Gelu Popescu

Noncommutative interpolation and Poisson transforms

General results of interpolation (e.g., Nevanlinna-Pick) by elements in the noncommutative analytic Toeplitz algebraF∞ (resp., noncommutative disc algebraAn) with consequences to the interpolation by bounded operator-valued analytic functions in the unit ball of ℂn are obtained. Noncommutative Poisson transforms are used to provide new von Neumann type inequalities. Completely isometric representations of the quotient algebraF∞/J on Hilbert spaces whereJ is anyw*-closed, 2-sided ideal ofF∞, are obtained and used to construct aw*-continuous,F∞/J-functional calculus associated to row contractionsT=[T1,…,Tn] whenf(T1, …, Tn)=0 for anyf∈J. Other properties of the dual algebraF∞/J are considered.

On Bohr's inequality

Bohrs inequality says that if is a bounded analytic function on the closed unit disc, then for 0 leq r ⩽ 1/3 and that 1/3 is sharp. In this paper we give an operator-theoretic proof of Bohrs inequality that is based on von Neumanns inequality. Since our proof is operator-theoretic, our methods extend to several complex variables and to non-commutative situations. We obtain Bohr type inequalities for the algebras of bounded analytic functions and the multiplier algebras of reproducing kernel Hilbert spaces on various higher-dimensional domains, for the non-commutative disc algebra An, and for the reduced (respectively full) group C*-algebra of the free group on n generators. We also include an application to Banach algebras. We prove that every Banach algebra has an equivalent norm in which it satisfies a non-unital version of von Neumanns inequality. 2000 Mathematical Subject Classification: 47A20, 47A56.

FACTORIZATION AND REFLEXIVITY ON FOCK SPACES

AbstractThe framework of the paper is that of the full Fock space

Free holomorphic automorphisms of the unit ball of B(ℋ)n

\mathcal{F}^2 (\mathcal{H}_n )

Multivariable Bohr inequalities

and the Banach algebraF∞ which can be viewed as non-commutative analogues of the Hardy spacesH2 andH∞ respectively.An inner-outer factorization for any element in

Commutant lifting, tensor algebras, and functional calculus

\mathcal{F}^2 (\mathcal{H}_n )