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.

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


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