Rings of frame maps from $\mathcal{P}(\mathbb{R})$ to frames which vanish at infinity

Authors : Ali Akbar ESTAJİ, Ahmad Mahmoudi DARGHADAM

Pages : 854-868

Doi:10.15672/hujms.624015

View : 16 | Download : 5

Publication Date : 2020-04-02

Article Type : Research Paper

Abstract :Let $\mathcal F_{\mathcal{P}}insert ignore into journalissuearticles values( L);$ be the set of all frame maps from $\mathcal Pinsert ignore into journalissuearticles values(\mathbb R);$ to $L$, which is an $f$-ring. In this paper, we introduce the subrings $\mathcal F_{{\mathcal{P}}_{\infty}}insert ignore into journalissuearticles values( L);$ of all frame maps from $\mathcal Pinsert ignore into journalissuearticles values(\mathbb R);$ to $L$ which vanish at infinity and $\mathcal F_{{\mathcal{P}}_{K}}insert ignore into journalissuearticles values( L);$ of all frame maps from $\mathcal Pinsert ignore into journalissuearticles values(\mathbb R);$ to $L$ with compact support. We prove $\mathcal F_{{\mathcal{P}}_{\infty}}insert ignore into journalissuearticles values( L);$ is a subring of $\mathcal F_{\mathcal{P}}insert ignore into journalissuearticles values(L);$ that may not be an ideal of $\mathcal F_{\mathcal{P}}insert ignore into journalissuearticles values(L);$ in general and we obtain necessary and sufficient conditions for $\mathcal F_{{\mathcal{P}}_{\infty}}insert ignore into journalissuearticles values( L);$ to be an ideal of $\mathcal F_{\mathcal{P}}insert ignore into journalissuearticles values( L);$. Also, we show that $\mathcal F_{{\mathcal{P}}_{K}}insert ignore into journalissuearticles values( L);$ is an ideal of $\mathcal F_{\mathcal{P}}insert ignore into journalissuearticles values( L);$ and it is a regular ring. For $f\in\mathcal F_{\mathcal{P}}insert ignore into journalissuearticles values( L);$, we obtain a sufficient condition for $f$ to be an element of $F_{{\mathcal{P}}_{\infty}}insert ignore into journalissuearticles values( L);$ insert ignore into journalissuearticles values($\mathcal F_{{\mathcal{P}}_{K}}insert ignore into journalissuearticles values( L);$);. Next, we give necessary and sufficient conditions for a frame to be compact. We introduce $\mathcal F_{\mathcal{P}}$-pseudocompact and next we establish equivalent condition for an $\mathcal F_{\mathcal{P}}$-pseudocompact frame to be a compact frame. Finally, we study when for some frame $L$ with $\mathcal F_{{\mathcal{P}}_{\infty}} insert ignore into journalissuearticles values(L);\neqinsert ignore into journalissuearticles values(0);$, there is a locally compact frame $M$ such that $\mathcal F_{{\mathcal{P}}_{\infty}}insert ignore into journalissuearticles values( L);\cong\mathcal F_{{\mathcal{P}}_{\infty}}insert ignore into journalissuearticles values(M);$ and $\mathcal F_{{\mathcal{P}}_{K}}insert ignore into journalissuearticles values( L);\cong\mathcal F_{{\mathcal{P}}_{K}}insert ignore into journalissuearticles values(M);$.
Keywords : Frame, compact frame, locally compact frame, zero dimensional frame, vanish at infinity