Event Details
- IMAG Functional Analysis Seminar
- Title: Is it possible to identify/distinguish unital Banach algebras by their invertible elements?
- By Antonio Peralta (UGR)
- Abstract: It is natural to ask whether the multiplicative subgroup, $A^{-1},$ of invertible elements in a unital (associative) Banach algebra $A$ determine uniquely the structure of $A$. The answer depends, in general, on the structure you consider on $A^{-1}$. For example, there exist non-isomorphic unital Banach algebras $A$ and $B$ whose groups of invertible elements are topologically isomorphic. However, if we also consider the metric structure induced by the norm on $A^{-1}$, the answer is different. O. Hatori established in 2009--2011 a positive answer to this question in the case of a surjective isometry $\Delta$ from an open subgroup of the group of invertible elements in an associative unital (semisimple commutative) Banach algebra $A$ onto an open subgroup of the group of invertible elements in an associative unital Banach algebra $B$.%, showing that such a mapping induces an isometric real-linear algebra isomorphism from $A$ onto $B$. In this talk we shall try to understand the general form of a bijection preserving distances between the sets, $M^{-1}$ and $N^{-1}$, of invertible elements of two unital Jordan-Banach algebras $M$ and $N,$ respectively.
