Huai Xin CAO(1),Zong Ben XU(2)
In this paper, let (K, d) be a given compact metric space, A a unital Banach
algebra 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 and
of 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 algebras
with 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 when
φ(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 of
a metric space is introduced and is applied to give necessary condition and sufficient
condition for lim_(α→+∞)l~α(K, A) =A. Conditions for lim_(α→0~+)L~α(K, A) = C(K, A) are
also obtained.