Wavelet and Linear Algebra

Abstract Let $\mathcal{A}$ and $\mathcal{B}$
be unital prime algebras and $\mathcal{A}$ contains a non-trivial idempotent $P_1$.
We consider a bijective map $\phi: \mathcal{A} \rightarrow \mathcal{B}$ which satisfies
\begin{equation*}
\phi (A.BoA)= \phi (A). \phi(B)o \phi(A)
\end{equation*}
for every element $A,B\in \mathcal{A}$, where '.' is a usual product and "$\circ$" is a Jordan product.

Abstract Let $\mathcal{B(X)}$ be the algebra of all bounded linear operators on a complex Banach space $\mathcal{X}$. In this paper, we determine the form of a surjective additive map $\phi: \mathcal{B(X)} \rightarrow \mathcal{B(X)}$ preserving the fixed points of Jordan products of operators, i.e., $F(AoB) \subseteq F(\phi(A) o\phi(B))$, for every $A,B \in \mathcal{B(X)}$, where $AoB=AB+BA$, and $F(A)$ denotes the set of all fixed points of operator $A$.