$$C^*$$ C ∗ -algebras of Bergman Type Operators with Piecewise Continuous Coefficients Over Domains with Dini-Smooth Corners

C*-Algebras of Poly-Bergman Type Operators With Piecewise Slowly Oscillating Coefficients

“…By the proof of [13,Theorem 7.1(v)] for z = 1 , for every sequence {t 𝜈 } ⊂ Υ with lim →∞ t = 0 , we obtain Since the functions a k in (6.19) belong to SO (U) and hence a k • ∈ SO (Π) , we infer that for every ∈ M 0 (SO (Π)) there exists a sequence of numbers t 𝜈 > 0 such that lim →∞ t = 0 and where the character ∈ M z (SO (U) is related to ∈ M 0 (SO (Π)) . Consequently, for a ∈ ( ) , we deduce from (6.19), (6.20) and (6.21) that where the piecewise constant function a z, ∈ L ∞ (Π) is defined by Further, for every coset J 0 ∈ J ,Π , where we infer that (6.17) (V t f )(w) = tf (tw) for all f ∈ L 2 (Π) and all w ∈ Π.…”

“…Such C * -algebras for bounded domains U with smooth boundaries were studied in [24] and [25]. The papers [12,13,15] (see also [20]) dealt with studying C * -algebras generated by the multiplication operators by piecewise continuous functions and by the Bergman and anti-Bergman projections B U,1 and BU,1 on the space L 2 (U) over domains U with piecewise Dini-smooth boundaries admitting Dini-smooth corners. These C * -algebras with a wider class of multiplication operators by piecewise slowly oscillating functions were studied in [14] by applying results of [43] on the C * -algebra Q = VMO ( ) ∩ L ∞ ( ) .…”