Involutions and trivolutions in algebras related to second duals of group algebras (Q470856)

The authors define a trivolution on a complex algebra \(A\) to be a non-zero conjugate-linear anti-homomorphism \(\tau:A\to A\) such that \(\tau^3=\tau\). When an algebra \(A\) is equipped with the trivolution \(\tau\), the pair \((A, \tau)\) is called a trivolutive algebra. In this paper, some equivalent conditions for a non-zero map to be a trivolution are given. The authors study also homomorphisms which are at the same time also trivolutions (in the case of Banach algebras). They also consider extensions of trivolutions from a trivolutive algebra to its unitization. The last parts of the paper are devoted to the applications in the case of several group algebras and topologically introverted spaces.