2015/11/05 by Huang, Hansong, Zheng, Dechao
#32H35 #47A13 #47B35 #47B91 #Complex Variables (math.CV) #FOS: Mathematics #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.1511.01678
In this paper we study multiplication operators on Bergman spaces of high dimensional bounded domains and those von Neumann algebras induced by them via the geometry of domains and function theory of their symbols. In particular, using local inverses and L2a-removability, we show that for a holomorphic proper map Φ=(ϕ1, ϕ2, ⋯ , ϕd) on a bounded domain Ω in ℂd, the dimension of the von Neumann algebra V^*(Φ,Ω) consisting of bounded operators on the Bergman space La2(Ω), which commute with both Mϕj and its adjoint Mϕj^* for each j, equals the number of components of the complex manifold SΦ= \(z,w)∈ Ω2: Φ(z)=Φ(w), z\not∈ Φ-1(Φ(Z))\, where Z is the zero variety of the Jacobian JΦ of Φ. This extends the main result in \citeDSZ in high dimensional complex domains. Moreover we show that the von Neumann algebra V^*(Φ,Ω) may not be abelian in general although Douglas, Putinar and Wang \citeDPW showed that V^*(Φ,\mathbbD) for the unit disk \mathbbD is abelian.