Noncommutative \(C^k\) functions and Fr\'{e}chet derivatives of operator functions

• Published in: arXiv.org 2023-12
• Main Author: Nikitopoulos, Evangelos A
• Format: Article
• Language: English
• Subjects: Algebra; Derivatives; Function space; Hilbert space; Integrals; Mathematical analysis; Operators (mathematics); Polynomials; Tensors
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.2011.03126

Fix a unital \(C^*\)-algebra \(\mathscr{A}\), and write \(\mathscr{A}_{sa}\) for the set of self-adjoint elements of \(\mathscr{A}\). Also, if \(f:\mathbb{R}\to\mathbb{C}\) is a continuous function, then write \(f_\mathscr{A}:\mathscr{A}_{sa}\to\mathscr{A}\) for the operator function \(a\mapsto f(a)\) defined via functional calculus. In this paper, we introduce and study a space \(NC^k(\mathbb{R})\) of \(C^k\) functions \(f:\mathbb{R}\to\mathbb{C}\) such that, no matter the choice of \(\mathscr{A}\), the operator function \(f_\mathscr{A}:\mathscr{A}_{sa}\to\mathscr{A}\) is \(k\)-times continuously Fréchet differentiable. In other words, if \(f\in NC^k(\mathbb{R})\), then \(f\) "lifts" to a \(C^k\) map \(f_\mathscr{A}:\mathscr{A}_{sa}\to\mathscr{A}\), for any (possibly noncommutative) unital \(C^*\)-algebra \(\mathscr{A}\). For this reason, we call \(NC^k(\mathbb{R})\) the space of noncommutative \(C^k\) functions. Our proof that \(f_\mathscr{A}\in C^k(\mathscr{A}_{sa};\mathscr{A})\), which requires only knowledge of the Fréchet derivatives of polynomials and operator norm estimates for "multiple operator integrals" (MOIs), is more elementary than the standard approach; nevertheless, \(NC^k(\mathbb{R})\) contains all functions for which comparable results are known. Specifically, we prove that \(NC^k(\mathbb{R})\) contains the homogeneous Besov space \(\dot{B}_1^{k,\infty}(\mathbb{R})\) and the H\"older space \(C_{loc}^{k,\varepsilon}(\mathbb{R})\). We highlight, however, that the results in this paper are the first of their type to be proven for arbitrary unital \(C^*\)-algebras, and that the extension to such a general setting makes use of the author's recent resolution of certain "separability issues" with the definition of MOIs. Finally, we prove by exhibiting specific examples that \(W_k(\mathbb{R})_{loc}\subsetneq NC^k(\mathbb{R})\subsetneq C^k(\mathbb{R})\), where \(W_k(\mathbb{R})_{loc}\) is the "localized" \(k^{th}\) Wiener space.