Linking algebra - Imported Knowlpedia Content

Definition

Let ${}_AE_B$ be an imprimitivity bimodule. Its linking algebra is the $C^*$ -algebra

L(E)= \begin{pmatrix} A&E\\ \widetilde E&B \end{pmatrix},

where $\widetilde E$ is the conjugate $B$ - $A$ bimodule and multiplication uses the left and right inner products of $E$ . Concretely,

L(E)\cong \mathcal K_B(E\oplus B).

Under this realization, $A\cong\mathcal K_B(E)$ and $B\cong\mathcal K_B(B)$ are the diagonal corners, while $E$ and $\widetilde E$ are the off-diagonal corners. These four pieces are closed under block multiplication and involution.

Block operations

Writing an element as $\left(\begin{smallmatrix}a&\xi\\ \widetilde\eta&b\end{smallmatrix}\right)$ , the mixed products recover the module actions, and the products of off-diagonal entries recover the inner products:

\xi\widetilde\eta={}_A\langle\xi,\eta\rangle,\qquad \widetilde\xi\eta=\langle\xi,\eta\rangle_B.

The adjoint interchanges the two off-diagonal corners. Thus the entire imprimitivity structure is encoded by ordinary multiplication and involution inside one $C^*$ -algebra.

Full complementary corners

In the multiplier algebra $M(L(E))$ , the two diagonal projections $p$ and $q=1-p$ satisfy

pL(E)p\cong A,\qquad qL(E)q\cong B,\qquad pL(E)q\cong E.

They are full: the ideals generated by either projection are all of $L(E)$ . Conversely, complementary full corners of a $C^*$ -algebra determine an imprimitivity bimodule. This provides a corner characterization of strong Morita equivalence Raeburn–Williams, Chapter 2.

Conventions and scope

References

Iain Raeburn and Dana P. Williams, Morita Equivalence and Continuous-Trace C-Algebras*, American Mathematical Society, 1998. AMS DOI record. Relevant: Chapter 2 on linking algebras and full corners.
E. Christopher Lance, Hilbert C-Modules: A Toolkit for Operator Algebraists*, Cambridge University Press, 1995. Publisher record. Relevant: Chapter 1 on compact operators over direct sums of Hilbert modules.