In this note, we show that cite[Corollary 3.2]{sad} is not always true. In fact, we characterize essential left $phi$-contractibility of the group algebras in terms of compactness of its related locally compact group. Also, we show that for any compact commutative group $G$, $L^{2}(G)$ is always essentially left $phi$-contractible. We discuss the essential left $phi$-contractibility of some Fou...

We give sufficient conditions that allow contractible (resp., reflexive amenable) Banach algebras to be finite-dimensional and semisimple algebras. Moreover, we show that any contractible (resp., reflexive amenable) Banach algebra in which every maximal left ideal has a Banach space complement is indeed a direct sum of finitely many full matrix algebras. Finally, we characterize Hermitian ∗-alg...

We consider the Dirac operators with singular potentials 1 $$\begin{aligned} D_{\varvec{A},\Phi ,m,\Gamma \delta _{\Sigma }}=\mathfrak {D}_{\varvec{A},\Phi ,m}+\Gamma } \end{aligned}$$ where 2 \mathfrak ,m}= \sum \limits _{j=1}^{n} \alpha _{j}\left( -i\partial _{x_{j}}+A_{j}\right) +\alpha _{n+1}m+\Phi I_{N} is a operator on $$\mathbb {R}^{n}$$ variable magnetic and electrostatic $$\varvec{A=}(...

A unital Fréchet algebra A is called contractible if there exists an element d ∈ A⊗̂A such that πA(d) = 1 and ad = da for all a ∈ A where πA : A⊗̂A → A is the canonical Fréchet A-bimodule morphism. We give a sufficient condition for an infinite-dimensional contractible Fréchet algebra A to be a direct sum of a finite-dimensional semisimple algebra M and a contractible Fréchet algebra N without an...

Abstract Let f be a Morse function on smooth compact manifold $$M$$ M with boundary. The path component $$\mathrm {PH}^{-1}_f(D)$$ PH f - 1 ( D )</m...

in this work, we obtain the fekete-szegö inequalities for the class $p_{sigma }left( lambda ,phi right) $ of bi-univalent functions. the results presented in this paper improve the recent work of prema and keerthi [11].

In this work, we obtain the Fekete-Szegö inequalities for the class $P_{Sigma }left( lambda ,phi right) $ of bi-univalent functions. The results presented in this paper improve the recent work of Prema and Keerthi [11].

Abstract We consider quasilinear elliptic problems of the form − div ( ϕ ( ∣ ∇ u ) stretchy="false">) +<...