Tensor Product of operator algebras

Question

I want to clearly know the notion of 'algebraic tensor product' for Hilbert spaces or $C^*$-algebras or von Neumann algebras. Take any two objects $A$ and $B.$ In many places people have told to define the algebraic tensor product and then complete it suitably. For example for Hilbert spaces one define the inner product $<a\otimes b,c\otimes d>:=<a,c><b,d>$ and then complete it. But how is it well-defined? I am assuming that by 'algebraic tensor product' one means the usual vector space tensor product. What about the $C^*$ algebra? I have same confusion for min tensor product for $C^*$-algebra.

Answer 1

To see that this inner product is well-defined, see theorem 6.3.1 in Murphy's book "$C^*$-algebras and operator theory". This has a very accessible exposition of the tensor product of Hilbert spaces. All you need for prerequisites is understanding how the algebraic tensor product of vector spaces is made.

The $C^*$-algebraic case is much more complicated. Let $A$ and $B$ be $C^*$-algebras and write $A \odot B$ to denote their algebraic tensor product. This becomes a $*$-algebra for the obvious multiplication and involution maps. We want to put a norm on $A \odot B$ such that it becomes a $C^*$-algebra, or at least a $C^*$-algebra after taking a completion. In general, many suitable norms are possible, and different choices of norms may give different completions so it must always be specified with which $C^*$-norm one is working.

In practice, one always works with either the minimal or the maximal $C^*$-norms for the reason that these are much-better behaving than the intermediate norms.