Analysis & PDE

  • Anal. PDE
  • Volume 9, Number 5 (2016), 1185-1234.

Free pluriharmonic functions on noncommutative polyballs

Gelu Popescu

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We study free k-pluriharmonic functions on the noncommutative regular polyball Bn, n = (n1,,nk) k , which is an analogue of the scalar polyball (n1)1 × × (nk)1. The regular polyball has a universal model S := {Si,j} consisting of left creation operators acting on the tensor product F2(Hn1) F2(Hnk) of full Fock spaces. We introduce the class Tn of k-multi-Toeplitz operators on this tensor product and prove that T n = span{AnAn} - SOT, where An is the noncommutative polyball algebra generated by S and the identity. We show that the bounded free k-pluriharmonic functions on Bn are precisely the noncommutative Berezin transforms of k-multi-Toeplitz operators. The Dirichlet extension problem on regular polyballs is also solved. It is proved that a free k-pluriharmonic function has continuous extension to the closed polyball Bn if and only if it is the noncommutative Berezin transform of a k-multi-Toeplitz operator in span{AnAn} -.

We provide a Naimark-type dilation theorem for direct products Fn1+ × × Fnk+ of unital free semigroups, and use it to obtain a structure theorem which characterizes the positive free k-pluriharmonic functions on the regular polyball with operator-valued coefficients. We define the noncommutative Berezin (resp. Poisson) transform of a completely bounded linear map on C(S), the C-algebra generated by Si,j, and give necessary and sufficient conditions for a function to be the Poisson transform of a completely bounded (resp. completely positive) map. In the last section of the paper, we obtain Herglotz–Riesz representation theorems for free holomorphic functions on regular polyballs with positive real parts, extending the classical result as well as the Korányi–Pukánszky version in scalar polydisks.

Article information

Anal. PDE, Volume 9, Number 5 (2016), 1185-1234.

Received: 3 December 2015
Revised: 29 February 2016
Accepted: 12 April 2016
First available in Project Euclid: 12 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47A13: Several-variable operator theory (spectral, Fredholm, etc.) 47A56: Functions whose values are linear operators (operator and matrix valued functions, etc., including analytic and meromorphic ones)
Secondary: 47B35: Toeplitz operators, Hankel operators, Wiener-Hopf operators [See also 45P05, 47G10 for other integral operators; see also 32A25, 32M15] 46L52: Noncommutative function spaces

noncommutative polyball Berezin transform Poisson transform Fock space multi-Toeplitz operator Naimark dilation completely bounded map pluriharmonic function free holomorphic function Herglotz–Riesz representation


Popescu, Gelu. Free pluriharmonic functions on noncommutative polyballs. Anal. PDE 9 (2016), no. 5, 1185--1234. doi:10.2140/apde.2016.9.1185.

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  • W. B. Arveson, “Subalgebras of $C\sp{\ast} $-algebras”, Acta Math. 123 (1969), 141–224.
  • J. A. Ball and V. Vinnikov, “Formal reproducing kernel Hilbert spaces: the commutative and noncommutative settings”, pp. 77–134 in Reproducing kernel spaces and applications, edited by D. Alpay, Operator Theory: Advances and Applications 143, Birkhäuser, Basel, 2003.
  • J. A. Ball, G. Marx, and V. Vinnikov, “Noncommutative reproducing kernel Hilbert spaces”, preprint, 2016.
  • F. A. Berezin, “\cyr Kovariantnye i kontravariantnye simvoly operatorov”, Izv. Akad. Nauk SSSR Ser. Mat. 36:5 (1972), 1134–1167. Translated as “Covariant and contravariant symbols of operators” in Math. USSR Izv. 6:5 (1972), 1117–1151.
  • A. Brown and P. R. Halmos, “Algebraic properties of Toeplitz operators”, J. Reine Angew. Math. 213 (1963), 89–102.
  • K. R. Davidson and D. R. Pitts, “The algebraic structure of non-commutative analytic Toeplitz algebras”, Math. Ann. 311:2 (1998), 275–303.
  • K. R. Davidson, E. Katsoulis, and D. R. Pitts, “The structure of free semigroup algebras”, J. Reine Angew. Math. 533 (2001), 99–125.
  • K. R. Davidson, J. Li, and D. R. Pitts, “Absolutely continuous representations and a Kaplansky density theorem for free semigroup algebras”, J. Funct. Anal. 224:1 (2005), 160–191.
  • R. G. Douglas, Banach algebra techniques in operator theory, 2nd ed., Graduate Texts in Mathematics 179, Springer, New York, 1998.
  • E. G. Effros and Z.-J. Ruan, Operator spaces, London Mathematical Society Monographs. New Series 23, Oxford University Press, New York, 2000.
  • G. Herglotz, “Über Potenzreien mit positivem, reelen Teil im Einheitskreis”, Ber. Verh. Sächs. Gesellsch. Wiss. Leipzig Math. Phys. Kl. 63 (1911), 501–511.
  • K. Hoffman, Banach spaces of analytic functions, Prentice-Hall, Englewood Cliffs, NJ, 1962.
  • M. Kennedy, “Wandering vectors and the reflexivity of free semigroup algebras”, J. Reine Angew. Math. 653 (2011), 47–73.
  • M. Kennedy, “The structure of an isometric tuple”, Proc. Lond. Math. Soc. $(3)$ 106:5 (2013), 1157–1177.
  • A. Korányi and L. Pukánszky, “Holomorphic functions with positive real part on polycylinders”, Trans. Amer. Math. Soc. 108 (1963), 449–456.
  • J. von Neumann, “Eine Spektraltheorie für allgemeine Operatoren eines unitären Raumes”, Math. Nachr. 4 (1951), 258–281.
  • M. A. Neumark, “\cyr Polozhitelp1no-opredelennye operatornye funktsii na kommutativno\u i gruppe”, Izv. Akad. Nauk SSSR Ser. Mat. 7:5 (1943), 237–244.
  • V. I. Paulsen, Completely bounded maps and dilations, Pitman Research Notes in Mathematics Series 146, Longman Scientific & Technical, Harlow, 1986.
  • G. Pisier, Similarity problems and completely bounded maps, 2nd ed., Lecture Notes in Mathematics 1618, Springer, Berlin, 2001.
  • G. Popescu, “Multi-analytic operators and some factorization theorems”, Indiana Univ. Math. J. 38:3 (1989), 693–710.
  • G. Popescu, “Multi-analytic operators on Fock spaces”, Math. Ann. 303:1 (1995), 31–46.
  • G. Popescu, “Poisson transforms on some $C\sp *$-algebras generated by isometries”, J. Funct. Anal. 161:1 (1999), 27–61.
  • G. Popescu, Entropy and multivariable interpolation, Memoirs of the American Mathematical Society 868, American Mathematical Society, Providence, RI, 2006.
  • G. Popescu, “Noncommutative transforms and free pluriharmonic functions”, Adv. Math. 220:3 (2009), 831–893.
  • G. Popescu, “Berezin transforms on noncommutative varieties in polydomains”, J. Funct. Anal. 265:10 (2013), 2500–2552.
  • G. Popescu, “Euler characteristic on noncommutative polyballs”, J. Reine Angew. Math. (online publication December 2014).
  • G. Popescu, “Curvature invariant on noncommutative polyballs”, Adv. Math. 279 (2015), 104–158.
  • G. Popescu, “Holomorphic automorphisms of noncommutative polyballs”, preprint, 2015. To appear in J. Operator Theory.
  • G. Popescu, “Berezin transforms on noncommutative polydomains”, Trans. Amer. Math. Soc. 368:6 (2016), 4357–4416.
  • G. Popescu, “Hyperbolic geometry of noncommutative polyballs”, in preparation.
  • F. Riesz, “Sur certains systèmes singuliers d'équations intégrales”, Ann. Sci. École Norm. Sup. $(3)$ 28 (1911), 33–62.
  • W. Rudin, Function theory in polydiscs, W. A. Benjamin, New York, 1969.
  • J. Schur, “Über Potenzreihen, die im Innern des Einheitskreises beschränkt sind”, J. Reine Angew. Math. 148 (1918), 122–145.
  • W. F. Stinespring, “Positive functions on $C\sp *$-algebras”, Proc. Amer. Math. Soc. 6 (1955), 211–216.
  • B. Sz.-Nagy, C. Foias, H. Bercovici, and L. Kérchy, Harmonic analysis of operators on Hilbert space, 2nd ed., Springer, New York, 2010.
  • G. Wittstock, “Ein operatorwertiger Hahn–Banach Satz”, J. Funct. Anal. 40:2 (1981), 127–150.
  • G. Wittstock, “On matrix order and convexity”, pp. 175–188 in Functional analysis: surveys and recent results, III (Paderborn, 1983), edited by K. D. Bierstedt and B. Fuchssteiner, North-Holland Mathematics Studies 90, North-Holland, Amsterdam, 1984.