Central Haagerup Tensor Products and Completely Bounded Maps under Strong Morita Equivalence
Abstract
Let A and B be strongly Morita equivalent C*-algebras, implemented by a nonzero imprimitivity bimodule X=AXB with its canonical operator-space structure. Denote by ZX their common multiplier centre, canonically identified through X. The coefficient action gives a complete contraction ΘX:A⊗ZX,hB→CB(X), where A⊗ZX,hB is the central Haagerup tensor product obtained by balancing over ZX and CB(X) is the space of completely bounded maps on X. We show that ΘX 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 ℓ, norm preservation for sums of at most ℓ elementary tensors is equivalent to (ℓ2+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.
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BibTeX
@misc{gogic2026central,
title = {{Central Haagerup Tensor Products and Completely Bounded Maps under Strong Morita Equivalence}},
author = {Ilja Gogić},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.39241},
eprint = {2609.39241},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.39241},
}
RIS
TY - UNPB
TI - Central Haagerup Tensor Products and Completely Bounded Maps under Strong Morita Equivalence
AU - Ilja Gogić
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.39241
UR - https://arxiv.org/abs/2609.39241
AB - Let A and B be strongly Morita equivalent C^*-algebras, implemented by a nonzero imprimitivity bimodule X=_AX_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 Θ_X:A⊗_Z_X,hB→CB(X), where A⊗_Z_X,hB is the central Haagerup tensor product obtained by balancing over Z_X and CB(X) is the space of completely bounded maps on X. We show that Θ_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 ℓ, norm preservation for sums of at most ℓ elementary tensors is equivalent to (ℓ^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.
ER -