Glasnik Matematicki, Vol. 59, No. 1 (2024), 171-191. \( \)

ON THE VALIDITY OF THE CAUCHY–SCHWARZ INEQUALITY FOR THE BRACKET MAP

Ivana Slamić

Faculty of Mathematics, University of Rijeka, 51 000 Rijeka, Croatia
e-mail:islamic@math.uniri.hr

Abstract.   Due to its properties, the bracket map associated with a dual integrable unitary representation of a locally compact group can be viewed as a certain operator-valued inner product; however, in the non-commutative setting, the Cauchy–Schwarz property for bracket is no longer present in its full strength. In this paper, we show that fulfillment of the property, even in weaker forms, has strong consequences on the underlying group \(G\) and the corresponding von Neumann algebra \(\mbox{VN}(G)\). In particular, we show that for unimodular group \(G\), positive elements of the \(L^1\) space over \(\mbox{VN}(G)\) which are affiliated with the commutant of \(\mbox{VN}(G)\) are precisely those for which the weaker variant of the inequality is fulfilled, and that the validity of the Cauchy–Schwarz property for the appropriate set of elements indicates the existence of a closed abelian subgroup or abelian von Neumann subalgebra of \(\mbox{VN}(G)\).

2020 Mathematics Subject Classification.   43A65, 43A15

Key words and phrases.   Dual integrable representation, bracket map, Cauchy–Schwarz inequality

Full text (PDF) (access from subscribing institutions only)

https://doi.org/10.3336/gm.59.1.08

References:

  1. D. Barbieri, E. Hernández and A. Mayeli, Bracket map for Heisenberg group and the characterization of cyclic subspaces, Appl. Comput. Harmon. Anal. 37 (2014), 218–234.
    MathSciNet    CrossRef

  2. D. Barbieri, E. Hernández and J. Parcet, Riesz and frame systems generated by unitary actions of discrete groups, Appl. Comput. Harmon. Anal. 39 (2015), 369–399.
    MathSciNet    CrossRef

  3. D. Barbieri, E. Hernández and V. Paternostro, The Zak transform and the structure of spaces invariant by the action of an LCA group, J. Funct. Anal. 269 (2015), 1327–1358.
    MathSciNet    CrossRef

  4. D. Barbieri, E. Hernández and V. Paternostro, Spaces invariant under unitary representations of discrete groups, J. Math. Anal. Appl. 492 (2020), 124357, 32 pp.
    MathSciNet    CrossRef

  5. B. Bekka and P. Harpe, Unitary representations of groups, duals, and characters, American Mathematical Society, Providence, RI, 2020.
    MathSciNet

  6. S. J. Bernau, The square root of a positive self-adjoint operator, J. Austral. Math. Soc. 8 (1968), 17–36.
    MathSciNet    CrossRef

  7. A. Connes, On the spatial theory of von Neumann algebras, J. Functional Analysis 35 (1980), 153–164.
    MathSciNet    CrossRef

  8. T. Furuta, On the polar decomposition of an operator, Acta Sci. Math. 46 (1983), 261–268.
    MathSciNet

  9. M. Hilsum, Les espaces \(L^p\) d'une algèbre de von Neumann définies par la derivée spatiale, J. Functional Analysis 40 (1981), 151–169.
    MathSciNet    CrossRef

  10. E. Hernández, P. M. Luthy, H. Šikić, F. Soria and E. N. Wilson, Spaces generated by orbits of unitary representations: a tribute to Guido Weiss, J. Geom. Anal. 31 (2021), 8735–8761.
    MathSciNet    CrossRef

  11. E. Hernández, H. Šikić, G. L. Weiss and E. N. Wilson, Cyclic subspaces for unitary representations of LCA groups; generalized Zak transform, Colloq. Math. 118 (2010), 313–332.
    MathSciNet    CrossRef

  12. J. W. Iverson, Subspaces of \(L^2\) invariant under translation by an abelian subgroup, J. Funct. Anal., 269 (2015), 865–913.
    MathSciNet    CrossRef

  13. J. W. Iverson, Frames generated by compact group actions, Trans. Amer. Math. Soc. 370 (2018), 509–551.
    MathSciNet    CrossRef

  14. R. V. Kadison and J. R. Ringrose, Fundamentals of the theory of operator algebras, Academic Press, New York, 1983.
    MathSciNet

  15. E. Kaniuth and A. T. -M. Lau, Fourier and Fourier-Stieltjes algebras on locally compact groups, American Mathematical Society, Providence, RI, 2018.
    MathSciNet    CrossRef

  16. R. A. Kunze, \(L^p\) Fourier transforms on locally compact unimodular groups, Trans. Amer. Math. Soc, 89 (1958), 519-540.
    MathSciNet    CrossRef

  17. P. M. Luthy, H. Šikić, F. Soria, G. L. Weiss and E. N. Wilson, One-dimensional dyadic wavelets, Mem. Amer. Math. Soc. 280 (2022), 1–152.
    MathSciNet    CrossRef

  18. E. Nelson, Notes on non-commutative integration, J. Functional Analysis 15 (1974), 103–116.
    MathSciNet    CrossRef

  19. N. W. Rickert, Convolution of \(L^2\)-functions, Colloq. Math. 19 (1968), 301–303.
    MathSciNet    CrossRef

  20. I. E. Segal, A noncommutative extension of abstract integration, Ann. Math. 57 (1953), 401–457.
    MathSciNet    CrossRef

  21. H. Šikić and I. Slamić, Dual integrable representations on locally compact groups, J. Geom. Anal. 34 (2024), paper no. 91, 52 pp.
    MathSciNet    CrossRef

  22. H. Šikić and I. Slamić, Maximal cyclic subspaces for dual integrable representations, J. Math. Anal. Appl. 511 (2022), 25 pp.
    MathSciNet    CrossRef

  23. W. F. Stinespring, Integration theorems for gages and duality for unimodular groups, Trans. Amer. Math. Soc. 90 (1959), 15–56.
    MathSciNet    CrossRef

  24. M. Terp, \(L^p\)-Fourier transformation on non-unimodular locally compact groups, Adv. Oper. Theory 2 (2017), 547–583.
    MathSciNet    CrossRef

  25. M. Terp, \(L^p\)-spaces associated with von Neumann algebras, Math. Institute, Copenhagen Univ., 1981.

  26. M. Takesaki, Theory of operator algebras II, Springer-Verlag, Berlin, 2003.
    MathSciNet    CrossRef
Glasnik Matematicki Home Page