JB-algebra

Remarks

Example: the real symmetric $n \times n$ matrices with $a \circ b = \frac{1}{2}(ab + ba)$ and the operator norm. Clause 1 holds in finite dimension, and clause 2 follows from $\norm{ab} \le \norm{a}\norm{b}$. For clauses 3 and 4, $\norm{a}$ is the largest absolute value of an eigenvalue, so $\norm{a^2} = \norm{a}^2$, and $a^2 \le a^2 + b^2$ as positive semidefinite matrices.

Non-example: $\R^2$ with the componentwise product and the norm $\norm{(x_1, x_2)} = \abs{x_1} + \abs{x_2}$ fails clause 3. For $a = (1, 1)$ one has $\norm{a^2} = 2$ but $\norm{a}^2 = 4$. With the norm $\max(\abs{x_1}, \abs{x_2})$ instead, it is a JB-algebra.

The self-adjoint part of a C*-algebra, with $a \circ b = \frac{1}{2}(ab + ba)$, is a JB-algebra.