An extension of compact operators by compact operators with no nontrivial multipliers

Abstract

We construct a noncommutative, separably represented, type I and approximately finite dimensional $C^{\ast }$-algebra such that its multiplier algebra is equal to its unitization. This algebra is an essential extension of the algebra $\mathcal{K}({\ell }_2(\mathfrak{c}))$ of compact operators on a nonseparable Hilbert space by the algebra $\mathcal{K}({\ell }_2)$ of compact operators on a separable Hilbert space, where $\mathfrak{c}$ denotes the cardinality of continuum. Although both $\mathcal{K}({\ell }_2(\mathfrak{c}))$ and $\mathcal{K}({\ell }_2)$ are stable, our algebra is not. This sheds light on the permanence properties of the stability in the nonseparable setting. Namely, unlike in the separable case, an extension of a nonseparable $C^{\ast }$-algebra by $\mathcal{K}({\ell }_2)$ does not have to be stable. Our construction can be considered as a noncommutative version of Mrówka’s $\Psi$-space; a space whose one point compactification is equal to its Cech–Stone compactification and is induced by a special uncountable family of almost disjoint subsets of $\mathbb{N}$.