Abstract
Admissible vectors for unitary representations of locally compact groups are the basis for group frame and covariant coherent state expansions. Main tools in the study of admissible vectors have been Plancherel and central integral decompositions, of applicability only under certain separability and semifiniteness restrictions. In this work, we present a study of admissible vectors in terms of convolution Hilbert algebras valid for arbitrary unitary representations of general locally compact groups.
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Notes
Given a von Neumann algebra \({\mathcal {M}}\) acting on a Hilbert space \({\mathcal {H}}\), if \(P\in {\mathcal {M}}\) is a projection, the reduced von Neumann algebra \({\mathcal {M}}_P\) is the set of all \(A\in {\mathcal {M}}\), such that \(PA=AP=A\). \({\mathcal {M}}_P\) is a von Neumann algebra on \(P{\mathcal {H}}\) and its commutant is the induced von Neumann algebra \([{\mathcal {M}}']_P\) whose elements are all the restrictions to \(P{\mathcal {H}}\) of elements of \({\mathcal {M}}'\).
Recall that for a convex cone \({\mathfrak {P}}\) in \(L^2(G)\), the dual cone \({\mathfrak {P}}^\circ \) is defined by \({\mathfrak {P}}^\circ :=\{g\in L^2(G):(f|g)\ge 0 \text { for }f\in {\mathfrak {P}}\}\). If \({\mathfrak {P}}={\mathfrak {P}}^\circ \), then \({\mathfrak {P}}\) is called self-dual.
Standard forms were introduced by Haagerup [22]. Every von Neumann algebra \({\mathcal {M}}\) can be represented on a Hilbert space \({\mathcal {H}}\) in which it is standard. In such standard form, every \(\omega \) in the predual \({\mathcal {M}}_*\) is of the form \(\omega =\omega _{\xi ,\eta }\), i.e., \(\omega (A)=\omega _{\xi ,\eta }(A):=(A\xi |\eta )_{\mathcal {H}}\), \(A\in {\mathcal {M}}\), for certain \(\xi ,\eta \in {\mathcal {H}}\); furthermore, every automorphism \(\alpha \) of \({\mathcal {M}}\) is implemented by a unique unitary operator U defined on \({\mathcal {H}}\), that is, \(\alpha (A)=UAU^*\), \(A\in {\mathcal {M}}\), such that \(UJ=JU\) and \(U{\mathfrak {P}}\subset {\mathfrak {P}}\). See also [36, Sect. IX.1].
References
Ali, S.T., Antoine, J.-P., Gazeau, J.-P.: Coherent states, wavelets, and their generalizations. Theoretical and Mathematical Physics, 2nd edn. Springer, New York (2014)
Ali, T.S., Führ, H., Krasowska, A.E.: Plancherel inversion as unified approach to wavelet transforms and Wigner functions. Ann. Henri Poincaré 4(6), 1015–1050 (2003)
Ambrose, W.: The \(L_2\)-system of a unimodular group. I. Trans. Am. Math. Soc. 65, 27–48 (1949)
Aronszajn, N.: Theory of reproducing kernels. Trans. Am. Math. Soc. 68, 337–404 (1950)
Barbieri, D., Hernández, E., Parcet, J.: Riesz and frame systems generated by unitary actions of discrete groups. Appl. Comput. Harmon. Anal. 39(3), 369–399 (2015)
Bekka, B.: Square integrable representations, von Neumann algebras and an application to Gabor analysis. J. Fourier Anal. Appl. 10(4), 325–349 (2004)
Bratteli, O., Robinson, D. W.: Operator algebras and quantum statistical mechanics. 1.\(C^*\)- and\(W^*\)-algebras, symmetry groups, decomposition of states, second ed. Texts and Monographs in Physics. Springer-Verlag, New York, (1987)
Carey, A.L.: Square-integrable representations of non-unimodular groups. Bull. Aust. Math. Soc. 15(1), 1–12 (1976)
Carey, A.L.: Induced representations, reproducing kernels and the conformal group. Comm. Math. Phys. 52(1), 77–101 (1977)
Carey, A.L.: Group representations in reproducing kernel Hilbert spaces. Rep. Math. Phys. 14(2), 247–259 (1978)
Christensen, O.: Frames and bases. Applied and numerical harmonic analysis. An introductory course. Birkhäuser Boston, Inc., Boston, MA, (2008).
Dixmier, J.: \(C^*\)-algebras. North-Holland Publishing Co., Amsterdam-New York-Oxford (1977)
Duflo, M., Moore, C.C.: On the regular representation of a nonunimodular locally compact group. J. Funct. Anal. 21(2), 209–243 (1976)
Enock, M., Schwartz, J.-M.: Kac algebras and duality of locally compact groups. Springer, Berlin (1992)
Eymard, P.: L’algèbre de Fourier d’un groupe localement compact. Bull. Soc. Math. France 92, 181–236 (1964)
Folland, G.B.: A course in abstract harmonic analysis. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL (1995)
Führ, H.: Admissible vectors and traces on the commuting algebra. In Group 24: Physical and Mathematical Aspects of Symmetries, vol. 173 of Institute of Physics Conference Series. IOP Publishing, London, UK, (2003), pp. 867–874
Führ, H.: Abstract harmonic analysis of continuous wavelet transforms. Lecture Notes in Mathematics, vol. 1863. Springer, Berlin (2005)
Gilmore, R.: Geometry of symmetrized states. Ann. Phys. 74, 391–463 (1972)
Godement, R.: Les fonctions de type positif et la théorie des groupes. Trans. Am. Math. Soc. 63, 1–84 (1948)
Grossmann, A., Morlet, J., Paul, T.: Transforms associated to square integrable group representations. I. Gen. Results. J. Math. Phys. 26(10), 2473–2479 (1985)
Haagerup, U.: The standard form of von Neumann algebras. Math. Scand. 37(2), 271–283 (1975)
Han, D., Larson, D. R.: Frames, bases and group representations. Mem. Amer. Math. Soc. 147, 697 (2000), x+94
Kadison, R. V., Ringrose, J. R.: Fundamentals of the theory of operator algebras. Vol. I. Elementary theory, vol. 100 of Pure and Applied Mathematics. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, (1983)
Kadison, R. V., Ringrose, J. R.: Fundamentals of the theory of operator algebras. Vol. II. Advanced theory, vol. 100 of Pure and Applied Mathematics. Academic Press, Inc., Orlando, FL, (1986)
Perdrizet, F.: Éléments positifs relatifs à une algèbre hilbertienne à gauche. Compos. Math. 23, 25–47 (1971)
Perelomov, A.M.: Coherent states for arbitrary Lie group. Comm. Math. Phys. 26, 222–236 (1972)
Perelomov, A.M.: Generalized coherent states and their applications. Texts and Monographs in Physics. Springer, Berlin (1986)
Phillips, J.: Positive integrable elements relative to a left Hilbert algebra. J. Funct. Anal. 13, 390–409 (1973)
Phillips, J.: A note on square-integrable representations. J. Funct. Anal. 20(1), 83–92 (1975)
Rieffel, M.A.: Square-integrable representations of Hilbert algebras. J. Functi. Anal. 3, 265–300 (1969)
Rieffel, M.A.: Integrable and proper actions on \(C^*\)-algebras, and square-integrable representations of groups. Expo. Math. 22(1), 1–53 (2004)
Segal, I. E.: An extension of Plancherel’s formula to separable unimodular groups. Ann. of Math. (2) 52, 272–292 (1950)
Stetkær, H.: Representations, convolution square roots and spherical vectors. J. Funct. Anal. 252(1), 1–33 (2007)
Takesaki, M.: Tomita’s theory of modular Hilbert algebras and its applications. Lecture Notes in Mathematics, vol. 128. Springer, Berlin-New York (1970)
Takesaki, M.: Theory of operator algebras. II, vol. 125 of Encyclopaedia of Mathematical Sciences. Springer, Berlin, (2003)
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This work was supported by research project MTM2014-57129-C2-1-P (Secretaría General de Ciencia, Tecnología e Innovación, Ministerio de Economía y Competitividad, Spain).
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Gómez-Cubillo, F., Wickramasekara, S. Admissible Vectors and Hilbert Algebras. Mediterr. J. Math. 18, 16 (2021). https://doi.org/10.1007/s00009-020-01687-0
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DOI: https://doi.org/10.1007/s00009-020-01687-0