In this short article we answer the question posed in Ghadermazi M., Karamzadeh O.A.S., Namdari M., On the functionally countable subalgebra of $C\left(X\right)$, Rend. Sem. Mat. Univ. Padova 129 (2013), 47–69. It is shown that $C_c\left(X\right)$ is isomorphic to some ring of continuous functions if and only if ${\upsilon }_{0}X$ is functionally countable. For a strongly zero-dimensional space $X$, this is equivalent to say that $X$ is functionally countable. Hence for every $P$-space it is equivalent to pseudo-${\aleph }_0$-compactness.

Pointwise interpolation inequalities, in particular, ku(x)c(Mu(x)) 1-k/m (Mmu(x))k/m, k<m, and |Izf(x)|c (MIf(x))Re z/Re (Mf(x))1-Re z/Re , 0<Re z<Re<n, where ${\nabla }_k$ is the gradient of order $k$, $ℳ$ is the Hardy-Littlewood maximal operator, and $I_z$ is the Riesz potential of order $z$, are proved. Applications to the theory of multipliers in pairs of Sobolev spaces are given. In particular, the maximal algebra in the multiplier space $M(W_p^m\left({ℝ}^n\right)\to W_p^l\left({ℝ}^n\right))$ is described.

We study the geometrical properties of a unit vector field on a Riemannian 2-manifold, considering the field as a local imbedding of the manifold into its tangent sphere bundle with the Sasaki metric. For the case of constant curvature $K$, we give a description of the totally geodesic unit vector fields for $K=0$ and $K=1$ and prove a non-existence result for $K\ne 0,1$. We also found a family ${\xi }_{\omega }$ of vector fields on the hyperbolic 2-plane $L^2$ of curvature $-c^2$ which generate foliations on $T_1{L}^2$ with leaves of constant intrinsic...

In this article, we investigate new topological descriptions for two well-known mappings ${𝒵}_A$ and ${\Im }_A$ defined on intermediate rings $A\left(X\right)$ of $C\left(X\right)$. Using this, coincidence of each two classes of $z$-ideals, ${𝒵}_A$-ideals and ${\Im }_A$-ideals of $A\left(X\right)$ is studied. Moreover, we answer five questions concerning the mapping ${\Im }_A$ raised in [J. Sack, S. Watson, $C$ and $C^{*}$ among intermediate rings, Topology Proc. 43 (2014), 69–82].

Let C(Ω) be the algebra of all complex-valued continuous functions on a topological space Ω where C(Ω) contains unbounded functions. First it is shown that C(Ω) cannot have a Banach algebra norm. Then it is shown that, for certain Ω, C(Ω) cannot possess an (incomplete) normed algebra norm. In particular, this is so for $\Omega ={ℝ}^n$ where ℝ is the reals.

We describe the topological reflexive closure of the isometry group of the suspension of B(H).

We characterize unitary equivalence of quasi-free Hilbert modules, which complements Douglas and Misra's earlier work [New York J. Math. 11 (2005)]. We first confine our arguments to the classical setting of reproducing Hilbert spaces and then relate our result to equivalence of Hermitian vector bundles.