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

DOI10.1016/J.EXMATH.2022.12.004arXiv2011.03126MaRDI QIDQ6353104

Publication date: 5 November 2020

Abstract: Fix a unital

C^{*}

-algebra

m a t h c a l A

. If

f : m a t h b b R o m a t h b b C

is a continuous function, then we write

f_{m a t h c a l A} : m a t h c a l A_{m a t h r m s a} o m a t h c a l A

for the operator function

a m a p s t o f (a)

defined via functional calculus. In this paper, we introduce and study a space

N C^{k} (m a t h b b R)

of

C^{k}

functions

f : m a t h b b R o m a t h b b C

such that, no matter the choice of

m a t h c a l A

, the operator function

f_{m a t h c a l A} : m a t h c a l A_{m a t h r m s a} o m a t h c a l A

is

k

-times continuously Fr'{e}chet differentiable. In other words, if

f i n N C^{k} (m a t h b b R)

, then

f

"lifts" to a

C^{k}

map

f_{m a t h c a l A} : m a t h c a l A_{m a t h r m s a} o m a t h c a l A

, for any (possibly noncommutative) unital

C^{*}

-algebra

m a t h c a l A

. For this reason, we call

N C^{k} (m a t h b b R)

the space of noncommutative

C^{k}

functions. Our proof that

f_{m a t h c a l A} i n C^{k} (m a t h c a l A_{m a t h r m s a}; m a t h c a l A)

, which requires only knowledge of the Fr'{e}chet derivatives of polynomials and operator norm estimates for "multiple operator integrals" (MOIs), is more elementary than the standard approach; nevertheless,

N C^{k} (m a t h b b R)

contains all functions for which comparable results are known. Specifically, we prove that

N C^{k} (m a t h b b R)

contains the homogeneous Besov space

d o t B_{1}^{k, i n f t y} (m a t h b b R)

and the H"{o}lder space

C_{l o c}^{k, v a r e p s i l o n} (m a t h b b 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} (m a t h b b R)_{l o c} s u b s e t n e q N C^{k} (m a t h b b R) s u b s e t n e q C^{k} (m a t h b b R)

, where

W_{k} (m a t h b b R)_{l o c}

is the "localized"

k^{e x t t h}

Wiener space.

Mathematics Subject Classification ID

Sobolev spaces and other spaces of ``smooth functions, embedding theorems, trace theorems (46E35) Functional calculus for linear operators (47A60) Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones) (47A56) Other ``noncommutative mathematics based on (C^*)-algebra theory (46L89) Calculus of functions on infinite-dimensional spaces (26E15)

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