Berezin transforms on noncommutative varieties in polydomains

Popescu, Gelu

J. Funct. Anal. 265 (2013), no. 10, 2500–2552

Let Q be a set of polynomials in noncommutative indeterminates Zi,j, i∈{1,…,k}, j∈{1,…,ni}. In this paper, we study noncommutative varieties

VQ(H):={X={Xi,j}∈D(H):g(X)=0 for all g∈Q},

where D(H) is a regular polydomain in B(H)n1+⋯+nk and B(H) is the algebra of bounded linear operators on a Hilbert space H. Under natural conditions on Q, we show that there is a universal model S={Si,j} such that g(S)=0, g∈Q, acting on a subspace of a tensor product of full Fock spaces. We characterize the variety VQ(H) and its pure part in terms of the universal model and a class of completely positive linear maps. We obtain a characterization of those elements in VQ(H) which admit characteristic functions and prove that the characteristic function is a complete unitary invariant for the class of completely non-coisometric elements. We study the universal model S, its joint invariant subspaces and the representations of the universal operator algebras it generates: the variety algebra A(VQ), the Hardy algebra F∞(VQ), and the C⁎-algebra C⁎(VQ). Using noncommutative Berezin transforms associated with each variety, we develop an operator model theory and dilation theory for large classes of varieties in noncommutative polydomains. This includes various commutative cases which are closely connected to the theory of holomorphic functions in several complex variables and algebraic geometry.