I am new to learning tensor products. I encountered examples like $L(\mathbb R^2)=L(\mathbb R)\otimes L(\mathbb R)$ or for compact spaces $X,Y$ the identity $C(X\times Y)=C(X)\otimes C(Y)$. I don't know any details on those things (actually I even don't know in which sense we take those tensor products and equalities; I would guess the first one as Hilber-spaces and the second one as C*-algebras).
A question that arose directly for me is: What is the reason why in several examples where we have structures on sets we have that the product transforms to a tensor product? I think one would first expect that products translate into products. My guess is that in some sense the tensor product structure is a completion of the pure product structure?
Kind regards, Sebastian
Do you know any category theory? Then you see, that $L$ and $C$ are contravariant functors, so that it is only natural (if they are respecting limits), that they would transform products to coproducts. And the coproduct in the category of $\mathbb{C}$-Algebras is just the tensorproduct.
If you are not familiar with category theory, recall that the product has a certain universial property, which revolves around morphisms going into the product. Recall also, that the tensorproduct has a universial property, which revolves around morphisms going out of the tensorproduct. So lets look for example at $C(\cdot)$. Let $f: K_1\rightarrow K_2$ be a continous morphism between compact topological spaces. Then $C$ turns the arrow around, which means that we get a morphism $C(f):C(K_2)\rightarrow C(K_1)$ by pre-composing with $f$. This means, that $C$ would "take" an object, which fulfills the universial property of the product and "transforms" it into an object, which fulfills the universial property of the tensorproduct, since arrows that went into the product $X\times Y$ will be transformed to go out of $C(X\times Y)$.
Be wary though, that not every such so-called functor, which turns around the arrows has to transform products to coproducts!