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

### Saeed Ghasemi

Czech Academy of Sciences, Prague, Czech Republic, and Polish Academy of Sciences, Warsaw, Poland### Piotr Koszmider

Polish Academy of Sciences, Warsaw, Poland

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## Abstract

We construct a noncommutative, separably represented, type I and approximately finite dimensional $C^*$-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 $C^*$-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 $\mathbb{N}$.

## Cite this article

Saeed Ghasemi, Piotr Koszmider, An extension of compact operators by compact operators with no nontrivial multipliers. J. Noncommut. Geom. 12 (2018), no. 4, pp. 1503–1529

DOI 10.4171/JNCG/316