Speaker: Ajit Iqbal Singh, Indian National Science Academy

Title: Involutions and Trivolutions in Algebra and Analysis

Abstract: The natural involutions in a group of taking inverse and in the complex plane of taking reflection in the real axis are usually combined to give an involution in the algebra of complex functions on groups thus forcing it to be conjugate linear. For finite or infinite matrices, taking inverse is replaced by taking transpose, the reflection in the diagonal, which forces it to be an anti-homomorphism as well. This is the basis of the definition of an involution in Algebra. Certain restrictions, continuity conditions and scaling are called for when we pass on to the context of Functional Analysis or Harmonic  Analysis on groups, semi-groups and hypergroups. This opens up possibilities of many new involutions. However when we go to the biduals with Arens products, the availability starts being choosy. While the situation is fine for Arens regular algebras like C*-algebras,  surprisingly so,  it is  impossible to have any involution for biduals of  infinite amenable group algebras! Just like generalised inverses help in solving systems of equations, trivolutions T spring up as conjugate linear anti-homomorphisms T that are their own generalised inverses, i.e., T.T.T=T. This expository talk will give a gentle account including recent work by the speaker and her collaborators Mahmoud Filali and Mehdi Monfared.

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