In 1996, Harris and Kadison posed the following problem: show that a linear bijection between C∗-algebras that preserves the identity and the set of invertible elements is a Jordan isomorphism. In this paper, we show that if A and B are semisimple Banach algebras andΦ : A→ B is a linear map onto B that preserves the spectrum of elements, thenΦ is a Jordan isomorphism if either A or B is a C∗-al...

We give a complete description of degenerations $3$-dimensional nilpotent algebras, $4$-dimensional commutative algebras and $5$-dimensional anticommutative over $ \mathbb C$. In particular, we correct several mistakes from the paper `Contractions low-dimensional Jordan algebras' by Ancochea Berm\'{u}dez, Fres\'{a}n Margalef Bentabol.

In this article, it is proved that under some conditions every bijective Jordan triple product homomorphism from generalized matrix algebras onto rings is additive. As a corollary, we obtain that every bijective Jordan triple product homomorphism from Mn(A) (A is not necessarily a prime algebra) onto an arbitrary ring R is additive.