bibkey: krishna2022cstarschoenberg authors: K. Mahesh Krishna year: 2022 title: “C*-algebraic Schoenberg Conjecture” doi: 10.48550/arXiv.2206.06653 url: https://arxiv.org/abs/2206.06653v1 claim: “Section 2 defines the ordered derivative P′ and states Conjectures 2.1, 2.3 and 2.4; Theorem 2.2 proves the degree-two case of Conjecture 2.1.” strata_touched:

• D5/S3/Quantum/Algebra/CStarSchoenberg
• D5/S3/Quantum/Algebra/CStarDeBruinSharma
• D5/S3/Quantum/Algebra/CStarKushelTyaglov license: citation-only triage: anchor

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DOI: https://doi.org/10.48550/arXiv.2206.06653

Source: https://arxiv.org/abs/2206.06653v1

C*-algebraic Schoenberg Conjecture

K. Mahesh Krishna, “C*-algebraic Schoenberg Conjecture”, arXiv:2206.06653v1 (14 June 2022; single version), Section 2.

Let $\mathcal{A}$ be a C*-algebra. Given $P(z):=(z-a_1)(z-a_2)\cdots (z-a_d)$ for all $z\in \mathcal{A}$ with $a_1,a_2,\dots ,a_d\in \mathcal{A}$, we define $P^{\prime }(z)={\sum }_{j=1}^d(z-a_1)\cdots \widehat{(z-a_j)}\cdots (z-a_d),\forall z\in \mathcal{A}$ where the term with cap is missing.

Conjecture 2.1 (C-algebraic Schoenberg Conjecture).* Let $\mathcal{A}$ be a C*-algebra. Let $d\in \mathbb{N}\setminus \{1\}$, $P(z):=(z-a_1)(z-a_2)\cdots (z-a_d)$ be a polynomial over $\mathcal{A}$ with $a_1,a_2,\dots ,a_d\in \mathcal{A}$. If $P^{\prime }$ can be written as $P^{\prime }(z)=d(z-b_1)(z-b_2)\cdots (z-b_{d-1})$ on $\mathcal{A}$ with $b_1,b_2,\dots ,b_{d-1}\in \mathcal{A}$, then $$\sum _{k=1}^{d-1}{b}_k{b}_k^{\ast }\le \frac{1}{d^2}\left[\sum _{j=1}^d{a}_j\right]{\left[\sum _{j=1}^d{a}_j\right]}^{\ast }+\frac{d-2}{d}\sum _{j=1}^d{a}_j{a}_j^{\ast }$$ and $$\sum _{k=1}^{d-1}{b}_k^{\ast }{b}_k\le \frac{1}{d^2}{\left[\sum _{j=1}^d{a}_j\right]}^{\ast }\left[\sum _{j=1}^d{a}_j\right]+\frac{d-2}{d}\sum _{j=1}^d{a}_j^{\ast }{a}_j.$$

Conjecture 2.3 (C-algebraic de Bruin-Sharma Conjecture).* Let $\mathcal{A}$ be a C*-algebra, $d\in \mathbb{N}\setminus \{1\}$ and let $P(z):=(z-a_1)(z-a_2)\cdots (z-a_d)$ be a polynomial over $\mathcal{A}$ with $a_1,a_2,\dots ,a_d\in \mathcal{A}$. Assume that $P^{\prime }$ can be written as $P^{\prime }(z):=d(z-b_1)\cdots (z-b_{d-1})$ on $\mathcal{A}$ with $b_1,b_2,\dots ,b_{d-1}\in \mathcal{A}$. If ${\sum }_{j=1}^d{a}_j=0,$ then $$\sum _{k=1}^{d-1}(b_k{b}_k^{\ast }{)}^2\le \frac{2}{d^2}{\left(\sum _{j=1}^d{a}_j{a}_j^{\ast }\right)}^2+\frac{d-4}{d}\sum _{j=1}^d(a_j{a}_j^{\ast }{)}^2$$ and $$\sum _{k=1}^{d-1}(b_k^{\ast }{b}_k{)}^2\le \frac{2}{d^2}{\left(\sum _{j=1}^d{a}_j^{\ast }{a}_j\right)}^2+\frac{d-4}{d}\sum _{j=1}^d(a_j^{\ast }{a}_j{)}^2.$$

Conjecture 2.4 (C-algebraic Kushel-Tyaglov Conjecture).* Let $\mathcal{A}$ be a C*-algebra, $n\in \mathbb{N}\setminus \{1\}$ and let $P(z):=(z-a_1)(z-a_2)\cdots (z-a_d)$ be a polynomial over $\mathcal{A}$ with $a_1,a_2,\dots ,a_d\in \mathcal{A}$. Assume that $P^{\prime }$ can be written as $P^{\prime }(z):=d(z-b_1)\cdots (z-b_{d-1})$ on $\mathcal{A}$ with $b_1,b_2,\dots ,b_{d-1}\in \mathcal{A}$. Then $$\sum _{k=1}^{d-1}(b_k{b}_k^{\ast }{)}^2\le \frac{d-6}{d}\sum _{j=1}^n(a_j{a}_j^{\ast }{)}^2+\frac{1}{d^2}{\left(\sum _{j=1}^d{a}_j{a}_j^{\ast }\right)}^2+\frac{1}{d^2}\left[\sum _{j=1}^d{a}_j^2-\frac{1}{d^2}{\left(\sum _{k=1}^d{a}_k\right)}^2\right]{\left[\sum _{j=1}^d{a}_j^2-\frac{1}{d^2}{\left(\sum _{k=1}^d{a}_k\right)}^2\right]}^{\ast }+\frac{2}{d}\sum _{j=1}^d{a}_j\left[a_j+\frac{1}{d}\sum _{k=1}^d{a}_k\right]{\left[a_j+\frac{1}{d}\sum _{k=1}^d{a}_k\right]}^{\ast }{a}_j^{\ast }-\frac{4}{d^3}\sum _{j=1}^d{a}_j\left[\sum _{k=1}^d{a}_k\right]{\left[\sum _{k=1}^d{a}_k\right]}^{\ast }{a}_j^{\ast }$$ and $$\sum _{k=1}^{d-1}(b_k^{\ast }{b}_k{)}^2\le \frac{d-6}{d}\sum _{j=1}^d(a_j^{\ast }{a}_j{)}^2+\frac{1}{d^2}{\left(\sum _{j=1}^d{a}_j^{\ast }{a}_j\right)}^2+\frac{1}{d^2}{\left[\sum _{j=1}^d{a}_j^2-\frac{1}{d^2}{\left(\sum _{k=1}^d{a}_k\right)}^2\right]}^{\ast }\left[\sum _{j=1}^d{a}_j^2-\frac{1}{d^2}{\left(\sum _{k=1}^d{a}_k\right)}^2\right]+\frac{2}{d}\sum _{j=1}^n{a}_j^{\ast }{\left[a_j+\frac{1}{d}\sum _{k=1}^d{a}_k\right]}^{\ast }\left[a_j+\frac{1}{d}\sum _{k=1}^d{a}_k\right]a_j-\frac{4}{d^3}\sum _{j=1}^d{a}_j^{\ast }{\left[\sum _{k=1}^d{a}_k\right]}^{\ast }\left[\sum _{k=1}^d{a}_k\right]a_j.$$

Theorem 2.2. Conjecture 2.1 holds for C*-algebraic polynomials of degree 2.

The source proves Conjecture 2.1 for d = 2 (Theorem 2.2) and states that Conjecture 2.3 holds for d = 2. In Conjecture 2.4 the index $n$ is bound, and $d$ is otherwise free while the sums run to $n$ in two places. The encoding reads n as d.

The definition of P′ and Conjecture 2.1 begin on page 2; Conjecture 2.1 continues on page 3. Theorem 2.2 and Conjecture 2.3 occur on page 3. Conjecture 2.4 spans pages 3–4. In Conjecture 2.4, the encoding reads the printed n as d. Indices become zero-based Fin types, while the product order is retained.

The source quantifies over C*-algebras. The encoding quantifies over unital C*-algebras in Type, with PartialOrder and StarOrderedRing, and d ≥ 2. These are restrictions of the universal assertion, so a counterexample in this subclass refutes the source assertion. The counterexample has degree three, leaving the source’s degree-two result intact.