Normed algebra with multiplicative norm, non-isomorphic to $R$, $C$, $H$. Is there an infinite dimensional real normed algebra $A$  such that $\|xy\|=\|x\|\cdot \|y\|$ for all $x,y \in A$?
Thanks.
 A: It depends on whether you assume $A$ to have a unit element or not and whether $A$ is associative or not.
If $A$ has a unit element then the answer is no, independent of associativity of $A$. This is the main result in the article 
Kazimierz Urbanik and Fred B. Wright, Absolute-valued algebras, Proc. Amer. Math. Soc. 11 (1960), 861-866.
Urbanik and Wright give the following example of an infinite-dimensional non-associative and non-unital Banach algebra with the desired property: Fix a bijection $\phi\colon \mathbb{N} \times \mathbb{N} \to \mathbb{N}$.
For $x, y \in \ell^2(\mathbb{N})$ define $z = xy$ for $n = \phi(k,l)$ by $z_n = x_k \cdot y_l$. Then
$$
\lVert z \rVert^2 = \sum_{n=1}^\infty z_n^2 = \sum_{k,l = 1}^\infty x_k^2 y_l^2 = \left(\sum_{k=1}^\infty x_k^2\right)\left(\sum_{l=1}^\infty y_k^2\right) = \lVert x\rVert^2 \lVert y\rVert^2
$$
shows that with this multiplication $\ell^2(\mathbb{N})$ has the desired property.
See also this thread on MathOverflow where the above paper is in the answer by Faisal and the answer by Andreas Thom points out that Mazur proved that the only real, associative and unital normed division algebras are the familiar $\mathbb{R},\mathbb{C},\mathbb{H}$.
I don't know whether there is an associative non-unital example or not since the property is destroyed upon passing to the unitization.
