JournalsjncgVol. 12, No. 4pp. 1503–1529

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

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Abstract

We construct a noncommutative, separably represented, type I and approximately finite dimensional CC^*-algebra such that its multiplier algebra is equal to its unitization. This algebra is an essential extension of the algebra K(2(c))\mathcal K(\ell_2(\mathfrak{c})) of compact operators on a nonseparable Hilbert space by the algebra K(2)\mathcal K(\ell_2) of compact operators on a separable Hilbert space, where c\mathfrak{c} denotes the cardinality of continuum. Although both K(2(c))\mathcal K(\ell_2(\mathfrak{c})) and K(2)\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 CC^*-algebra by K(2)\mathcal K(\ell_2) does not have to be stable. Our construction can be considered as a noncommutative version of Mrówka’s Ψ\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 N\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