非交换Lipschitz-φ算子代数

本文引入由紧距离空间(K,d)到给定Banach代数A中的Lipschitz-φ算子构成的非交换Banach代数L~φ(K,A)与l~φ(K,A),证明了它们都是由K到A的全体连续算子构成的非交换Banach代数C(K,A)的子代数,并且关于范数||f||φ=L_φ(f)+||f||∞是Banach代数,研究了不同 Lipschitz尺度函数φ对应的大(小)Lipschitz代数之间的关系。特别当φ(t)=t~α时,引入了极限代数lim_(α→0+)l~α(K,A),lim_(α→+∞)l~α(K,A),lim_(α→0+)L~α(K,A)与lim_(α→+∞)L~α(K,A)以及距离空间的Lipschitz连通性,得到了lim_(α→+∞)l~α(K,A)=A的充要条件,也给出了lim_(α→0+)L~α(K,A)=C(K,A)的条件。

Non-Commutative Lipschitz-φ Operator Algebras

In this paper, let (K, d) be a given compact metric space, A a unital Banachalgebra and φ: [0, +∞)→ [0, ∞) a continuous function with ker φ= {0}. Noncommu-tative Banach algebras L~φ(K, A) and l~φ(K, A), consisting of Lipschitz-φ operators andof little Lipschitz-φ operators from K into A, respectively are introduced and discussed.It is proved that these algebras are both subalgebras of noncommutative Banach alge-bra C(K, A), consisting of all continuous operators from K into A, and Banach algebraswith respect to norm ||f||_φ = L_φ(f) +||f||_∞. Secondly, inclusion relationships between L~φ(K,A) and L~ψ(K,A) (resp. l~φ(K,A) and l~ψ(K,A)) are given. Especially whenφ(t) = t~α, the limit algebras lim_(α→0~+)l~α(K, A), lim_(α→+∞)l~α(K, A), lim_(α→-o~+)L~α(K, A)and lim_(α→+∞)L~α(K,A) are defined and studied. Thirdly, Lipschitz connectedness ofa metric space is introduced and is applied to give necessary condition and sufficientcondition for lim_(α→+∞)l~α(K, A) =A. Conditions for lim_(α→0~+)L~α(K, A) = C(K, A) arealso obtained.