Products of C*-algebras that do not embed into the Calkin algebra
Abstract
We consider the Calkin algebra $\mathcal Q(\ell _2)$, i.e., the quotient of the algebra $\mathcal B(\ell _2)$ of all bounded linear operators on the separable Hilbert space $\ell _2$ divided by the ideal $\mathcal K(\ell _2)$ of all compact operators on $\ell _2$. We show that in the Cohen model of set theory $\mathsf{ZFC}$ there is no embedding of the product $(c_0(2^\omega ))^{\mathbb N}$ of infinitely many copies of the abelian ${\rm C}^*$-algebra $c_0(2^\omega )$ into $\mathcal Q(\ell _2)$ (while $c_0(2^\omega )$ always embeds into $\mathcal Q(\ell _2)$). This enlarges the collection of the known examples due to Vaccaro and to McKenney and Vignati of abelian algebras, asymptotic sequence algebras, reduced products and coronas of stabilizations which consistently do not embed into the Calkin algebra. As in the Cohen model the rigidity of quotient structures fails in general, our methods do not rely on these rigidity phenomena as is the case of most examples mentioned above. The results should be considered in the context of the result of Farah, Hirshberg and Vignati which says that consistently all ${\rm C}^*$-algebras of density up to $2^\omega $ do embed into $\mathcal Q(\ell _2)$. In particular, the algebra $(c_0(2^\omega ))^{\mathbb N}$ consistently embeds into the Calkin algebra as well.
