What is the 'complexification' of an algebra?

Question

I'm studying a book named: Mathematical Topics Between Classical and Quantum Mechanics by professor Landsman. In the Theorem 1.1.9 of the book he says, and I quote, that

Given a Jordan-Lie-Banach algebra $\mathfrak{U}_{\mathbb{R}}$ for which $\hbar \geq 0$, its complexification $\mathfrak{U}$ is a $C^\star$ algebra.

Question: what does he means by 'complexification'? The Jordan-Lie-Banach algebra is an algebra over the field $\mathbb{R}$, so the complexification will be :

We have that the elements of $\mathfrak{U}$ will be of the form

$$A+iB\,\,\,\,\,\,\,\,A,B \in \mathfrak{U}_{\mathbb{R}}$$

in other words (thanks to a clarifying comment) we change to a $\mathbb{C}$-vector field.

---

Infinitylord comment (very accurate) saying the equivalence of the above statements.

$$(x_1 + iy_1)A + (x_2 + iy_2)B = (x_1 + x_2)(A + B) + i(y_1 + y_2)(A+B)$$

Answer 1

Both (1) and (2). You then have a structure which (most likely) is not contained in any structure you have seen previously, so you can't assume that it is a vector space. There is rather a lot of work to do, but fortunately most of it is defining things according to the "wishlist" principle - just define them they way you would like them to work out - and then checking that it works.

For a start: define equality of vectors, addition and scalar multiplication for $\mathfrak U$ in terms of $\mathfrak U_{\Bbb R}$, which is assumed given. By the "wishlist" principle this would be

$A+iB=C+iD$ iff $A=C$ and $B=D$;

addition... too easy, leave it to you, and multiplication

$(u+iv)(A+iB)=(uA-vB)+i(uB+vA)$.

You then need to show that you have a vector space, and you need to do it by checking all the axioms, as $\mathfrak U$ is (probably) not a subset of any vector space you already know. The proofs will be long boring algebra, making sure you scrupulously apply the definitions.

You will probably want to go further and define the complexification of linear transformations, inner products etc. Again the "wishlist" principle applies and we define

the complexification of a linear transformation $T:\mathfrak U_{\Bbb R}\to\mathfrak V_{\Bbb R}$ is the function $$\hat T:\mathfrak U\to\mathfrak V\ ,\quad \hat T(A+iB)=T(A)+iT(B)\ .$$

You will need to prove that this is linear but again it's just long boring algebra.

You may also need to know that $\mathfrak U_{\Bbb R}$ is a subspace of $\mathfrak U$. This is slightly more tricky, mainly because it's not actually true (can't be, as the spaces have different scalar fields). But you can show that $$\mathfrak U_0=\{A+i{\bf0}\mid A\in \mathfrak U_{\Bbb R}\}$$ is a real vector space and is isomorphic to $\mathfrak U_{\Bbb R}$.

Good luck!