Abstract
We investigate involutions and trivolutions in the second dual of algebras related to a locally compact topological semigroup and the Fourier algebra of a locally compact group. We prove, among the other things, that for a large class of topological semigroups namely, compactly cancellative foundation \(*\)-semigroup S when it is infinite non-discrete cancellative, \(M_a(S)^{**}\) does not admit an involution, and \(M_a(S)^{**}\) has a trivolution with range \(M_a(S)\) if and only if S is discrete. We also show that when G is an amenable group, the second dual of the Fourier algebra of G admits an involution extending one of the natural involutions of A(G) if and only if G is finite. However, \(A(G)^{**}\) always admits trivolution.
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References
Arens R, The adjoint of a bilinear operation, Proc. Am. Math. Soc. 2 (1951) 839–848
Arens R, Operations induced in function classes, Monatsh. Math. 55 (1951) 1–19
Baker J W, Convolution measure algebras with involution, Proc. Am. Math. Soc. 27 (1971) 91–96
Baker A C and Baker J W, Algebra of measures on locally compact semigroups III, J. Lond. Math. Soc. 4 (1972) 685–695
Baker J, Lau A T-M and Pym J, Module homomorphisms and topological centres associated with weakly sequentially complete Banach algebras, J. Funct. Anal. 158 (1998) 186–208
Bonsall F F and Duncan J, Complete normed algebras (1973) (Berlin: Springer)
Civin P and Yood B, The second conjugate space of a Banach algebra as an algebra, Pac. J. Math. 11 (1961) 847–870
Conway J B, The compact operators are not complemented in \(\cal{B}(\cal{H})\), Proc. Am. Math. Soc. 32 (1975) 549–550
Dales H G, Banach algebras and automatic continuity (2000) (Oxford: Oxford University Press)
Dzinotyiweyi H A M, The analogue of the group algebra for topological semigroup (1984) (Boston: Pitman) (London, 1984)
Eymard P, L’algèbre de Fourier d’un groupe localement compact, Bull. Soc. Math. France 92 (1964) 181–236
Farhadi H and Ghahramani F, Involutions on the second duals of group algebras and a multiplier problem, Proc. Edinb. Math. Soc. 50 (2007) 153–161
Filali M, Sangani Monfared M and Singh A I, Involutions and trivolutions in algebras related to second duals of group algebras, Ill. J. Math. 57(3) (2013) 755–773
Granirer E E, Density theorems for some linear subspaces and some \(C^*\), Sympos. Mat., vol. XXII, Istit. Nazionale di Alta Mat. (1977) 61–70
Granirer E E, On group representations whose \(C^*\)-algebra is an ideal in its von Neumann algebra, Ann. Inst. Fourier (Grenoble) 29 (1979) 37–52
Grosser M, Algebra involutions on the bidual of a Banach algebra, Manuscr. Math. 48 (1984) 291–295
Hewitt E and Ross K A, Abstract harmonic analysis, Volume II (1970) (Berlin-Heidelberg-New York: Springer)
Lashkarizadeh Bami M, Function algebras on weighted topological semigroups, Math. Japonica 48 (1998) 217–227
Lashkarizadeh Bami M, The topological centers of \({LUC}({S})^*\) of certain foundation semigroups, Glasgow Math. J. 42 (2000) 335–343
Lashkarizadeh Bami M, Mohammadzadeh B and Nasr-Isfahani R, Inner invariant extensions of Dirac measures on compactly cancellative topological semigroups, Bull. Belg. Math. Soc. 14 (2007) 699–708
Lau A T-M, Uniformly continuous functionals on the Fourier algebra of any locally compact group, Trans. Am. Math. Soc. 251 (1979) 39–59
Lau A T-M, The second conjugate algebra of the Fourier algebra of a locally compact group, Trans. Am. Math. Soc. 267 (1981) 53–63
Lau A T-M, Continuity of Arens multiplication on the dual space of bounded uniformly continuous functions on locally compact groups and topological semigroups, Math. Proc. Camb. Philos. Soc. 99 (1986) 273–283
Lau A T-M and Losert V, The \(C^*\)-algebra generated by operators with compact support on a locally compact group, J. Funct. Anal. 112 (1993) 1–30
Lau A T-M, Loy R J and Willis G A, Amenability of Banach and \(C^*\)-algebras on locally compact groups, Stud. Math. 119(2) (1996) 161–178
Lau A T-M and Pym J, Concerning the second dual of the group algebra of a locally compact group, J. Lond. Math. Soc. 41 (1990) 445–460
Lau A T-M and Ülger A, Topological centers of certain dual algebras, Trans. Am. Math. Soc. 384(3) (1996) 1191–1212
Maghsoudi S and Nasr-Isfahani R, The Arens regularity of certain Banach algebras related to compactly cancellative foundation semigroups, Bull. Belg. Math. Soc. 16 (2009) 205–221
Maghsoudi S and Nasr-Isfahani R, On the second conjugate of the weighted semigroup algebra for a locally compact semigroup, Quaest. Math. 35 (2012) 219–227
Maghsoudi S and Nasr-Isfahani R, On the topological center of a Banach algebra related to a foundation topological semigroup, Semigroup Forum 83 (2012) 33–38
Neufang M, Solution to Farhadi–Ghahramani’s multiplier problem, Proc. Am. Math. Soc. 138 (2010) 553–555
Renauld P F, Invariant means on a class of von Neumann algebras, Trans. Am. Math. Soc. 170 (1972) 285–291
Singh A I, Involutions on the second duals of group algebras versus subamenable groups, Stud. Math. 206 (2011) 51–62
Sleijpen G L G, The dual of the space of measures with continuous translations, Semigroup Forum 22 (1981) 139–150
Sleijpen G L G, Convolution measure algebras on semigroups, Ph.D. Thesis (1976) (The Netherlands: Katholike Universiteit)
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Alinejad, A., Ghaffari, A. Involutions and trivolutions on second dual of algebras related to locally compact groups and topological semigroups. Proc Math Sci 127, 689–705 (2017). https://doi.org/10.1007/s12044-017-0346-3
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DOI: https://doi.org/10.1007/s12044-017-0346-3
Keywords
- Arens multiplication
- foundation semigroup
- second dual
- compactly cancellative
- involution
- trivolution
- Fourier algebra