Products of C*-algebras that do not embed into the Calkin algebra

TL;DR

This paper demonstrates, within a specific set-theoretic model, that certain infinite products of abelian C*-algebras cannot embed into the Calkin algebra, expanding understanding of the algebra's embedding limitations.

Contribution

It shows that in the Cohen model, the product of infinitely many copies of c_0(2^ω) does not embed into the Calkin algebra, contrasting with known embedding results.

Findings

No embedding of (c_0(2^ω))^N into the Calkin algebra in Cohen model

c_0(2^ω) always embeds into the Calkin algebra

Expands examples of non-embeddable abelian algebras

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 ZFC there is no embedding of the product $(c_0(2^\omega))^{\mathbb{N}}$ of infinitely many copies of the abelian 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 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.