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Transactions of the American Mathematical Society

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Unitary invariants on the unit ball of $ B(\mathcal{H})^n$

Author: Gelu Popescu
Journal: Trans. Amer. Math. Soc. 365 (2013), 6243-6267
MSC (2010): Primary 47A45, 47A13; Secondary 43A65, 47A48
Published electronically: August 13, 2013
MathSciNet review: 3105750
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Abstract: In this paper, we introduce a unitary invariant

$\displaystyle \Gamma :[B(\mathcal {H})^n]_1^-\to \mathbb{N}_\infty \times \math... ...imes \mathbb{N}_\infty ,\qquad \mathbb{N}_\infty :=\mathbb{N}\cup \{ \infty \},$

defined in terms of the characteristic function $ \Theta _T$, the noncommutative Poisson kernel $ K_T$, and the defect operator $ \Delta _T$ associated with $ T\in [B(\mathcal {H})^n]_1^-$. We show that the map $ \Gamma $ detects the pure row isometries in the closed unit ball of $ B(\mathcal {H})^n$ and completely classify them up to a unitary equivalence. We also show that $ \Gamma $ detects the pure row contractions with polynomial characteristic functions and completely noncoisometric row contractions, while the pair $ (\Gamma , \Theta _T)$ is a complete unitary invariant for these classes of row contractions.

The unitary invariant $ \Gamma $ is extracted from the theory of characteristic functions and noncommutative Poisson transforms, and from the geometric structure of row contractions with polynomial characteristic functions which are studied in this paper. As an application, we characterize the row contractions with constant characteristic function. In particular, we show that any completely noncoisometric row contraction $ T$ with constant characteristic function is homogeneous, i.e., $ T$ is unitarily equivalent to $ \varphi (T)$ for any free holomorphic automorphism $ \varphi $ of the unit ball of $ B(\mathcal {H})^n$.

Under a natural topology, we prove that the free holomorphic automorphism group $ \text {\rm Aut}(B(\mathcal {H})^n_1)$ is a metrizable, $ \sigma $-compact, locally compact group, and provide a concrete unitary projective representation of it in terms of noncommutative Poisson kernels.

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Additional Information

Gelu Popescu
Affiliation: Department of Mathematics, The University of Texas at San Antonio, San Antonio, Texas 78249

Keywords: Unitary invariant, row contraction, characteristic function, Poisson kernel, automorphism, projective representation, Fock space.
Received by editor(s): February 6, 2012
Published electronically: August 13, 2013
Additional Notes: This research was supported in part by an NSF grant
Article copyright: © Copyright 2013 American Mathematical Society

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