Noncommutative [formula omitted] functions and Fréchet derivatives of operator functions

Abstract

Fix a unital C∗-algebra A, and write Asa for the set of self-adjoint elements of A. Also, if f:R→ℂ is a continuous function, then write fA:Asa→A for the operator functiona↦f(a) defined via functional calculus. In this paper, we introduce and study a space NCk(R) of Ck functions f:R→ℂ such that, no matter the choice of A, the operator function fA:Asa→A is k-times continuously Fréchet differentiable. In other words, if f∈NCk(R), then f “lifts” to a Ck map fA:Asa→A, for any (possibly noncommutative) unital C∗-algebra A. For this reason, we call NCk(R) the space of noncommutativeCkfunctions. Our proof that fA∈Ck(Asa;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, NCk(R) contains all functions for which comparable results are known. Specifically, we prove that NCk(R) contains the homogeneous Besov space Ḃ1k,∞(R) and the Hölder space Clock,ɛ(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 Wk(R)loc⊊NCk(R)⊊Ck(R), where Wk(R)loc is the “localized” kth Wiener space.