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

Abstract: Let $A$ be a unital $C<sup>*$-algebra and $H$ a Hilbert space. The cone $\CP(A,B(H))$ of completely positive maps carries the Bures metric $β$, closely related to the cb-norm. We introduce a family of Bures--Kuratowski (BK) metrics on $\CB(A,B(H))$ that extend $β$ 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 $\ell<sup>p$-wedge decomposition, splitting $\CB(A,B(H))$ into the Bures cone and a non-CP component attached at $θ$. 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.

• The paper develops a family of Bures–Kuratowski metrics on completely bounded maps that extend the classical Bures metric from CP maps through an explicit ℓ^p wedge decomposition.
• It introduces a novel decomposition of the non-CP sector using regular-representation Hausdorff geometry and Kuratowski embeddings, providing computable topological invariants.
• The framework applies Vietoris–Rips and Čech simplicial complexes to reveal distinct homological features between CP and non-CP regions, advancing operator-algebraic topology.

Bures–Kuratowski Metrics and Simplicial Geometry for Completely Bounded Maps

Introduction and Objectives

This paper presents a novel metric framework on the space of completely bounded (CB) maps between a unital $C^*$-algebra $A$ and bounded operators $B(H)$, designated as $CB(A, B(H))$. The traditional Bures metric $\beta$, defined via Stinespring representations, endows the cone of completely positive (CP) maps $CP(A, B(H))$ with an operator-algebraic metric structure closely related to the cb-norm. However, the extension of Bures geometry to all CB maps is nontrivial, as global compatibility with the cb-norm is unattainable due to the intricate structure of non-CP maps. The central contribution of the paper is a family of Bures–Kuratowski (BK) metrics on $CB(A, B(H))$ that retain the exact Bures geometry on the CP cone and introduce a new, explicit decomposition of the non-CP sector via regular-representation Hausdorff geometry and Kuratowski-type embeddings. This construction is tied to advances in the theory of regular representations of CB maps and has strong implications for the topological analysis of operator-algebraic metric spaces via simplicial complexes.

BK Metric Construction and Structure

The extension proceeds by identifying canonical universal regular models $(K, \tau)$ for completely contractive maps, leveraging the Bhat–Mallick–Sumesh theorem which guarantees every CB map factors through a regular homomorphism. For each regular model, $\mathscr{R}_\tau(\phi)$ collects all (scaled) isometric implementers, and the Hausdorff distance between such sets yields the model-dependent metric $\delta_{\text{reg}}^\tau$. The supremum of normalized Hausdorff coordinates across all universal models defines the intrinsic regular metric $A$0 on $A$1.

To glue the Bures geometry on the CP cone to the regular non-CP geometry, the authors introduce a Kuratowski embedding based at an anchor $A$2: for CP maps, $A$3 records the Bures distance to other CP maps, shifted by the reference point $A$4.

The final BK metric is parameterized by anchor $A$5, scaling $A$6, exponent $A$7, and $A$8: $A$9 This metric reduces to the Bures distance on the CP cone and splits the space canonically into two sectors via an explicit $B(H)$0 wedge decomposition: the classical Bures-metric CP cone, and a pointed non-CP sector with regular geometry, attached uniquely at the anchor $B(H)$1.

Metric Topology and Isometric Embeddings

The dependence of the BK metric on the parameters $B(H)$2 is limited: variations do not alter the induced topology, while choice of anchor $B(H)$3 yields genuinely different metric topologies. Notably, the assignment

$B(H)$4

is an isometric embedding of the pointed Bures CP cone into the space of all metrics on $B(H)$5 with respect to uniform convergence. The authors explicitly demonstrate that this construction does not generally recover the cb-norm topology, nor does it admit global KSW-type upper bounds on $B(H)$6, establishing a sharp distinction between the extension to non-CP maps and classical CP theory.

Simplicial Complexes: Vietoris–Rips and Čech Analyses

The metric wedge structure induced by the BK metric allows for explicit computation of topological invariants via Vietoris–Rips and Čech constructions. For the BK space as an $B(H)$7 wedge, the Vietoris–Rips complex at scale $B(H)$8 admits an explicit join decomposition of mixed simplices, governed by maximal radial functions measuring distance to the glued basepoints. In the case $B(H)$9, for finite point clouds,

$CB(A, B(H))$0

where $CB(A, B(H))$1 and $CB(A, B(H))$2 denote the CP and non-CP vertices of $CB(A, B(H))$3. This gives an explicit, algorithmically tractable description of the complex’s topology and demonstrates the emergence of nontrivial homology (e.g., loops $CB(A, B(H))$4, higher-dimensional spheres) arising purely from combinatorial interactions between CP and non-CP sectors. For instance, isolated points on each side of the wedge can generate non-contractible $CB(A, B(H))$5-type subcomplexes absent from either sector alone.

For Čech complexes, the situation is more delicate: mixed simplicity is governed by ball intersection criteria in one sector coupled to radial constraints from the other, and the distinction between intrinsic and ambient constructions becomes essential. The ambient Čech complex can become contractible at small scales due to the presence of the glued basepoint, potentially collapsing homological features that persist in the Rips filtration.

Operator Algebra Examples and Explicit Computations

The paper provides explicit computations for the Bures metric and the resulting BK simplicial geometry in finite-dimensional commutative and matrix cases, such as $CB(A, B(H))$6 (where the Bures metric reduces to Euclidean distance between root-vectors on $CB(A, B(H))$7), and $CB(A, B(H))$8, $CB(A, B(H))$9 (where the ray generated by the completely depolarizing channel $\beta$0 is Bures-isometric to $\beta$1 via $\beta$2). On these clouds, the structural results translate directly into explicit topological computations, e.g., for certain scales, the Rips complex realizes a $\beta$3-cycle ($\beta$4), and adding non-CP points can create wedge-induced homology that is absent in the CP subcomplex.

Implications and Further Directions

This framework provides a unified metric-topological structure for the analysis of CB maps, clarifying the geometric and algebraic interactions between CP and non-CP regions. The BK metrics facilitate new, computable simplicial/topological invariants sensitive to the regular representation geometry of operator algebraic maps, with applications to structure theory, classification, and numerical topology of quantum channels and linear maps.

The construction suggests that domains where operator-algebraic data is analyzed via topological methods—such as quantum information theory, classification of operator spaces, noncommutative geometry, and persistent topology for operator-valued data—may benefit from this wedge-metric approach. Future developments could include persistent homology for CB spaces, extensions to more general non-self-adjoint cones, and invariants for dynamical or categorical quantum systems encoding both CP and non-CP behaviors.

Conclusion

The paper establishes a concrete and tractable metric and topological framework for completely bounded maps that exactly respects the CP geometry and elucidates the structure of the non-CP sector via Hausdorff geometry in universal regular models. The wedge construction yields explicit, computable criteria for combinatorial topology (Vietoris–Rips and Čech complexes) on finite clouds, reveals new homological phenomena, and links deep aspects of operator algebra geometry to applied topological analysis. The separation between CP and non-CP sectors, parametrization by anchors, and explicit homological consequences represent a substantive advancement in the metric-topological study of noncommutative linear maps (2604.04248).