Example of a non-associative algebra

This page presents and discusses an example of a non-associative Template:Lw over the real numbers.

The multiplication is defined by taking the Template:Lw of the usual multiplication: $a*b=\overline{ab}$. This is a commutative, non-associative division algebra of dimension 2 over the reals, and has no unit element.

Proof that (C,*) is a division algebra

For a proof that $ℝ$ is a field, see real number. Then, the complex numbers themselves clearly form a vector space.

It remains to prove that the binary operation given above satisfies the requirements of a division algebra

• (x + y)z = x z + y z;
• x(y + z) = x y + x z;
• (a x)y = a(x y); and
• x(b y) = b(x y);

for all scalars a and b in $ℝ$ and all vectors x, y, and z (also in $ℂ$).

For distributivity:

$x*(y+z)=\overline{x(y+z)}=\overline{xy+xz}=\overline{xy}+\overline{xz}=x*y+x*z,$

(similarly for right distributivity); and for the third and fourth requirements

$(ax)*y=\overline{(ax)y}=\overline{a(xy)}=\bar{a}\cdot \overline{xy}=\bar{a}(x*y).$

Non associativity of (C,*)

• $a*(b*c)=a*\overline{bc}=\overline{a\stackrel{‾}{bc}}=\bar{a}bc$

  $(a*b)*c=\overline{ab}*c=\overline{\stackrel{‾}{ab}c}=ab\bar{c}$

So, if a, b, and c are all non-zero, and if a and c do not differ by a real multiple, $a*(b*c)\ne (a*b)*c$.