nLab internal tensor product

Definition

The internal tensor product of Hilbert $C^*$-modules is a generalisation of the algebraic notion of a tensor product of modules.

Let $A$ and $B$ be $C^*$-algebras, and let $E$ be a right Hilbert $A$-module and $F$ be a right Hilbert $B$-module, and finally let $\varphi \colon A\to \mathcal{L}_B(F)$ be a $*$-homomorphism.

Form the algebraic tensor product $E \otimes_A F$ which is the quotient of the vector space tensor product $E\otimes F$ by the subspace $N$ spanned by elements of the form $x a \otimes y - x \otimes \varphi(a)(y)$ for $a\in A, x\in E, y\in F$. We define a $B$-valued inner product by

$$\langle x_1\otimes y_1, x_2\otimes y_2 \rangle = \langle y_1, \varphi(\langle x_1,x_2\rangle) y_2 \rangle.$$

It is easy to check that this respects the quotient by $N$. Less obvious is:

Lemma

If $z\in E\otimes F$ is such that $\langle z,z \rangle=0$ then $z\in N$.

Proof

To do.

It follows that we have a $B$-valued inner product on $E \otimes_A F$ which turns $E \otimes_A F$ into a Hilbert C$^*$-module over $B$, which is denoted by $E \otimes_\varphi F$. This is the internal tensor product of $E$ and $F$ (using $\varphi$).

Properties

As a right $B$ module

Action of adjointable operators on $E$

To do.

References

Lance’s book, Chapter 4.