Does it make sense to define a structure that is an inner product space and a K-algebra? And would this be useful in any way?

Question

I've recently encountered K-algebras which I understand are different from inner product spaces in that one is equipped with a billinear product and the other with a bilinear form.

Is it possible and/or useful to define a space that has both of these operations? If so, what are some examples?

Answer 1

It's perfectly fine to define both structures on some ring. A well-known example of such an object are the $\newcommand{\b}{\mathbb} \b{R}$-algebra of complex numbers with the conventional multiplication \begin{align} \b{C} \times \b{C} &\to \b{C} \\ (z, w) &\mapsto zw \end{align}

and the inner product

\begin{align} \b{C} \times \b{C} &\to \b{R} \\ (z, w) &\mapsto \operatorname{Re}(\bar{z} w) \end{align} Using polarization, the inner product is equivalent to the norm $|\cdot|: \b{C} \to \b{R}$. All these structures have been known to be useful.

However it should be noted that, as is usual when defining multiple different structures on an object, these structures on $\b{C}$ are so useful, because they are compatible. In this particular case compatibility takes the form of the multiplication rule for absolute values $|z w| = |z| |w|$.

Other examples I can think of:

• Quaternions also are an $\b{R}$-algebra with an $\b{R}$-inner product satisfying the same compatibility as the complex numbers
• real or complex quadratic matrices matrices form a (noncommutative!) $\b{R}$-algebra with inner product $\left< A, B \right> = \operatorname{Re}(\operatorname{tr}(A^* B))$. The corresponding norm here is the Frobenius norm $\|\cdot\|_F$. The compatibility here is weaker than in the above cases since the Frobenius norm is only submultiplicative: $$ \|AB\|_F \leq \|A\|_F \|B\|_F $$