22 @@
 % a (bilinear) algebra multiplication in any other context.
 \newcommand*{\jp}[2]{{#1} \circ {#2}}
 
+% The Jordan-algebraic trace (sum of eigenvalues) of its argument.
+% Use a blank argument to get the "tr" operator itself.
+\newcommand*{\tr}[1]{%
+  \operatorname{tr}%
+    \if\relax\detokenize{#1}\relax\else
+    \of{#1}%
+  \fi%
+}
+\ifdefined\newglossaryentry
+  \newglossaryentry{tr}{
+    name={\ensuremath{\tr{x}}},
+    description={the Jordan-algebraic trace of $x$},
+    sort=tr
+  }
+\fi
+
 % The "quadratic representation" of the ambient space applied to its
 % argument. We have standardized on the "P" used by Faraut and Korányi
 % rather than the "U" made popular by Jacobson.
-- 
2.49.0