Applications of compact multipliers to algebrability of $(\ell_{\infty}\setminus c_0)\cup\{0\}$ and $(B(\ell_2(\mathbb{N}))\setminus K(\ell_2(\mathbb{N}))\cup \{ 0\}.$

Abstract

In present work we deal with the class $\mathcal{C}=\mathcal{C}_1\cup \mathcal{C}_2$ where $\mathcal{C}_1$ (respectively, $\mathcal{C}_2$) is formed by all separable Uniform algebras (respectively, separable commutative C$^*$-algebras) with no compact elements. For a given algebra $A$ in $\mathcal{C}_1$ (respectively, $A$ in $\mathcal{C}_2$) we show that $A$ is isometrically isomorphic as algebra (respectively, as C$^*$-algebra) to a subalgebra $M$ of $\ell_{\infty}$ with $M\subset (\ell_{\infty}\setminus c_0)\cup\{0\}.$ Under the additional assumption that $A$ is non-unital we verify that there exists a copy of $M(A)$ (the multipliers algebra of $A$ which is non-separable) inside $(\ell_{\infty}\setminus c_0)\cup\{0\}$. For an infinitely generated abelian C$^*$-algebra $B,$ we study the least cardinality possible of a system of generators ($gen_{C^*}(B)$). In fact we deduce that $gen_{C^*}(B)$ coincides with the smallest cardinal number $n$ such that an embedding of $\Delta(B)$ (= the spectrum of $B$) in $\mathbb{R}^n$ exists - The finitely generated version of this result was proved by Nagisa. In addition, we introduce new concepts of algebrability in terms of $gen_{C^*}(B)$ ($(C^*)$-genalgebrability) and its natural variations. From our methods we infer that there is $^*$-isomorphic copy of $\ell_{\infty}$ in $(\ell_{\infty}\setminus c_0)\cup\{0\}$. In particular, $(\ell_{\infty}\setminus c_0)\cup\{0\}$ contains a copy of every separable Banach space. Moreover, all the positive answers of this work holds if we replace the set $(\ell_{\infty}\setminus c_0)\cup\{0\}$ with $(B(\ell_2(\mathbb{N}))\setminus K(\ell_2(\mathbb{N}))\cup \{ 0\}.$