## Illinois Journal of Mathematics

### Involutions and trivolutions in algebras related to second duals of group algebras

#### Abstract

We define a trivolution on a complex algebra $A$ as a non-zero conjugate-linear, anti-homomorphism $\tau$ on $A$, which is a generalized inverse of itself, that is, $\tau^{3}=\tau$. We obtain characterizations of trivolutions and show with examples that they appear naturally on many Banach algebras, particularly those arising from group algebras. We give several results on the existence or non-existence of involutions on the dual of a topologically introverted space. We investigate conditions under which the dual of a topologically introverted space admits trivolutions.

#### Article information

Source
Illinois J. Math., Volume 57, Number 3 (2013), 755-773.

Dates
First available in Project Euclid: 3 November 2014

https://projecteuclid.org/euclid.ijm/1415023509

Digital Object Identifier
doi:10.1215/ijm/1415023509

Mathematical Reviews number (MathSciNet)
MR3275737

Zentralblatt MATH identifier
1308.46059

#### Citation

Filali, M.; Sangani Monfared, M.; Singh, Ajit Iqbal. Involutions and trivolutions in algebras related to second duals of group algebras. Illinois J. Math. 57 (2013), no. 3, 755--773. doi:10.1215/ijm/1415023509. https://projecteuclid.org/euclid.ijm/1415023509

#### References

• R. Arens, The adjoint of a bilinear operation, Proc. Amer. Math. Soc. 2 (1951), 839–848.
• R. Arens, Operations induced in function classes, Monatsh. Math. 55 (1951), 1–19.
• J. Baker, A. T.-M. Lau and J. Pym, Module homomorphisms and topological centres associated with weakly sequentially complete Banach algebras, J. Funct. Anal. 158 (1998), 186–208.
• T. Budak, N. Iş\ik and J. Pym, Minimal determinants of topological centres for some algebras associated with locally compact groups, Bull. Lond. Math. Soc. 43 (2011), 495–506.
• C. Chou, On topologically invariant means on a locally compact group, Trans. Amer. Math. Soc. 151 (1970), 443–456.
• C. Chou, The exact cardinality of the set of invariant means on a group, Proc. Amer. Math. Soc. 55 (1976), 103–106.
• C. Chou, Topological invariant means on the von Neumann algebra ${\mathit{VN}(G)}$, Trans. Amer. Math. Soc. 273 (1982), 207–229.
• P. Civin and B. Yood, The second conjugate space of a Banach algebra as an algebra, Pacific J. Math. 11 (1961), 847–870.
• H. G. Dales, Banach algebras and automatic continuity, Oxford Univ. Press, Oxford, 2000.
• H. G. Dales and A. T.-M. Lau, The second duals of Beurling algebras, Mem. Amer. Math. Soc. 836 (2005), 1–191.
• H. G. Dales, A. T.-M. Lau and D. Strauss, Banach algebras on semigroups and their compactifications, Mem. Amer. Math. Soc. 205 (2010), 1–165.
• J. W. Degen, Some aspects and examples of infinity notions, Math. Logic Quart. 40 (1994), 111–124.
• A. Derighetti, M. Filali and M. Sangani Monfared, On the ideal structure of some Banach algebras related to convolution operators on ${L}^p(G)$, J. Funct. Anal. 215 (2004), 341–365.
• P. Eymard, L'algèbre de Fourier d'un groupe localement compact, Bull. Soc. Math. France 92 (1964), 181–236.
• H. Farhadi and F. Ghahramani, Involutions on the second duals of group algebras and a multiplier problem, Proc. Edinb. Math. Soc. (2) 50 (2007), 153–161.
• M. Filali, Finite-dimensional left ideals in some algebras associated with a locally compact group, Proc. Amer. Math. Soc. 127 (1999), 2325–2333.
• M. Filali, M. Neufang and M. Sangani Monfared, On ideals in the bidual of the Fourier algebra and related algebras, J. Funct. Anal. 258 (2010), 3117–3133.
• M. Filali and J. S. Pym, Right cancellation in $\mathit{LUC}$-compactification of a locally compact group, Bull. Lond. Math. Soc. 35 (2003), 128–134.
• M. Filali and P. Salmi, One-sided ideals and right cancellation in the second dual of the group algebra and similar algebras, J. Lond. Math. Soc. (2) 75 (2007), 35–46.
• E. E. Granirer, On some spaces of linear functionals on the algebra ${A}_p(G)$ for locally compact groups, Colloq. Math. 52 (1987), 119–132.
• E. E. Granirer, On the set of topologically invariant means on an algebra of convolution operators on ${L}^p(G)$, Proc. Amer. Math. Soc. 124 (1996), 3399–3406.
• M. Gromov, Endomorphisms of symbolic algebraic varieties, J. Eur. Math. Soc. (JEMS) 1 (1999), 109–197.
• M. Grosser, Algebra involutions on the bidual of a Banach algebra, Manuscripta Math. 48 (1984), 291–295.
• M. Grosser and V. Losert, The norm-strict bidual of a Banach algebra and the dual of $C_u (G)$, Manuscripta Math. 45 (1984), 127–146.
• C. Herz, Harmonic synthesis for subgroups, Ann. Inst. Fourier (Grenoble) 23 (1973), 91–123.
• Z. Hu, On the set of topologically invariant means on the von Neumann algbera ${\mathit{VN}}(G)$, Illinois J. Math. 39 (1995), 463–490.
• A. T.-M. Lau, Uniformly continuous functionals on the Fourier algebra of any locally compact group, Trans. Amer. Math. Soc. 251 (1979), 39–59.
• A. T.-M. Lau, Uniformly continuous functionals on Banach algebras, Colloq. Math. 51 (1987), 195–205.
• A. T.-M. Lau and A. L. T. Paterson, The exact cardinality of the set of topological left invariant means on an amenable locally compact group, Proc. Amer. Math. Soc. 98 (1986), 75–80.
• A. T.-M. Lau and J. Pym, Concerning the second dual of the group algebra of a locally compact group, J. Lond. Math. Soc. (2) 41 (1990), 445–460.
• A. T.-M. Lau and W. Takahashi, Invariant submeans and semigroups of nonexpansive mappings on Banach spaces with normal structure, J. Funct. Anal. 142 (1996), 79–88.
• A. T.-M. Lau and W. Takahashi, Nonlinear submeans on semigroups, Topol. Methods Nonlinear Anal. 22 (2003), 345–353.
• M. Neufang, Solution to Farhadi–Ghahramani's multiplier problem, Proc. Amer. Math. Soc. 138 (2010), 553–555.
• T. W. Palmer, Banach algebras and the general theory of $*$-algebras, vol. 1, Cambridge Univ. Press, Cambridge, 1994.
• T. W. Palmer, Banach algebras and the general theory of $*$-algebras, vol. 2, Cambridge Univ. Press, Cambridge, 2001.
• A. I. Singh, $L_0^{\infty}(G)^{\ast}$ as the second dual of the group algebra $L^1(G)$ with a locally convex topology, Michigan Math. J. 46 (1999), 143–150.
• A. I. Singh, Involutions on the second duals of group algebras versus subamenable groups, Studia Math. 206 (2011), 51–62.
• N. Spronk and R. Stokke, Matrix coefficients of unitary representations and associated compactifications, Indiana Univ. Math. J. 62 (2013), 99–148.
• N. J. Young, The irregularity of multiplication in group algebras, Quart. J. Math. 24 (1973), 59–62.