CStarKushelTyaglov

Abstract

A cubic polynomial over two by two complex matrices refutes Krishna’s C*-algebraic conjecture.

Definition 1.1 (Krishna’s Conjecture 2.4).

$$claimKushelTyaglov\iff \forall A:Type,[\mathrm{CStarAlgebra}\left(A\right)][\mathrm{PartialOrder}\left(A\right)][\mathrm{StarOrderedRing}\left(A\right)]\forall d:\mathbb{N},(2\le d)\Rightarrow \forall a:\mathrm{Fin}\left(d\right)\to A,\forall b:\mathrm{Fin}\left(d-1\right)\to A,(\forall z:A,\sum _{j:\mathrm{Fin}\left(d\right)}(\mathrm{List}.\mathrm{prod}\left(\mathrm{List}.\mathrm{eraseIdx}\left(\mathrm{List}.\mathrm{ofFn}\left((i:\mathrm{Fin}\left(d\right))\mapsto z-a\left(i\right)\right),\mathrm{val}\left(j\right)\right)\right))=(d)\cdot (\mathrm{List}.\mathrm{prod}\left(\mathrm{List}.\mathrm{ofFn}\left((k:\mathrm{Fin}\left(d-1\right))\mapsto z-b\left(k\right)\right)\right)))\Rightarrow \mathrm{let}S=\sum _{j:\mathrm{Fin}\left(d\right)}(a\left(j\right));\mathrm{let}T=\sum _{j:\mathrm{Fin}\left(d\right)}((a\left(j\right){)}^2)-(\frac{1}{(d:\mathbb{C}{)}^2})\cdot ((S{)}^2);(\sum _{k:\mathrm{Fin}\left(d-1\right)}((b\left(k\right)\cdot \mathrm{star}\left(b\left(k\right)\right){)}^2)\le (\frac{(d:\mathbb{C})-6}{(d:\mathbb{C})})\cdot (\sum _{j:\mathrm{Fin}\left(d\right)}((a\left(j\right)\cdot \mathrm{star}\left(a\left(j\right)\right){)}^2))+(\frac{1}{(d:\mathbb{C}{)}^2})\cdot ((\sum _{j:\mathrm{Fin}\left(d\right)}(a\left(j\right)\cdot \mathrm{star}\left(a\left(j\right)\right)){)}^2)+(\frac{1}{(d:\mathbb{C}{)}^2})\cdot (T\cdot \mathrm{star}\left(T\right))+(\frac{2}{(d:\mathbb{C}{)}^1})\cdot (\sum _{j:\mathrm{Fin}\left(d\right)}(a\left(j\right)\cdot \left(a\left(j\right)+(\frac{1}{(d:\mathbb{C}{)}^1})\cdot (S)\right)\cdot \mathrm{star}\left(a\left(j\right)+(\frac{1}{(d:\mathbb{C}{)}^1})\cdot (S)\right)\cdot \mathrm{star}\left(a\left(j\right)\right)))-(\frac{4}{(d:\mathbb{C}{)}^3})\cdot (\sum _{j:\mathrm{Fin}\left(d\right)}(a\left(j\right)\cdot S\cdot \mathrm{star}\left(S\right)\cdot \mathrm{star}\left(a\left(j\right)\right))))\wedge (\sum _{k:\mathrm{Fin}\left(d-1\right)}((\mathrm{star}\left(b\left(k\right)\right)\cdot b\left(k\right){)}^2)\le (\frac{(d:\mathbb{C})-6}{(d:\mathbb{C})})\cdot (\sum _{j:\mathrm{Fin}\left(d\right)}((\mathrm{star}\left(a\left(j\right)\right)\cdot a\left(j\right){)}^2))+(\frac{1}{(d:\mathbb{C}{)}^2})\cdot ((\sum _{j:\mathrm{Fin}\left(d\right)}(\mathrm{star}\left(a\left(j\right)\right)\cdot a\left(j\right)){)}^2)+(\frac{1}{(d:\mathbb{C}{)}^2})\cdot (\mathrm{star}\left(T\right)\cdot T)+(\frac{2}{(d:\mathbb{C}{)}^1})\cdot (\sum _{j:\mathrm{Fin}\left(d\right)}(\mathrm{star}\left(a\left(j\right)\right)\cdot \mathrm{star}\left(a\left(j\right)+(\frac{1}{(d:\mathbb{C}{)}^1})\cdot (S)\right)\cdot \left(a\left(j\right)+(\frac{1}{(d:\mathbb{C}{)}^1})\cdot (S)\right)\cdot a\left(j\right)))-(\frac{4}{(d:\mathbb{C}{)}^3})\cdot (\sum _{j:\mathrm{Fin}\left(d\right)}(\mathrm{star}\left(a\left(j\right)\right)\cdot \mathrm{star}\left(S\right)\cdot S\cdot a\left(j\right))))$$

Formalization. D5/S3/Quantum/Algebra/CStarKushelTyaglov.claimKushelTyaglov (✓ std3).

Citation. K. Mahesh Krishna (2022). C-algebraic Schoenberg Conjecture*. DOI: 10.48550/arXiv.2206.06653. URL: https://arxiv.org/abs/2206.06653v1.

Commentary.

Conjecture 2.4 (C*-algebraic Kushel-Tyaglov Conjecture), pp. 3–4: “Let 𝒜 be a C*-algebra, n∈ℕ{1} and let P(z) ≔ (z−a₁)(z−a₂)⋯(z−a_d) be a polynomial over 𝒜 with a₁, a₂, …, a_d ∈ 𝒜. Assume that P′ can be written as P′(z) ≔ d(z−b₁)⋯(z−b_{d−1}) on 𝒜 with b₁, b₂, …, b_{d−1} ∈ 𝒜. Then” the two inequalities displayed below. The printed n is read as d throughout. The encoding uses zero-based Fin d and Fin (d−1), complex scalar multiplication, star and the C*-order. It restricts the source to d≥2 and unital C*-algebras in Type with PartialOrder and StarOrderedRing; this weakens the assertion, so its negation refutes the source statement.

Theorem 1.2 (The conjecture is false).

$$\neg claimKushelTyaglov$$

Proof. Machine-checked in Lean as D5/S3/Quantum/Algebra/CStarKushelTyaglov.result (✓ std3). ∎

Resolves. Problems/krishna-2022-cstar-kushel-tyaglov (refuted) by D5/S3/Quantum/Algebra/CStarKushelTyaglov.result.

Source. Repository-derived.

Acknowledgement. K. Mahesh Krishna (2022). C-algebraic Schoenberg Conjecture*. DOI: 10.48550/arXiv.2206.06653. URL: https://arxiv.org/abs/2206.06653v1.

Commentary.

At degree three the matrices a and b satisfy the factorization hypothesis. The first inequality has right side minus left side equal to −67/27 times Matrix.single 0 0 1. Its first diagonal entry is negative, contradicting the C*-order.

References

• Truth anchor: D5/S3/Quantum/Algebra/CStarKushelTyaglov.claimKushelTyaglov
• Truth anchor: D5/S3/Quantum/Algebra/CStarKushelTyaglov.result
• Dependency: D5/S3/Quantum/Algebra/CStarSchoenberg