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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?

Mike Izbicki
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    According to the answer to this question, the quaternions do form a real Banach algebra but not a complex one. – shalop Aug 17 '17 at 18:59
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    Putting the two quoted sentences together, "The quaternions form a 4-dimensional real Banach algebra...", "...not a complex Banach algebra". –  Aug 17 '17 at 19:01
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    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 $Z(\mathbb{H}) = \mathbb{R}\cdot 1$. – Daniel Fischer Aug 17 '17 at 19:28

2 Answers2

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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 $\mathbb{C}$.

Noah Schweber
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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.

Eric Wofsey
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