• Corpus ID: 119234772

# The Calkin algebra is $\aleph_1$-universal

@article{Farah2017TheCA,
title={The Calkin algebra is \$\aleph\_1\$-universal},
author={Ilijas Farah and Ilan Hirshberg and Alessandro Vignati},
journal={arXiv: Operator Algebras},
year={2017}
}
• Published 6 July 2017
• Mathematics
• arXiv: Operator Algebras
We discuss the existence of (injectively) universal C*-algebras and prove that all C*-algebras of density character $\aleph_1$ embed into the Calkin algebra, $Q(H)$. Together with other results, this shows that each of the following assertions is relatively consistent with ZFC: (i) $Q(H)$ is a $2^{\aleph_0}$-universal C*-algebra. (ii) There exists a $2^{\aleph_0}$-universal C*-algebra, but $Q(H)$ is not $2^{\aleph_0}$-universal. (iii) A $2^{\aleph_0}$-universal C*-algebra does not exist. We…
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## References

SHOWING 1-10 OF 50 REFERENCES
Relative commutants of strongly self-absorbing C*-algebras
• Mathematics
• 2015
The relative commutant $A'\cap A^{\mathcal{U}}$ of a strongly self-absorbing algebra $A$ is indistinguishable from its ultrapower $A^{\mathcal{U}}$. This applies both to the case when $A$ is the
Logic and $\mathrm{C}^*$-algebras: set theoretical dichotomies in the theory of continuous quotients
Given a nonunital $\mathrm{C}^*$-algebra $A$ one constructs its corona algebra $\mathcal M(A)/A$. This is the noncommutative analog of the \v{C}ech-Stone remainder of a topological space. We analyze
EXACTNESS OF CUNTZ–PIMSNER C*-ALGEBRAS
• Mathematics
Proceedings of the Edinburgh Mathematical Society
• 2001
Abstract Let $H$ be a full Hilbert bimodule over a $C^*$-algebra $A$. We show that the Cuntz–Pimsner algebra associated to $H$ is exact if and only if $A$ is exact. Using this result, we give
Simple nuclear C*-algebras not isomorphic to their opposites
• Mathematics
Proceedings of the National Academy of Sciences
• 2017
It is shown that it is consistent with Zermelo–Fraenkel set theory with the axiom of choice (ZFC) that there is a simple nuclear nonseparable C∗-algebra, which is not isomorphic to its opposite algebra.
A Classification Theorem for Nuclear Purely Infinite Simple C -Algebras 1
Starting from Kirchberg's theorems announced at the operator algebra conference in Gen eve in 1994, namely O2 A = O2 for separable unital nuclear simple A and O1 A = A for separable unital nuclear
Filters in C*-Algebras
In this paper we analyze states on C*-algebras and their relationship to filter-like structures of projections and positive elements in the unit ball. After developing the basic theory we use this to
Dilations of C*-Correspondences and the Simplicity of Cuntz–Pimsner Algebras☆
We develop a dilation theory for C*-correspondences, showing that every C*-correspondence E over a C*-algebra A can be universally embedded into a Hilbert C*-bimodule XE over a C*-algebra AE such
On certain Cuntz-Pimsner algebras
Let A be a separable unital C.-algebra. Let ð : A ?? L(H) be a faithful representation of A on a separable Hilbert space H such that ð(A) ?? K(H) = {0}. We show that OE, the Cuntz- Pimsner algebra