Mehdi Monfared, University of Windsor
“Involutions and Trivolutions on Second Dual Algebras”
The involution of a C∗-algebra has a nautral extension to its second dual. This is due to the regularity of multiplication in C∗-algebras. In general, for an involutive Banach algebra, not much is known about involutions on A∗∗. Ghahramani and Farhadi have shown that if G has an infinite amenable subgroup, then L1(G)∗∗ does not have any involution extending that of L1(G). Nothing more was known if G is discrete and does not contain an infinite amenable group.
We show that the result of of Ghahramani and Farhadi is a particular case of a general phenomenon. If A∗∗ has at least two φ-topologically invariant means (where φ is non-zero character on A), then A∗∗ does not admit any involutions whose restriction to A is hermitian.
Motivated by the above results, we define a trivolution on a complex algebra A as a non- zero conjugate-linear, anti-homomorphism τ on A, which is a generalized inverse of itself, that is, τ3 = τ. We give several characterizations of trivolutions and show with examples that they appear naturally on many Banach algebras.
We show that if G is discrete, L1(G)∗∗ has a trivolution with range L1(G), extending the natural involution of L1(G).
This talk is based on joint results with M. Filali and Ajit Iqbal Singh.