zbMATH — the first resource for mathematics

Introverted subspaces of the duals of measure algebras. (English) Zbl 1398.43001
Summary: 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 $$\mathrm{C}^*$$-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 $$GL_0(\mathcal{G})^\bot$$. 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.
MSC:
 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 Linear operators defined by compactness properties
Full Text:
References:
 [1] C.A. Akemann, Some mapping properties of the group algebras of a compact group, Pacific J. Math. 22 (1967), 1–8. · Zbl 0158.14205 [2] J.W. Conway, A course in functinal analysis, Springer Sci. Bus. Media 96 (2013). [3] H.G. Dales and A.T.-M. Lau, The second duals of Beurling algebras, Mem. Amer. Math. Soc. 177 (2005), 1–191. · Zbl 1075.43003 [4] H.G. Dales, A.T.-M. Lau and D. Strauss, Second duals of measure algebras, Disser. Math. 481 (2012), 1–121. [5] R.E. Edwards, Functional analysis, Holt, Rinehart and Winston, New York, 1965. · Zbl 0182.16101 [6] G.H. Esslamzadeh, H. Javanshiri and R. Nasr-Isfahani, Locally convex algebras which determine a locally compact group, Stud. Math. 233 (2016), 197–207. · Zbl 1360.43001 [7] 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. · Zbl 0832.22007 [8] 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. · Zbl 0881.43004 [9] E. Hewitt and K. Ross, Abstract harmonic analysis, I, Springer, Berlin, 1970. · Zbl 0213.40103 [10] 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. · Zbl 1271.43001 [11] 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). [12] 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. · Zbl 0667.43004 [13] V. Losert, Weakly compact multipliers on group algebras, J. Funct. Anal. 213 (2004), 466–472. · Zbl 1069.43001 [14] V. Losert, M. Neufang, J. Pachl and J. Steprāns, Proof of the Ghahramani-Lau conjecture, Adv. Math. 290 (2016), 709–738. · Zbl 1336.43002 [15] G.J. Murphy, C$$^*$$-algebras and operator theory, Academic Press, London 1990. [16] 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). [17] 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. · Zbl 0345.43001 [18] ——–, Convolution and separate continuity, Pacific J. Math. 75 (1978), 601–611. · Zbl 0419.43003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.