nLab internal tensor product

Internal tensor product of Hilbert C*^*-modules

Definition

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

Let AA and BB be C *C^*-algebras, and let EE be a right Hilbert AA-module and FF be a right Hilbert BB-module, and finally let φ:A B(F)\varphi \colon A\to \mathcal{L}_B(F) be a **-homomorphism.

Form the algebraic tensor product E AFE \otimes_A F which is the quotient of the vector space tensor product EFE\otimes F by the subspace NN spanned by elements of the form xayxφ(a)(y) x a \otimes y - x \otimes \varphi(a)(y) for aA,xE,yFa\in A, x\in E, y\in F. We define a BB-valued inner product by

x 1y 1,x 2y 2=y 1,φ(x 1,x 2)y 2. \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 NN. Less obvious is:

Lemma

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

Proof

To do.

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

Properties

As a right BB module

Action of adjointable operators on EE

To do.

References

Lance’s book, Chapter 4.

Created on March 27, 2018 at 16:23:23. See the history of this page for a list of all contributions to it.