In this article we extended the contraction principle to the complete ℂ^* -algebra-valued semimetric spaces that fulfill an extra regularity property. Next, we study some properties of quasisymmetric maps in ℂ^* -algebra-valued regular semimetric spaces. Using very specific hypotheses we prove that every quasisymmetric map of a ℂ^* -algebra-valued regular semimetric space admits at least one fixed point.
Using Fourier-analytic methods, we derive a Hardy-type oscillatory identity for the family of planar domains {x∈ℝ^2| |x_1|^p + |x_2|^p ≤ r^p} with positive p satisfying (2/p) ∈ℕ . This identity generalizes the classical Hardy’s identity for the Euclidean disk, arising from the Poisson summation formula. For these domains of non-smooth curves (including the astroid), the rotational symmetry of the Fourier transform is lost, so it cannot be expressed using classical Bessel functions. To address this, we introduce a family of generalized Bessel functions naturally arising from the Fourier transform of p-radial functions. These functions capture the oscillatory structure induced by the loss of rotational symmetry and provide the analytic framework for Hardy-type identities between these domains. While classical motivations relate to estimating lattice point errors in number theory, we focus here on deriving the identities themselves and understanding their structure.
Let ℰ be the set of positive odd numbers that cannot be represented as the sum of a prime and a power of two. In 1950, Erdős proved that ℰ contains an infinite arithmetic progression. In this paper, it is proved that if { mh+a: h=0,1,…}⊆ℰ , then m≥ 11184810 and m has at least seven distinct prime divisors. Furthermore, these lower bounds are best possible.
Al-Rawashdeh studied the induced map between the projections and he proved that it is an orthoisomorphism for a large class of C^* -algebras. In this paper, we extend the results of Al-Rawashdeh by replacing 𝒰(𝒜) with GL(𝒜), and 𝒫(𝒜) with ℐ(𝒜) . We prove that the induced map of idempotents is an orthoisomorphism for large class of simple, unital C^* -algebras of real rank zero with cancellation and having 2-divisible K_0 -groups, Cuntz algebras, and for simple, unital purely infinite C^* -algebras having 2-divisible K_0 -groups.
