Misleading wikipedia entry on Banach algebras

Question

On the wikipedia page for banach algebras, under examples it states:

The quaternions form a 4-dimensional real Banach algebra, with the norm being given by the absolute value of quaternions.

and under counterexamples it states

The algebra of the quaternions $\mathbb {H}$ is not a complex Banach algebra (for any norm on $\mathbb {H}$)

These statements seem contradictory to me. Is that the case?

Answer 1

The phrases

real Banach algebra

and

complex Banach algebra

are very important. In general, we define a Banach algebra over a certain field, and the choice of field matters. You can't ignore the specific field, here: $\mathbb{H}$ is a Banach algebra over $\mathbb{R}$, but not over $\mathbb{C}$.

Answer 2

There's nothing contradictory. The first sentence says the quaternions are a real Banach algebra, and the second sentence says the quaternions are not a complex Banach algebra. So they're a real Banach algebra but not a complex Banach algebra.