Speaker
Description
The Cauchy functional equation plays a fundamental role in the study of additive and linear structures arising from numerical and norm-related information in functional analysis. In this talk, we investigate preserver problems on positive cones of commutative C$^{*}$-algebras, where a norm identity of Fischer--Muszély type, arising from the Cauchy functional equation, determines the underlying algebraic and geometric structure.
Let $A_i$ $(i=1,2)$ be commutative C$^{*}$-algebras, and let $A_i^{+}$ denote their positive cones. We consider surjective mappings $T:A_1^{+}\to A_2^{+}$, not assumed to be continuous, satisfying the norm identity
$$
\|T(a+b)\|=\|T(a)+T(b)\| \qquad (a,b\in A_1^{+}).
$$ We show that every such mapping is necessarily additive and positive homogeneous. Moreover, in the unital case, if $T$ is injective, then it can be normalized to a composition operator, which in turn induces an isometric isomorphism between the underlying commutative C*-algebras $A_1$ and $A_2$.
These results demonstrate that norm identities of Fischer--Muszély type on positive cones contain rich numerical and order-theoretic information, which completely determines the structure of commutative $\text{C}^*$-algebras.