Rocky Mountain Journal of Mathematics

Introverted subspaces of the duals of measure algebras

Hossein Javanshiri and Rasoul Nasr-Isfahani

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Let $\mathcal{G} $ be a locally compact group. In continuation of our studies on the first and second duals of measure algebras by the use of the theory of generalized functions, here we study the C$^*$-subalgebra $GL_0(\mathcal{G})$ of $GL(\mathcal{G})$ as an introverted subspace of $M(\mathcal{G} )^*$. In the case where $\mathcal{G} $ is non-compact, we show that any topological left invariant mean on $GL(\mathcal{G} )$ lies in $GL_0(\mathcal{G} )^\perp $. We then endow $GL_0(\mathcal{G} )^*$ with an Arens-type product, which contains $M(\mathcal{G} )$ as a closed subalgebra and $M_a(\mathcal{G} )$ as a closed ideal, which is a solid set with respect to absolute continuity in $GL_0(\mathcal{G} )^*$. Among other things, we prove that $\mathcal{G} $ is compact if and only if $GL_0(\mathcal{G} )^*$ has a non-zero left (weakly) completely continuous element.

Article information

Rocky Mountain J. Math., Volume 48, Number 4 (2018), 1171-1189.

First available in Project Euclid: 30 September 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 43A10: Measure algebras on groups, semigroups, etc. 43A15: $L^p$-spaces and other function spaces on groups, semigroups, etc. 43A20: $L^1$-algebras on groups, semigroups, etc. 47B07: Operators defined by compactness properties

Measure algebra generalized functions vanishing at infinity introverted subspace topological invariant mean completely continuous element


Javanshiri, Hossein; Nasr-Isfahani, Rasoul. Introverted subspaces of the duals of measure algebras. Rocky Mountain J. Math. 48 (2018), no. 4, 1171--1189. doi:10.1216/RMJ-2018-48-4-1171.

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  • C.A. Akemann, Some mapping properties of the group algebras of a compact group, Pacific J. Math. 22 (1967), 1–8.
  • J.W. Conway, A course in functinal analysis, Springer Sci. Bus. Media 96 (2013).
  • H.G. Dales and A.T.-M. Lau, The second duals of Beurling algebras, Mem. Amer. Math. Soc. 177 (2005), 1–191.
  • H.G. Dales, A.T.-M. Lau and D. Strauss, Second duals of measure algebras, Disser. Math. 481 (2012), 1–121.
  • R.E. Edwards, Functional analysis, Holt, Rinehart and Winston, New York, 1965.
  • G.H. Esslamzadeh, H. Javanshiri and R. Nasr-Isfahani, Locally convex algebras which determine a locally compact group, Stud. Math. 233 (2016), 197–207.
  • F. Ghahramani and A.T.-M. Lau, Multipliers and ideal in second conjugate algebra related to locally compact groups, J. Funct. Anal. 132 (1995), 170–191.
  • F. Ghahramani and J.P. McClure, The second dual algebra of the measure algebra of a compact group, Bull. Lond. Math. Soc. 29 (1997), 223–226.
  • E. Hewitt and K. Ross, Abstract harmonic analysis, I, Springer, Berlin, 1970.
  • H. Javanshiri and R. Nasr-Isfahani, The strong dual of measure algebras with certain locally convex topologies, Bull. Austral. Math. Soc. 87 (2013), 353–365.
  • A.T.-M. Lau, Fourier and Fourier-Stieltjes algebras of a locally compact group and amenability, in Topological vector spaces, algebras and related areas, Pitman Res. Notes Math. 316 (1994).
  • A.T.-M. Lau and J. Pym, Concerning the second dual of the group algebra of a locally compact group, J. Lond. Math. Soc. 41 (1990), 445–460.
  • V. Losert, Weakly compact multipliers on group algebras, J. Funct. Anal. 213 (2004), 466–472.
  • V. Losert, M. Neufang, J. Pachl and J. Steprāns, Proof of the Ghahramani-Lau conjecture, Adv. Math. 290 (2016), 709–738.
  • G.J. Murphy, C$^*$-algebras and operator theory, Academic Press, London 1990.
  • Yu.A. Šreĭder, The structure of maximal ideals in rings of measures with convolution, Math. Sbor. 27 (1950), 297–318 (in Russian); Math. Soc. Transl. 81 (1953), 365–391 (in English).
  • J.C. Wong, Abstract harmonic analysis of generalised functions on locally compact semigroups with applications to invariant means, J. Austral. Math. Soc. 23 (1977), 84–94.
  • ––––, Convolution and separate continuity, Pacific J. Math. 75 (1978), 601–611.