It is shown that every almost linear bijection $h : A \rightarrow B$ of a unital $C^*$-algebra $A$ onto a unital $C^*$-algebra $B$ is a $C^*$-algebra isomorphism when $h(3^n u y) = h(3^n u) h(y)$ for all unitaries $u \in A$, all $y \in A$, and all $n\in \mathbb Z$, and that almost linear continuous bijection $h : A \rightarrow B$ of a unital $C^*$-algebra $A$ of real rank zero onto a unital $C^*$-algebra $B$ is a $C^*$-algebra isomorphism when $h(3^n u y) = h(3^n u) h(y)$ for all $u \in \{ v \in A \mid v = v^*, \|v\|=1, v \text{ is invertible} \}$, all $y \in A$, and all $n\in \mathbb Z$.
Assume that $X$ and $Y$ are left normed modules over a unital $C^*$-algebra $A$. It is shown that every surjective isometry $T : X \rightarrow Y$, satisfying $T(0) =0$ and $T(ux) = u T(x)$ for all $x \in X$ and all unitaries $u \in A$, is an $A$-linear isomorphism. This is applied to investigate $C^*$-algebra isomorphisms in unital $C^*$-algebras.
Park, C., Rassias, T. (2010). Isomorphisms in unital $C^*$-algebras. International Journal of Nonlinear Analysis and Applications, 1(2), 1-10. doi: 10.22075/ijnaa.2010.62
MLA
C. Park; Th. M. Rassias. "Isomorphisms in unital $C^*$-algebras". International Journal of Nonlinear Analysis and Applications, 1, 2, 2010, 1-10. doi: 10.22075/ijnaa.2010.62
HARVARD
Park, C., Rassias, T. (2010). 'Isomorphisms in unital $C^*$-algebras', International Journal of Nonlinear Analysis and Applications, 1(2), pp. 1-10. doi: 10.22075/ijnaa.2010.62
VANCOUVER
Park, C., Rassias, T. Isomorphisms in unital $C^*$-algebras. International Journal of Nonlinear Analysis and Applications, 2010; 1(2): 1-10. doi: 10.22075/ijnaa.2010.62
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