[Paper Review] Modular Paulsen Problem and Modular Projection Problem

K. Mahesh Krishna|arXiv (Cornell University)|Jul 26, 2022

Advanced Operator Algebra Research4 citations

TL;DR

This paper extends the Paulsen and Projection Problems to Hilbert C*-modules over commutative C*-algebras, proving that a solution to the Modular Paulsen Problem implies a solution to the Modular Projection Problem. It introduces operator scaling over C*-algebras and formulates conjectures on restricted invertibility and Johnson-Lindenstrauss flattening in the modular setting, advancing noncommutative frame theory in functional analysis and operator algebras.

ABSTRACT

Based on the solution of extbf{Paulsen Problem} by Kwok, Lau, Lee, and Ramachandran [ extit{STOC'18-Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, 2018}] and independently by Hamilton, and Moitra [ extit{Isr. J. Math., 2021}] we study Paulsen Problem and Projection Problem in the context of Hilbert C*-modules. We show that for commutative C*-algebras, if Modular Paulsen Problem has a solution, then Modular Projection Problem also has a solution. We formulate the problem of operator scaling for matrices over C*-algebras.

Motivation & Objective

• To generalize the Paulsen Problem and Projection Problem from finite-dimensional Hilbert spaces to Hilbert C*-modules over commutative C*-algebras.
• To investigate whether a solution to the Modular Paulsen Problem implies a solution to the Modular Projection Problem in the context of commutative C*-algebras.
• To introduce and study the operator scaling problem for matrices over C*-algebras as a foundational tool for modular frame theory.
• To formulate and analyze modular analogues of classical results such as the Bourgain-Tzafriri restricted invertibility theorem and Johnson-Lindenstrauss flattening.
• To propose conjectures on modular restricted invertibility and flattening for noncommutative C*-algebras, particularly W*-algebras and algebras with invariant basis number (IBN) property.

Proposed method

• Formulates the Modular Paulsen Problem and Modular Projection Problem in the setting of Hilbert C*-modules over commutative C*-algebras.
• Uses the frame operator and its inverse square root to define transformations from nearly Parseval frames to exact Parseval frames in the modular setting.
• Introduces the concept of operator scaling for matrices over C*-algebras, generalizing classical scaling techniques to noncommutative operator algebras.
• Applies Manin matrices and column-wise determinants to formulate noncommutative versions of the restricted invertibility conjecture.
• Proposes the Modular Johnson-Lindenstrauss Flattening Conjecture, asserting that for sufficiently large $ m > C rac{1}{ ilde{ ho}^2} \ olimits \log M $, a matrix $ M \in \mathbb{M}_{m \times N}(\mathcal{A}) $ preserves pairwise distances up to $ (1 \pm \varepsilon) $.
• Employs the notion of Hilbert C*-module homomorphisms and norms to define the operator norm $ \|M\| $ used in the cardinality bounds of the restricted invertibility conjectures.

Experimental results

Research questions

• RQ1 Does a solution to the Modular Paulsen Problem imply a solution to the Modular Projection Problem in the case of commutative C*-algebras?
• RQ2 Can the operator scaling problem for matrices over C*-algebras be formulated and solved in a way that preserves frame-like properties?
• RQ3 What are the modular analogues of the Bourgain-Tzafriri restricted invertibility theorem in the context of Hilbert C*-modules?
• RQ4 Does the Modular Johnson-Lindenstrauss Flattening Conjecture hold for W*-algebras or C*-algebras with the invariant basis number (IBN) property?
• RQ5 What is the minimal dimension $ m $ required for a matrix $ M \in \mathbb{M}_{m \times N}(\mathcal{A}) $ to preserve pairwise distances in $ \mathcal{A}^N $ up to $ (1 \pm \varepsilon) $?

Key findings

• For commutative C*-algebras, if the Modular Paulsen Problem has a solution, then the Modular Projection Problem also has a solution.
• The paper establishes a direct implication between the solvability of the Modular Paulsen Problem and the Modular Projection Problem in the commutative setting.
• The Modular Johnson-Lindenstrauss Flattening Conjecture is proposed with a dimension bound $ m > \frac{C}{\varepsilon^2} \log M $, suggesting that such flattening is possible in the modular setting.
• The conjecture on modular restricted invertibility is formulated using Manin matrices and column determinants, with a lower bound on the size of the subset $ \sigma $ of indices satisfying $ \text{Card}(\sigma) \geq \frac{cd}{\|M\|^2} $.
• The paper shows that the distance between an $ \varepsilon $-nearly equal norm Parseval frame and its closest equal norm Parseval frame is bounded by $ \frac{d\varepsilon^2}{4} $, extending a classical result to the modular setting.
• The framework allows for the derivation of modular Welch bounds and the formulation of the Modular Zauner Conjecture, extending known results in frame theory to operator algebras.