Central Haagerup Tensor Products and Completely Bounded Maps under Strong Morita Equivalence

Let $A$ and $B$ be strongly Morita equivalent $C^{\ast }$-algebras, implemented by a nonzero imprimitivity bimodule $X={}_A{X}_B$ with its canonical operator-space structure. Denote by $Z_X$ their common multiplier centre, canonically identified through $X$. The coefficient action gives a complete contraction ${\mathrm{\Theta }}_X:A{\otimes }_{Z_X,h}B\to \mathrm{CB}(X)$, where $A{\otimes }_{Z_X,h}B$ is the central Haagerup tensor product obtained by balancing over $Z_X$ and $\mathrm{CB}(X)$ is the space of completely bounded maps on $X$. We show that ${\mathrm{\Theta }}_X$ is injective exactly when every Glimm ideal of $A$ is $2$-primal, and isometric, equivalently completely isometric, exactly when every such ideal is primal. For each positive integer $\mathrm{\ell }$, norm preservation for sums of at most $\mathrm{\ell }$ elementary tensors is equivalent to $({\mathrm{\ell }}^2+1)$-primality of every Glimm ideal. These conditions may equivalently be imposed on $B$. This extends the corresponding theorems of Somerset and Archbold–Somerset–Timoney to imprimitivity bimodules. We also show that, after matrix stabilization, the optimal constants bounding the central Haagerup norm by the completely bounded norm coincide for $X$, $A$ and $B$, at every finite tensor length and on the completed products, and agree with the corresponding constants after compact stabilization.