This paper introduces a new concept, the quasi-range-preserving operator, and gives necessary and sufficient conditions for a linear operator to be quasi-range-preserving. As a special case Glicksberg's problem is affirmatively answered.

To study arithmetic structures of natural numbers, we introduce a notion of entropy of arithmetic functions, called anqie entropy. This entropy possesses some crucial properties common to both Shannon's and Kolmogorov's entropies. We show that all arithmetic functions with zero anqie entropy form a C*-algebra. Its maximal ideal space defines our arithmetic compactification of natural numbers, which is totally disconnected but not extremely disconnected. We also compute the $K$-groups of the space of all continuous functions on the arithmetic compactification. As an application, we show that any topological dynamical system with topological entropy $\lambda$, can be approximated by symbolic dynamical systems with entropy less than or equal to $\lambda$.

In this note, we will discuss what kind of operators between C*-algebras preserves Jordan triple products

We look for bounded periodic solutions for a parametric fractional problem involving a continuous nonlinearity with subcritical growth. By using a variant of Caffarelli and Silvestre extension method adapted to the periodic case and variational tools we prove the existence of at least three bounded periodic solutions when the parameter varies in an appropriate range.

Let X, Y be realcompact spaces or completely regular spaces consisting of X, Y -points. Let X, Y be a linear bijective map from X, Y (resp. X, Y ) onto X, Y (resp. X, Y ). We show that if X, Y preserves nonvanishing functions, that is,