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

We introduce a generator-counting refinement of algebrability for abelian $C^*$-algebras and related Banach algebras.

Given an abelian $C^*$-algebra $A$, we define $(C^*)$-genalgebrability in terms of the minimal possible cardinality of a generating set, encoded by the invariants $gen_{C^*}(A)$ 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 $^*$-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 $C^*$-algebras: $gen_{C^*}(A)$ equals the smallest cardinal $n$ for which the spectrum $\Delta(A)$ embeds into $\mathbb R^n$, and we derive topological formulas for $gen_{C^*}(A)$ 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 $(d,\kappa)$-$(C^*)$-genalgebrable, and we discuss the connection with classical algebrability.