Internal tensor product of C*-correspondences

Definition

Let $E$ be a $C^*$ -correspondence from $A$ to $B$ , and let $F$ be one from $B$ to $C$ , with left action $\varphi_F:B\to\mathcal L_C(F)$ . Their internal tensor product $E\otimes_B F$ is the completion of the balanced algebraic tensor product $E\odot_B F$ , after quotienting vectors of zero length, for the $C$ -valued inner product

\langle\xi_1\otimes\eta_1,\xi_2\otimes\eta_2\rangle_C =\langle\eta_1,\varphi_F(\langle\xi_1,\xi_2\rangle_B)\eta_2\rangle_C.

The right $C$ -action comes from $F$ , and $A$ acts by $a\cdot(\xi\otimes\eta)=(a\cdot\xi)\otimes\eta$ . Thus $E\otimes_B F$ is a correspondence from $A$ to $C$ .

Construction and positivity

Balancing imposes $(\xi b)\otimes\eta=\xi\otimes\varphi_F(b)\eta$ . The displayed formula is compatible with this relation and defines a positive semidefinite form; its null space must be removed before completion. This quotient-and-completion step is essential: the ordinary algebraic tensor product is generally neither definite nor complete. Adjointable operators on $E$ induce adjointable operators $T\otimes 1$ on the completed product, which supplies the left $A$ -action Lance, chapter on tensor products.

Composition and unit correspondences

Internal tensor product is associative up to the canonical unitary

(E\otimes_B F)\otimes_C G\longrightarrow E\otimes_B(F\otimes_C G), \qquad (\xi\otimes\eta)\otimes\zeta\longmapsto\xi\otimes(\eta\otimes\zeta).

The standard correspondence $A$ from $A$ to itself is a unit: $A\otimes_A E\cong E$ and $E\otimes_B B\cong E$ . These canonical unitaries, rather than literal equalities of modules, provide the composition and identity laws for correspondences.

Examples and scope

When $B=\mathbb C$ , the construction reduces to the Hilbert-space tensor product, with any remaining coefficient algebra carried by $F$ . If the left $B$ -action on $F$ annihilates a nonzero ideal, tensors involving that ideal can become null; this is why the quotient cannot be omitted.

References

E. Christopher Lance, Hilbert C-Modules: A Toolkit for Operator Algebraists*, Cambridge University Press, 1995. Chapter DOI record. Relevant: Chapter 4, “Tensor products.”
Iain Raeburn and Dana P. Williams, Morita Equivalence and Continuous-Trace C-Algebras*, American Mathematical Society, 1998. DOI record. Relevant: Chapter 2 on Hilbert modules and internal tensor products.