Illinois Journal of Mathematics

Symmetry in tensor algebras over Hilbert space

Palle E. T. Jorgensen and Ilwoo Cho

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Abstract

This paper deals with three issues: (1) Unitary representations $U$ of a scale of (finite and infinite dimensional) non-compact Lie groups $G(H)$ built on a fixed complex Hilbert space $H$; and their covariant systems. Our computations for these representations make use of the associated Lie algebras. (2) The covariant representations involve the $C^{*}$-algebras going by the names, the Toeplitz algebras, and the Cuntz algebras. (3) An essential result which also is used throughout is our computation of the commutant of the unitary representation $U$ of $G(H)$ mentioned in (1). For a fixed Hilbert space $H$, we apportion the commutant as a specific projective limit-algebra of operators.

Article information

Source
Illinois J. Math., Volume 55, Number 3 (2011), 977-1013.

Dates
First available in Project Euclid: 29 May 2013

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1369841794

Digital Object Identifier
doi:10.1215/ijm/1369841794

Mathematical Reviews number (MathSciNet)
MR3069293

Zentralblatt MATH identifier
1268.05234

Subjects
Primary: 05E18: Group actions on combinatorial structures 11F70: Representation-theoretic methods; automorphic representations over local and global fields 37C45: Dimension theory of dynamical systems 37C80: Symmetries, equivariant dynamical systems 46L54: Free probability and free operator algebras 47L30: Abstract operator algebras on Hilbert spaces 47L90: Applications of operator algebras to physics

Citation

Jorgensen, Palle E. T.; Cho, Ilwoo. Symmetry in tensor algebras over Hilbert space. Illinois J. Math. 55 (2011), no. 3, 977--1013. doi:10.1215/ijm/1369841794. https://projecteuclid.org/euclid.ijm/1369841794


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References

  • D. Alpay, Some remarks on reproducing kernel Krein spaces, Rocky Mountain J. Math. 21 (1991), no. 4, 1189–1205.
  • W. B. Arveson, $p$-summable commutators in dimension $ d$, J. Operator Theory 54 (2005), no. 1, 101–117.
  • O. Bratteli, P. E. T. Jorgensen, A. Kishimoto and R. F. Werner, Pure states on $\mathcal{O}_{d}$, J. Operator Theory 43 (2000), no. 1, 97–143.
  • I. Cho and P. E. T. Jorgensen, $C^{*}$-algebras generated by partial isometries, J. Appl. Math. Comput. 26 (2008), 1–48.
  • I. Cho and P. E. T. Jorgensen, $C^{*}$-dynamical systems induced by partial isometries, Adv. Appl. Math. Sci. 2 (2010), 21–59.
  • I. Cho and P. E. T. Jorgensen, $C^{*}$-subalgebras generated by partial isometries, to appear in J. Math. Phys., DOI:\doiurl10.1063/1.3056588.
  • M. G. Crandall and R. S. Phillips, On the extension problem for dissipative operators, J. Funct. Anal. 2 (1968), 147–176.
  • K. R. Davidson, E. Katsoulis and D. R. Pitts, The structure of free semigroup algebras, J. Reine Angew. Math. 533 (2001), 99–125.
  • K. P. Davidson and D. R. Pitts, Nevanlinna–Pick interpolation for non-commutative analytic Toeplitz algebras, Integral Equation Operator Theory 31 (1998), no. 3, 321–337.
  • S. H. Ferguson and R. Rochberg, Higher order Hilbert–Schmidt Hankel forms and tensors of analytic kernels, Math. Scand. 96 (2005), no. 1, 117–146.
  • P. E. T. Jorgensen, Operators and representation theory, North-Holland Math. Stud., vol. 147, North-Holland, Amsterdam, 1988.
  • P. E. T. Jorgensen, D. P. Proskurin and Y. S. Samoulenko, The kernel of Fock representations of Wick algebras with braided operator of coefficients, Pacific J. Math. 198 (2001), 109–122.
  • P. E. T. Jorgensen, D. P. Proskurin and Y. S. Samoulenko, Generalized canonical commutation relations: Representations and stability of universal enveloping $C^{*}$-algebra, Pr. Inst. Mat. Nats. Akad. Nauk Ukr. Mat. Zastos. series, 43, Part 1, vol. 2, Nats\=\i onal. Akad. Nauk Ukraï ni \=Inst. Mat., Kiev, 2002, pp. 456–460.
  • P. E. T. Jorgensen, D. P. Proskurin and Y. S. Samoulenko, On $C^{*}$-algebras generated by pairs of $q$-commuting isometries, J. Phys. A 38 (2005), no. 12, 2669–2680.
  • P. E. T. Jorgensen, L. M. Schmitt and R. F. Werner, $q$-relations and stability of $C^{*}$-isomorphism classes, Algebraic methods in operator theory, Birkhäuser Boston, Boston, MA, 1994, pp. 261–271.
  • P. E. T. Jorgensen and R. F. Werner, Coherent states of the $q$-canonical commutation relations, Comm. Math. Phys. 164 (1994), no. 3, 455–471.
  • R. V. Kadison, Isometries of operator algebras, Ann. of Math. (2) 54 (1951), 325–338.
  • D. W. Kribs, The curvature invariant of a non-commuting $n$-tuple, Integral Equations Operator Theory 41 (2001), no. 4, 426–454.
  • D. W. Kribs, On bilateral weighted shifts in noncommutative multivariable operator theory, Indiana Univ. Math. J. 52 (2003), no. 6, 1595–1614.
  • G. Popescu, Free holomorphic functions and interpolation, Math. Ann. 342 (2008), no. 1, 1–30.
  • G. Popescu, Unitary invariants in multivariable operator theory, Mem. Amer. Math. Soc. 200 (2009), no. 941, vi+91.
  • G. Popescu, Noncommutative hyperbolic geometry on the unit ball of $B(H)^{n}$, J. Funct. Anal. 256 (2009), no. 12, 4030–4070.
  • F. Radulescu, Random matrices, amalgamated free products and subfactors of the von Neumann algebra of a free group, of noninteger index, Invent. Math. 115 (1994), 347–389.
  • D. Shale, Linear systems of free boson fields, Trans. Amer. Math. Soc. 103 (1962), 149–167.
  • S. J. Szarek and D. Voiculescu, Shannon's entropy power inequality via restricted Minkowski sum, Geometric aspects of functional analysis, Lect. Notes in Math., vol. 1745, Springer, Berlin, 2000, pp. 257–262.
  • D. Voiculescu, Symmetries of some reduced free product $C^{*}$-algebras, Operator algebras and their connections with topology and ergodic theory, Lect. Notes in Math., vol. 1132, Springer, Berlin, 1985, pp. 556–588.
  • D. Voiculescu, The analogues of entropy and of Fisher's information measure in free probability theory II, Invent. Math. 118 (1994), no. 3, 411–440.
  • D. Voiculescu, Free probability theory, Field Inst. Communicat., vol. 12, Amer. Math. Soc., Providence, RI, 1997.
  • D. Voiculescu, Symmetries arising from free probability theory, Frontiers in number theory, physics, and geometry, Springer, Berlin, 2006, pp. 231–243.
  • M. Wojtylak, A criterion for selfadjointness in Krein spaces, Bull. Lond. Math. Soc. 40 (2008), no. 5, 807–816.