Noncommutative Ck functions and Fréchet derivatives of operator functions
Noncommutative Ck functions and Fréchet derivatives of operator functions
复制标题
非交换 Ck 函数和算子函数的 Fréchet 导数
DOI：
10.1016/j.exmath.2022.12.004
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发表时间：
2023
影响因子：
0.7
通讯作者：
Nikitopoulos, Evangelos A.
中科院分区：
文献类型：
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作者：
Nikitopoulos, Evangelos A.
定一个单位C∗-代数A，记A为A的自伴元集；若f：R→ℂ为连续函数，则记f A：A Sa→A为由函数演算定义的算子函数a↦f(A)。本文引入并研究了由Ck函数f：R→ℂ构成的空间N Ck(R)，使得无论选择A，算子函数f A：a Sa→A都是k次连续Fréchet可微的.换言之，如果f∈NCk(R)，则f“提升”到Ck映射f A：a Sa→A，对于任何(可能是非交换的)单位C∗-代数A，我们称N Ck(R)为非交换Ck函数空间。我们的证明f A∈Ck(A Sa；A)只需要知道多项式的Fréchet导数和“多重算子积分”(MOI)的算子范数估计，比标准方法更基本；然而，N Ck(R)包含所有已知可比较结果的函数。证明了N C k(R)包含齐次Besov空间Ḃ1 k，∞(R)和Hölder空间C loc k，ɛ(R).然而，我们强调指出，本文中的结果是第一个在任意单位C∗-代数上被证明的结果，并且这种一般情形的推广利用了作者最近用MOI的定义解决了某些“可分性问题”。最后，通过具体的例子证明了Wk(R)loc⊊N Ck(R)⊊Ck(R)，其中Wk(R)loc是“局部化”的第k维纳空间.
Fix a unital C∗-algebra A, and write A sa for the set of self-adjoint elements of A. Also, if f: R→ ℂ is a continuous function, then write f A: A sa→ A for the operator function a↦ f (a) defined via functional calculus. In this paper, we introduce and study a space N C k (R) of C k functions f: R→ ℂ such that, no matter the choice of A, the operator function f A: A sa→ A is k-times continuously Fréchet differentiable. In other words, if f∈ N C k (R), then f “lifts” to a C k map f A: A sa→ A, for any (possibly noncommutative) unital C∗-algebra A. For this reason, we call N C k (R) the space of noncommutative C k functions. Our proof that f A∈ C k (A sa; 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, N C k (R) contains all functions for which comparable results are known. Specifically, we prove that N C k (R) contains the homogeneous Besov space B ̇ 1 k,∞(R) and the Hölder space C loc k, 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 W k (R) loc⊊ N C k (R)⊊ C k (R), where W k (R) loc is the “localized” k th Wiener space.