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Representations of noncommutative cubes and prisms

Douglas Farenick, Roghayeh Maleki, Sofia Medina Varela, Sushil Singla

TL;DR

The paper develops a comprehensive framework tying noncommutative convex geometry to operator-system theory for noncommutative cubes NC(d) and triangular/prismatic variants NCP(k). By leveraging dualities between NCConv and OpSys, it characterizes extreme points, representations (including factorial and boundary representations), and duality structures, and it establishes sharp separations among tensor products: R⊗min T ≠ R⊗c T ≠ R⊗max T for R,T in {NC(d),NCP(k)} with d,k≥3. A key highlight is the Halmos–Mirman correspondence for P(3), extended dilation results, and the explicit injective and minimal envelopes for low-dimensional cases, together with scaling constants and exactness/lifting properties that connect to the Connes Embedding Problem. Collectively, these results illuminate the noncommutative geometry of d-cubes and k-prisms, reveal intricate operator-system tensor-product phenomena, and provide concrete dilations, dualities, and CP-extension controls with potential CEP implications.

Abstract

Representations of the operator system determined by the canonical generators of the free product of two cyclic groups of order $2$ and $k$, or $d$ cyclic groups of order $2$, are studied for the purpose of shedding light on the noncommutative geometry of noncommutative $d$-cubes and $k$-prisms. By way of the duality of the categories NCConv and OpSys of noncommutative convex sets and operator systems, respectively, an analysis of noncommutative extreme points, exactness, the lifting property, automatic complete positivity, controlled completely positive extensions, tensor products, and operator system duality is undertaken. Of note is the pairing of two classical dilation theorems of Halmos and Mirman to give a complete description of the noncommutative triangular prism in terms of joint unitary dilations.

Representations of noncommutative cubes and prisms

TL;DR

The paper develops a comprehensive framework tying noncommutative convex geometry to operator-system theory for noncommutative cubes NC(d) and triangular/prismatic variants NCP(k). By leveraging dualities between NCConv and OpSys, it characterizes extreme points, representations (including factorial and boundary representations), and duality structures, and it establishes sharp separations among tensor products: R⊗min T ≠ R⊗c T ≠ R⊗max T for R,T in {NC(d),NCP(k)} with d,k≥3. A key highlight is the Halmos–Mirman correspondence for P(3), extended dilation results, and the explicit injective and minimal envelopes for low-dimensional cases, together with scaling constants and exactness/lifting properties that connect to the Connes Embedding Problem. Collectively, these results illuminate the noncommutative geometry of d-cubes and k-prisms, reveal intricate operator-system tensor-product phenomena, and provide concrete dilations, dualities, and CP-extension controls with potential CEP implications.

Abstract

Representations of the operator system determined by the canonical generators of the free product of two cyclic groups of order and , or cyclic groups of order , are studied for the purpose of shedding light on the noncommutative geometry of noncommutative -cubes and -prisms. By way of the duality of the categories NCConv and OpSys of noncommutative convex sets and operator systems, respectively, an analysis of noncommutative extreme points, exactness, the lifting property, automatic complete positivity, controlled completely positive extensions, tensor products, and operator system duality is undertaken. Of note is the pairing of two classical dilation theorems of Halmos and Mirman to give a complete description of the noncommutative triangular prism in terms of joint unitary dilations.
Paper Structure (28 sections, 40 theorems, 96 equations)

This paper contains 28 sections, 40 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. The Objects of Study
  3. Categorical structures
  4. Minimal and maximal noncommutative convex sets
  5. Noncommutative cubes
  6. Noncommutative prisms
  7. C$^*$- and injective covers
  8. Noncommutative extreme points, representations, and purity
  9. Representations of the Noncommutative Cube
  10. The noncommutative square
  11. Noncommutative cubes, with $d\geq3$
  12. Representations of Noncommutative Prisms
  13. Proof of the Halmos-Mirman Theorem
  14. Infinite-dimensional representations
  15. Finite-dimensional representations
  16. ...and 13 more sections

Key Result

Theorem 2.1

For every $d\geq 2$, and where $\mathfrak u=(u_1,\dots,u_d)$ is the $d$-tuple of canonical symmetries generating ${\rm C}^*(\mathbb Z_2*\cdots* \mathbb Z_2)$.

Theorems & Definitions (69)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3: Halmos-Mirman
  • Theorem 2.4
  • Theorem 2.5
  • proof
  • Definition 2.6
  • Proposition 2.7
  • proof
  • Corollary 2.8
  • ...and 59 more