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math.FA — Functional Analysis

Noncommutative Buzano-Dragomir Inequality and Applications

Contributed by K. MAHESH KRISHNA

Dragomir [\textit{Bull. Aust. Math. Soc., 2016}] showed that the Buzano inequality holds for orthogonal projections on Hilbert spaces. Dragomir [\textit{Linear Multilinear Algebra, 2016}] also derived the most general form of the Buzano inequality for bounded linear operators on Hilbert spaces. We show that Dragomir result extends to adjointable morphisms on Hilbert C*-modules. Using this generalization, we derive bounds for the roots of polynomials over commutative unital C*-algebras. We formulate the notion of noncommutative numerical range and derive numerical radius bounds for self-adjoint morphisms on Hilbert C*-modules over unital C*-algebras. We formulate several open problems, including the noncommutative Toeplitz-Hausdorff, von Neumann inequality, Ando inequality, Berger power dilation, Kittaneh inequality, Crouzeix problems.

Primarily human-written textHuman understanding: all partsBuzano InequalityCrouzeix inequality.Hilbert C*-modulesNumerical radiusPolynomial roots

Oct 07, 2026PDF ↗View work →

hexagon:2610.00134v2

math.OA — Operator Algebras

Poisson boundaries of type III factors and the relative bicentralizer conjecture

Contributed by Shuoxing Zhou

We develop the Poisson boundary theory for general von Neumann algebras with separable preduals. As applications, we generalize and give new proofs of the results on the weak Dixmier property for inclusions with expectations or operator-valued weights \cite{Pop99,Mar19,Iso25}. We also prove that every continuous factor admits irreducible embeddings into the hyperfinite factors of type $\mathrm{III}_\lambda$ , $0<\lambda\leq1$ , and into some hyperfinite factor of type $\mathrm{III}_0$ . Finally, using the Poisson boundary techniques developed above, we prove that for every inclusion with faithful normal conditional expectation $M\subset N$ , $M\subset c(N)$ has the weak Dixmier property. By Marrakchi's characterization \cite[Theorem~D]{Ma25}, this gives the relative bicentralizer conjecture for von Neumann algebras with separable preduals.

A mix of human-written and AI-generated textHuman understanding: all parts

Oct 06, 2026PDF ↗View work →

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Noncommutative Buzano-Dragomir Inequality and Applications

Poisson boundaries of type III factors and the relative bicentralizer conjecture

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