Tensor Products of C*-algebras

Let $A$ and $B$ be C*-algebras.  We can define their *-algebra tensor product as the standard tensor product of algebras $A\otimes B$ with product $(a\otimes b)(a'\otimes b')=aa'\otimes bb'$ and involution $(a\otimes b)^*=a^*\otimes b^*$.  There are a variety of norms one can impose on this tensor product to make $A\otimes B$ a Banach *-algebra.  For example we may define $\displaystyle\left\|\sum a_i\otimes b_i\right\|_\wedge=\sum \|a_i\|\|b_i\|$.

This seminorm becomes a norm on $A\otimes B$ modulo the appropriate subspace, and its completion is denoted $A\hat{\otimes}B$ and is called the projective tensor product of $A$ and $B$.  We also have $\left\|\left(\sum a_i\otimes b_i\right)^*\right\|_\wedge=\left\|\sum a_i^*\otimes b_i^*\right\|_\wedge=\sum \|a_i^*\|\|b_i^*\|=\sum \|a_i\|\|b_i\|=\left\|\sum a_i\otimes b_i\right\|,$

so $A\hat{\otimes}B$ is a Banach *-algebra.  But it fails to satisfy the C*-axiom ( $\|x^*x\|=\|x\|^2$): $\begin{array}{lcl}\left\|\left(\sum a_i\otimes b_i\right)^*\left(\sum a_i\otimes b_i\right)\right\|&=&\left\|\left(\sum a_i^*\otimes b_i^*\right)\left(\sum a_i\otimes b_i\right)\right\|\\&=&\left\|\sum a_i^*a_j\otimes b_i^*b_j\right\|\\&=&\sum\|a_i^*a_j\|\|b_i^*b_j\|\\&\leq&\sum \|a_i\|\|a_j\|\|b_i\|\|b_j\|\\&=&\left(\sum \|a_i\|\|b_i\|\right)^2\\&=&\left\|\sum a_i\otimes b_i\right\|^2\end{array}.$

It turns out that representations on $A$ and $B$ allow us to define norms on $A\otimes B$ that make it a C*-algebra.

Definition 1.  Let $\rho_A:A\to L(H_1)$ and $\rho_B:B\to L(H_2)$ be representations on $A$ and $B$.  We define the product representation $\rho=\rho_A\otimes\rho_B$ on $H_1\otimes H_2$ as $\rho(a\otimes b)=\rho_A(a)\otimes\rho_B(b)\in L(H_1)\otimes L(H_2)$.

Since we always have the trivial representations, the set of representations of $A$ on $H_1$ and $B$ on $H_2$ are never empty.  Let us define the minimal C*-norm on $A\otimes B$ by $\begin{array}{lcl}\displaystyle\left\|\sum a_i\otimes b_i\right\|_{\mbox{min}}&=&\displaystyle\sup_{\rho_A,\rho_B}\left\|\rho\left(\sum a_i\otimes b_i\right)\right\|\\&=&\displaystyle\sup_{\rho_A,\rho_B}\left\|\sum \rho_A(a_i)\otimes\rho_B(b_i)\right\|\end{array}$

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 $A\otimes B$ with this norm is a C*-algebra called the minimal (or spatial) tensor product of $A$ and $B$ with respect to $\rho_A$ and $\rho_B$, and is denoted $A\underline{\circledast} B$.

Definition 2.  Let $\rho_A:A\to L(H)$ be a representation and $N\leq H$ be the largest subspace of $H$ such that $\rho(a)(x)=0$ for all $a\in A$ and $x\in N$.  Then $N^\perp$ is called the essential subspace of $H$, and we will denote it $E(H)$.  If $E(H)=H$, then $\rho_A$ is said to be nondegenerate. $\rho_A$ is degenerate if it is not nondegenerate.

In other words, $\rho_A$ is nondegenerate if $N=0$.

Proposition 3.  If $\rho:A\otimes B\to L(H)$ is a nondegenerate representation, then there are unique nondegenerate representations $\rho_A:A\to L(H)$ and $\rho_B:B\to L(H)$ such that $\rho(a\otimes b)=\rho_A(a)\rho_B(b)=\rho_B(b)\rho_A(a)$.

But arbitrary representations of the tensor product of algebras cannot be broken into pieces.  This gives us the following.

Definition 4.  Let $H$ be a Hilbert space and $A,B$ be C*-algebras.  We define the maximal C*-norm on $A\otimes B$ as $\displaystyle\left\|\sum a_i\otimes b_i\right\|_{\mbox{max}}=\sup_{\rho}\left\|\rho\left(\sum a_i\otimes b_i\right)\right\|$

where $\rho:A\otimes B\to L(H)$.  This is also a C*-norm, and the completion of $A\otimes B$ under this norm is a C*-algebra called the maximal tensor product of $A$ and $B$ and is denoted $A\overline{\circledast}B$.

We also have that $\|\cdot\|_{\mbox{min}}\leq\|\cdot\|_*\leq\|\cdot\|_{\mbox{max}}\leq\|\cdot\|_\wedge$ where $\|\cdot\|_*$ is any C*-norm.  It follows that $\|(a\otimes b)\|_*=\|a\|\|b\|$.

Definition 5.  A functional on $A\otimes B$ is positive if $f(x^*x)\geq 0$ for all $x\in A\otimes B$.  A state on $A\otimes B$ where $A$ and $B$ are unital is a positive linear functional $f$ on $A\otimes B$ such that $f(1\otimes 1)=1$.  We denote the set of states by $S(A\otimes B)$.

As in the previous post, there is a GNS construction that gives a representation $\rho_f:A\overline{\circledast}B\to L(H_f)$ for a positive linear functional $f$, although one must show the left action on $H_f$ is by bounded operators.

Definition 6.  A C*-algebra $A$ is nuclear if for every C*-algebra $B$, there is a unique C*-norm on $A\otimes B$.

Hence in such a case, we would have $A\underline{\circledast} B=A\overline{\circledast} B$, and thus denote the product C*-algebra by $A\circledast B$.  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; $C^*(\mathbb{F}_2),$ the group C*-algebra of $\mathbb{F}_2$ (see next post), is an example.

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