Let and be C*-algebras. We can define their *-algebra tensor product as the standard tensor product of algebras with product and involution . There are a variety of norms one can impose on this tensor product to make a Banach *-algebra. For example we may define
This seminorm becomes a norm on modulo the appropriate subspace, and its completion is denoted and is called the projective tensor product of and . We also have
so is a Banach *-algebra. But it fails to satisfy the C*-axiom ():
It turns out that representations on and allow us to define norms on that make it a C*-algebra.
Definition 1. Let and be representations on and . We define the product representation on as
Since we always have the trivial representations, the set of representations of on and on are never empty. Let us define the minimal C*-norm on by
where the two norms on the right are operator norms. This is clearly finite (hence a norm) and satisfies the C*-axiom. The completion of with this norm is a C*-algebra called the minimal (or spatial) tensor product of and with respect to and , and is denoted .
Definition 2. Let be a representation and be the largest subspace of such that for all and . Then is called the essential subspace of , and we will denote it . If , then is said to be nondegenerate. is degenerate if it is not nondegenerate.
In other words, is nondegenerate if .
Proposition 3. If is a nondegenerate representation, then there are unique nondegenerate representations and such that .
But arbitrary representations of the tensor product of algebras cannot be broken into pieces. This gives us the following.
Definition 4. Let be a Hilbert space and be C*-algebras. We define the maximal C*-norm on as
where . This is also a C*-norm, and the completion of under this norm is a C*-algebra called the maximal tensor product of and and is denoted .
We also have that where is any C*-norm. It follows that .
Definition 5. A functional on is positive if for all . A state on where and are unital is a positive linear functional on such that . We denote the set of states by .
As in the previous post, there is a GNS construction that gives a representation for a positive linear functional , although one must show the left action on is by bounded operators.
Definition 6. A C*-algebra is nuclear if for every C*-algebra , there is a unique C*-norm on .
Hence in such a case, we would have , and thus denote the product C*-algebra by . The class of nuclear C*-algebras includes all of the commutative ones, finite ones, and is itself closed under inductive products and quotients. Non nuclear ones are exotic; the group C*-algebra of (see next post), is an example.
 Blackadar, Bruce. Operator Algebras. Encyclopedia of Mathematical Sciences. Vol. 122. Springer-Verlag. 2006.