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

HTML articles powered by AMS MathViewer

- 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.

## References

- William Arveson,
*The curvature invariant of a Hilbert module over $\textbf {C}[z_1,\cdots ,z_d]$*, J. Reine Angew. Math.**522**(2000), 173–236. MR**1758582**, DOI 10.1515/crll.2000.037 - Douglas N. Clark and Gadadhar Misra,
*On homogeneous contractions and unitary representations of $\textrm {SU}(1,1)$*, J. Operator Theory**30**(1993), no. 1, 109–122. MR**1302610** - Bhaskar Bagchi and Gadadhar Misra,
*Constant characteristic functions and homogeneous operators*, J. Operator Theory**37**(1997), no. 1, 51–65. MR**1438200** - Ciprian Foias and Jaydeb Sarkar,
*Contractions with polynomial characteristic functions I. Geometric approach*, Trans. Amer. Math. Soc.**364**(2012), no. 8, 4127–4153. MR**2912448**, DOI 10.1090/S0002-9947-2012-05450-X - Gelu Popescu,
*Isometric dilations for infinite sequences of noncommuting operators*, Trans. Amer. Math. Soc.**316**(1989), no. 2, 523–536. MR**972704**, DOI 10.1090/S0002-9947-1989-0972704-3 - Gelu Popescu,
*Characteristic functions for infinite sequences of noncommuting operators*, J. Operator Theory**22**(1989), no. 1, 51–71. MR**1026074** - Gelu Popescu,
*von Neumann inequality for $(B({\scr H})^n)_1$*, Math. Scand.**68**(1991), no. 2, 292–304. MR**1129595**, DOI 10.7146/math.scand.a-12363 - Gelu Popescu,
*Functional calculus for noncommuting operators*, Michigan Math. J.**42**(1995), no. 2, 345–356. MR**1342494**, DOI 10.1307/mmj/1029005232 - Gelu Popescu,
*Multi-analytic operators on Fock spaces*, Math. Ann.**303**(1995), no. 1, 31–46. MR**1348353**, DOI 10.1007/BF01460977 - Gelu Popescu,
*Non-commutative disc algebras and their representations*, Proc. Amer. Math. Soc.**124**(1996), no. 7, 2137–2148. MR**1343719**, DOI 10.1090/S0002-9939-96-03514-9 - Gelu Popescu,
*Poisson transforms on some $C^*$-algebras generated by isometries*, J. Funct. Anal.**161**(1999), no. 1, 27–61. MR**1670202**, DOI 10.1006/jfan.1998.3346 - Gelu Popescu,
*Curvature invariant for Hilbert modules over free semigroup algebras*, Adv. Math.**158**(2001), no. 2, 264–309. MR**1822685**, DOI 10.1006/aima.2000.1972 - Gelu Popescu,
*Entropy and multivariable interpolation*, Mem. Amer. Math. Soc.**184**(2006), no. 868, vi+83. MR**2263661**, DOI 10.1090/memo/0868 - Gelu Popescu,
*Operator theory on noncommutative varieties*, Indiana Univ. Math. J.**55**(2006), no. 2, 389–442. MR**2225440**, DOI 10.1512/iumj.2006.55.2771 - Gelu Popescu,
*Free holomorphic functions on the unit ball of $B(\scr H)^n$*, J. Funct. Anal.**241**(2006), no. 1, 268–333. MR**2264252**, DOI 10.1016/j.jfa.2006.07.004 - Gelu Popescu,
*Unitary invariants in multivariable operator theory*, Mem. Amer. Math. Soc.**200**(2009), no. 941, vi+91. MR**2519137**, DOI 10.1090/memo/0941 - Gelu Popescu,
*Free holomorphic automorphisms of the unit ball of $B(\scr H)^n$*, J. Reine Angew. Math.**638**(2010), 119–168. MR**2595338**, DOI 10.1515/CRELLE.2010.005 - Béla Sz.-Nagy, Ciprian Foias, Hari Bercovici, and László Kérchy,
*Harmonic analysis of operators on Hilbert space*, Revised and enlarged edition, Universitext, Springer, New York, 2010. MR**2760647**, DOI 10.1007/978-1-4419-6094-8 - Dan Voiculescu,
*Symmetries of some reduced free product $C^\ast$-algebras*, Operator algebras and their connections with topology and ergodic theory (Buşteni, 1983) Lecture Notes in Math., vol. 1132, Springer, Berlin, 1985, pp. 556–588. MR**799593**, DOI 10.1007/BFb0074909 - Johann von Neumann,
*Eine Spektraltheorie für allgemeine Operatoren eines unitären Raumes*, Math. Nachr.**4**(1951), 258–281 (German). MR**43386**, DOI 10.1002/mana.3210040124

## 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