发表机构
University of Zagreb（萨格勒布大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究强Morita等价C*-代数间中心Haagerup张量积到完全有界映射空间的系数作用，刻画其单射性、等距性与范数保持条件，推广Somerset等定理并证明稳定化后最优常数的一致性。
AI 中文摘要
设$A$和$B$是强Morita等价的$C^*$-代数，由非零的既约双模$X={}_AX_B$及其典范算子空间结构实现。记$Z_X$为它们的公共乘子中心，通过$X$典范地等同。系数作用给出一个完全压缩$\Theta_X:A\otimes_{Z_X,h}B\to\operatorname{CB}(X)$，其中$A\otimes_{Z_X,h}B$是通过在$Z_X$上平衡得到的中心Haagerup张量积，$\operatorname{CB}(X)$是$X$上的完全有界映射空间。我们证明$\Theta_X$是单射当且仅当$A$的每个Glimm理想都是$2$-素理想，并且是等距的（等价地完全等距的）当且仅当每个这样的理想都是素理想。对于每个正整数$\ell$，至多$\ell$个初等张量之和的范数保持等价于每个Glimm理想的$(\ell^2+1)$-素性。这些条件可以等价地施加在$B$上。这推广了Somerset和Archbold--Somerset--Timoney的相应定理到既约双模。我们还证明，在矩阵稳定化之后，控制中心Haagerup范数的最优常数与完全有界范数在$X$、$A$和$B$上一致，对每个有限张量长度以及在完备化乘积上，并且与紧稳定化后的相应常数一致。
英文摘要
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 $Θ_X:A\otimes_{Z_X,h}B\to\operatorname{CB}(X)$, where $A\otimes_{Z_X,h}B$ is the central Haagerup tensor product obtained by balancing over $Z_X$ and $\operatorname{CB}(X)$ 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 $\ell$, norm preservation for sums of at most $\ell$ elementary tensors is equivalent to $(\ell^2+1)$-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.
Comments20 pages
