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Transactions of the American Mathematical Society

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ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Unitary invariants on the unit ball of $B(\mathcal {H})^n$
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by Gelu Popescu PDF
Trans. Amer. Math. Soc. 365 (2013), 6243-6267 Request permission

Abstract:

In this paper, we introduce a unitary invariant \[ \Gamma :[B(\mathcal {H})^n]_1^-\to \mathbb {N}_\infty \times \mathbb {N}_\infty \times \mathbb {N}_\infty ,\qquad \mathbb {N}_\infty :=\mathbb {N}\cup \{ \infty \},\] 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 $\operatorname {Aut}(B(\mathcal {H})^n_1)$ 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
  • MR Author ID: 234950
  • Email: gelu.popescu@utsa.edu
  • Received by editor(s): February 6, 2012
  • Published electronically: August 13, 2013
  • Additional Notes: This research was supported in part by an NSF grant
  • © Copyright 2013 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 365 (2013), 6243-6267
  • MSC (2010): Primary 47A45, 47A13; Secondary 43A65, 47A48
  • DOI: https://doi.org/10.1090/S0002-9947-2013-05984-3
  • MathSciNet review: 3105750