For the Fr'{e}chet algebras $(A, (p_k))$ and $(B, (q_k))$ and $n in mathbb{N}$, $ngeq 2$, a linear map $T:A rightarrow B$ is called textit{almost $n$-multiplicative}, with respect to $(p_k)$ and $(q_k)$, if there exists $varepsilongeq 0$ such that$$q_k(Ta_1a_2cdots a_n-Ta_1Ta_2cdots Ta_n)leq varepsilon p_k(a_1) p_k(a_2)cdots p_k(a_n),$$for each $kin mathbb{N}$ and $a_1, a_2, ldots, a_nin A$. Th...

let $mathcal{a}$ be a banach algebra with bai and $e$ be an introverted subspace of $mathcal{a'}$.in this paper we study the quotient arens regularity of $mathcal{a}$ with respect to $e$ and prove that the group algebra $l^1(g)$ for a locally compact group $g$, is quotient arens regular with respect to certain introverted subspace $e$ of $l^infty(g)$.some related result are given as well.

There is a family of questions in relativized complexity theory|weak analogs of the Friedberg Jump-Inversion Theorem|that are resolved by 1-generic sets but which cannot be resolved by essentially any weaker notion of genericity. This paper deenes aw2-generic sets, i.e., sets which meet every dense set of strings that is r.e. in some incomplete r.e. set. Aw2-generic sets are very close to 1-gen...

We construct a Lipschitz-free space that is locally almost square but not weakly square; this the first example of such Banach space. also prove result, which indicates geodesic metric spaces are potential characterization for spaces. Lastly, we can have symmetric strong diameter 2 property.