Exterior tensor product of Hilbert C*-modules

Definition

Let $E$ be a right Hilbert $A$ -module and $F$ a right Hilbert $B$ -module. On the algebraic complex tensor product $E\odot F$ , define

(x\otimes y)(a\otimes b)=xa\otimes yb

and

\langle x_1\otimes y_1,x_2\otimes y_2\rangle =\langle x_1,x_2\rangle_A\otimes\langle y_1,y_2\rangle_B,

extending sesquilinearly. After quotienting by zero-length vectors and completing in the induced norm, one obtains the exterior tensor product $E\boxtimes F$ , a Hilbert module over the minimal tensor product $A\otimes_{\min}B$ .

Positivity and completion

The essential point is that the displayed algebra-valued form is positive on finite sums, not only on elementary tensors. Its null space is a submodule, and the quotient norm is

\|z\|=\|\langle z,z\rangle\|^{1/2}.

Completing produces a Hilbert $A\otimes_{\min}B$ -module independent of concrete faithful representations used to realize the minimal tensor norm Lance, Chapter 4.

Standard examples

Taking $E=A$ and $F=B$ with their standard module structures gives $A\boxtimes B\cong A\otimes_{\min}B$ . For column modules,

A^m\boxtimes B^n\cong(A\otimes_{\min}B)^{mn}.

When $A=B=\mathbb C$ , the construction reduces to the Hilbert-space tensor product. These examples show that both the vectors and their coefficient algebras are tensorized.

Exterior versus interior tensoring

The exterior product starts with modules over unrelated algebras and uses the complex tensor product. By contrast, the internal tensor product of correspondences balances over a shared intermediate algebra: relations of the form $xa\otimes y=x\otimes ay$ are imposed. The two constructions can interact, but they solve different composition problems.

References

E. Christopher Lance, Hilbert C-Modules: A Toolkit for Operator Algebraists*, London Mathematical Society Lecture Note Series 210, Cambridge University Press, 1995. Cambridge DOI record. Relevant: Chapter 4 on tensor products of Hilbert modules.