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.

[1] Blackadar, Bruce. *Operator Algebras.* Encyclopedia of Mathematical Sciences. Vol. 122. Springer-Verlag. 2006.