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

We introduce a generator-counting refinement of algebrability for abelian $C^{\ast }$-algebras and related Banach algebras. Given an abelian $C^{\ast }$-algebra $A$, we define $(C^{\ast })$-genalgebrability in terms of the minimal possible cardinality of a generating set, encoded by the invariants $ge{n}_{C^{\ast }}(A)$ and $gen(A)$. Using compact multipliers and the ideal $K(A)$ of compact elements, we develop embedding results into ${\ell }_{\infty }$ whose ranges avoid $c_0$ (except for the zero vector), and we obtain a universal $({\ell }_{\infty }\setminus c_0)$-embeddability phenomenon under the assumption $K(A)=\{0\}$. As an application, we construct a ${}^{\ast }$-isomorphic copy of ${\ell }_{\infty }$ inside $({\ell }_{\infty }\setminus c_0)\cup \{0\}$ and transfer the results to Calkin-type settings such as $(B({\ell }_2)\setminus K({\ell }_2))\cup \{0\}$ and their unitizations. We also establish a generator-counting theorem for abelian $C^{\ast }$-algebras: $ge{n}_{C^{\ast }}(A)$ equals the smallest cardinal $n$ for which the spectrum $\Delta (A)$ embeds into ${\mathbb{R}}^n$, and we derive topological formulas for $ge{n}_{C^{\ast }}(A)$ in the non-finitely generated case. Finally, we provide a complete classification of the pairs $(d,\kappa )$ for which $({\ell }_{\infty }\setminus c_0)\cup \{0\}$ is $(d,\kappa )$-$(C^{\ast })$-genalgebrable, and we discuss the connection with classical algebrability.