We show that free objects on sets do not exist in the category \({\varvec{ba}}\varvec{\ell }\) of bounded archimedean \(\ell \)-algebras. On other hand, we introduce weighted and prove }\). conclude by discussing several consequences this result.

Let $S$ be an inverse semigroup and let $E$ be its subsemigroup of idempotents. In this paper we define the $n$-th module cohomology group of Banach algebras and show that the first module cohomology group $HH^1_{ell^1(E)}(ell^1(S),ell^1(S)^{(n)})$ is zero, for every odd $ninmathbb{N}$. Next, for a Clifford semigroup $S$ we show that $HH^2_{ell^1(E)}(ell^1(S),ell^1(S)^{(n)})$ is a Banach sp...

In this article, we show that the Goldman–Iwahori metric on space of all norms a fixed vector satisfies Helly property for balls. On non-Archimedean side, deduce most classical Bruhat–Tits buildings may be endowed with natural piecewise ? ? $\ell ^\infty$ which is injective. We also prove semisimple groups over local fields act properly and cocompactly graphs. This gives another proof biautomat...

Let π $\pi$ be a cuspidal automorphic representation of GSp 4 ( A Q ) $\operatorname{GSp}_4(\mathbf {A}_\mathbf {Q})$ , whose archimedean component is holomorphic discrete series or limit representation. If not CAP endoscopic, then we show that its associated ℓ $\ell$ -adic Galois representations are irreducible and crystalline for 100 % $100\%$ primes . If, moreover, neither an induction nor s...

for $a,bin m_{nm},$ we say that $a$ is left matrix majorized (resp. left matrix submajorized) by $b$ and write $aprec_{ell}b$ (resp. $aprec_{ell s}b$), if $a=rb$ for some $ntimes n$ row stochastic (resp. row substochastic) matrix $r.$ moreover, we define the relation $sim_{ell s} $ on $m_{nm}$ as follows: $asim_{ell s} b$ if $aprec_{ell s} bprec_{ell s} a.$ this paper characterizes all linear p...

let $s$ be an inverse semigroup and let $e$ be its subsemigroup of idempotents. in this paper we define the $n$-th module cohomology group of banach algebras and show that the first module cohomology group $hh^1_{ell^1(e)}(ell^1(s),ell^1(s)^{(n)})$ is zero, for every odd $ninmathbb{n}$. next, for a clifford semigroup $s$ we show that $hh^2_{ell^1(e)}(ell^1(s),ell^1(s)^{(n)})$ is a banach space,...

Recently W. Holliday gave a choice-free construction of canonical extension boolean algebra B as the regular open subsets Alexandroff topology on poset proper filters B. We make this point-free by replacing space with free frame $$\mathcal {L}_B$$ generated bounded meet-semilattice all (ordered reverse inclusion) and prove that booleanization is Our main result generalizes approach to category ...