Operator commutation in Jordan algebras

Question

Let $(A,*,1)$ be a Jordan algebra and for an element $a\in A$ write $T_a:A\rightarrow A$ for the Jordan product map $T_a(b) = a*b$. We say that $a,b\in A$ operator commute when $T_aT_b = T_bT_a$. The Jordan identity can for instance be stated as "$a$ and $a^2$ operator commute".

I am interested in any resources that prove non-trivial results regarding operator commutation in particular for Euclidean Jordan algebras or JB/JBW-algebras.

Some particular questions I would like to know the answer to (counter-examples in pathological Jordan algebras are welcome, but again I am particularly interested in the truth of these statements for Jordan operator algebras):

1. If $a$ and $b$ operator commute, do $a$ and $b^2$ operator commute?
2. If $a$ and $b$ operator commute, do they span an associative algebra? What about the converse? I know the answer in general is false [1, 2.5.2]), but true when $b$ is idempotent. In the book [1] they claim that this equivalence holds in JB-algebras [1, 2.5.1], but I haven't been able to find a proof of this statement.
3. Given some subset $S\subseteq A$ we can construct the commutator subspace $S'$ consisting of all elements of $A$ that operator commute with every element in $S$. Is this a sub-algebra? In general this is false [2], but it is true if $S$ is itself a finite-dimensional semi-simple Jordan algebra [2]. It is also true if $S$ is a singleton of an idempotent [2]. What else can be said about this question?
4. The quadratic product $U_a := 2T_a - T_{a^2}$ models the operation $b\mapsto aba$ in an associative algebra. If $U_aU_b = U_bU_a$, do $a$ and $b$ operator commute?

I expect that for some of these questions we can find an affirmative answer using a known classification of algebras, such as is done in [3] for studying commutation of the quadratic products. For instance, a JBW-algebra always splits up as a direct sum of a special Jordan algebra and a purely exceptional algebra [4]. As these questions above are trivial for special algebras, the problem then reduces to studying purely exceptional algebras. I would of course hope that there is a more direct answer to these questions.

[1] Jordan operator algebras - Hanche-Olsen and Stormer

[2] https://link.springer.com/chapter/10.1007/978-1-4612-3694-8_13

[3] https://www.ams.org/journals/tran/2014-366-11/S0002-9947-2014-06054-6/home.html

[4] https://www.sciencedirect.com/science/article/pii/0022123679900107

Answer 1

I still don't know of any other papers that tackle these issues, but I have now published my own results with regards to this question: Commutativity in Jordan operator algebras.

The answer is that in JB-algebras 1., 2. and 3. are true, because $a$ and $b$ operator commute if and only if they generate a Jordan algebra of mutually operator commuting elements, and hence operator commutativity is as well-behaved as it is in a special Jordan algebra. 4. is only true if at least one of $a$ and $b$ is positive, as otherwise it isn't even true in the special case.