Bures--Kuratowski metrics and simplicial complexes for completely bounded maps

Abstract

Let $A$ be a unital $C^{\ast }$-algebra and $H$ a Hilbert space. The cone $\CP(A,B(H))$ of completely positive maps carries the Bures metric ${\hat{I}}^2$, closely related to the cb-norm. We introduce a family of Bures--Kuratowski (BK) metrics on $\CB(A,B(H))$ that extend ${\hat{I}}^2$ exactly on $\CP(A,B(H))$. The construction combines a Kuratowski embedding of the Bures cone, based at an anchor $θ\in\CP(A,B(H))$, with a regular-representation Hausdorff coordinate arising from universal regular models. Each BK metric admits an ${\ell }^p$-wedge decomposition, splitting $\CB(A,B(H))$ into the Bures cone and a non-CP component attached at $\hat{I}¸$. We then study Vietoris--Rips and Čech complexes of BK metric spaces. The wedge formula yields explicit criteria for mixed simplices, a join-type description of the mixed Rips complex, and ball-intersection criteria for mixed Čech simplices. For finite point clouds, this makes the mixed simplicial geometry computable from the two component metrics and reveals new homological features arising from the interaction between the CP and non-CP sectors.

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