On dual of group algebras under a locally convex topology

Document Type : Original Paper


1 Department of Mathematics, Semnan University, Semnan, Iran

2 Department of Mathematics, Velayat University, Iranshahr, Iran


For a locally compact group $G$, $L^1(G)$ is its group algebra and $L^\infty(G)$ is the dual of $L^1(G)$. We consider

on $L^\infty(G)$ the $\tau$-topology, i.e. the weak topology under all right multipliers induced

by measures in $L^1(G)$. For such an arbitrary $G$ the $\tau$-topology is not weaker than the

weak$^*$-topology and not stronger than the norm topology on $L^\infty(G)$. Among the other results we mention that except

for discrete $G$ the $\tau$-topology is always different from the norm-topology. The properties of $\tau$ are then studied further and we pay attention to the $\tau$-almost periodic elements of $L^\infty(G)$.


Main Subjects

  1. ع. غفاری، مقدمه ای بر آنالیز هارمونیک، انتشارات دانشگاه سمنان 1390.
  2. M. Amini, H. Nikpey and S.M. Tabatabaie, Crossed product of C∗-algebras by hypergroups, Math. Nachr. 292 (2019) 1897–1910.
    3. A. Aminpour, A. Dianatifar and R. Nasr Isfahani, Asymptotically non-expansive actions of strongly amenable semigroups and fixed points, J. Math. Anal. Appl. 461 (2018) 364–-377.
    4. J.F. Berglund, H.D. Junghenn and P. Milnes, Analysis on Semigroups: Function Spaces, Compatifications, Representations, Wiley-Interscience and Canadian Mathematics Series of Monographs and Texts, 1988.
    5. J. Baker, A.T. Lau and J. Pym, Module homomorphisms and topological centres associated with weakly sequentially complete Banach algebras, J. Funct. Anal. 158 (1998) 186-–208.
    6. H.G. Dales, Banach algebras and automatic continuity, London Mathematical Society Monographs. New Series 24, Oxford Univ. Press, 2000.
    7. M. Eshaghi, A. Ghaffari and M.B. Sahabi, Induced topologies on certain Banach algebras, Filomat 37 (2023) 1311–1318.
    8. A. Ghaffari, T. Hadadi, S. Javadi and M. Sheibani, On the structure of hypergroups with respect to the induced topology, Rocky Mountain J. Math. 52 (2022) 519–533.
  3. A. Ghaffari and S. Amirjan, Invariantly complemented and amenability in Banach algebras related to locally compact groups, Rocky Mountain J. Math. 47 (2017) 445–461.
    10. F. Ghahramani and A.T. Lau, Multipliers and Modulus on Banach algebras related to locally compact groups, J. Funct. Anal. 150 (1997) 478-–497.
    11. R.A. Kamyabi-Gol, Topological center of dual Banach algebras associated to hypergroups and invariant complemented subspaces, Ph.D. Thesis, University of Alberta, 1997.
    12. A. Medghalchi and A. Mollakhalili, Compact and weakly compact multipliers of locally compact quantum groups, Bull. Iran. Math. Soc. 44 (2018) 101–-136.
    13. A.M. Peralta, I. Villanueva, J.D.M. Wright and K. Ylinen, Weakly compact operators and the strong* topology for a Banach space, Proc. R. Soc. Edinb. A 140 (2020) 1249–1267.
    14. W. Rudin, Functional Analysis, McGraw Hill, New York, 1991.
    15. V. Runde, Amenable Banach Algebras, Springer Monographs in Mathematics, Springer Verlag, 2020.
    16. V. Runde, Lectures on Amenability, Springer-Verlag Berlin Heidelberg; 2002.
    17. J.M. Sepulcre and T. Vidal, A note on spaces of almost periodic functions with values in Banach spaces, Canad. Math. Bull. 65 (2022) 953–962.