0
$\begingroup$

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?

$\endgroup$
3
  • 1
    $\begingroup$ According to the answer to this question, the quaternions do form a real Banach algebra but not a complex one. $\endgroup$ – shalop Aug 17 '17 at 18:59
  • 3
    $\begingroup$ Putting the two quoted sentences together, "The quaternions form a 4-dimensional real Banach algebra...", "...not a complex Banach algebra". $\endgroup$ – user856 Aug 17 '17 at 19:01
  • 1
    $\begingroup$ If $A$ is a unital complex algebra, then $\mathbb{C}\cdot 1 \subset Z(A)$ (since $(z\cdot 1)\times a = 1 \times (z\cdot a) = z\cdot a = (z\cdot a)\times 1 = a \times (z\cdot 1)$). But $Z(\mathbb{H}) = \mathbb{R}\cdot 1$. $\endgroup$ – Daniel Fischer♦ Aug 17 '17 at 19:28
2
$\begingroup$

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 $\mathbb{C}$.

$\endgroup$
1
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.