The Calkin algebra is ℵ₁-universal

2017/07/06 by Farah, Ilijas, Hirshberg, Ilan, Vignati, Alessandro
#03E35 #03E75 #46L05 #FOS: Mathematics #Logic (math.LO) #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.1707.01782

Abstract

We discuss the existence of (injectively) universal C*-algebras and prove that all C*-algebras of density character ℵ₁ 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^ℵ₀-universal C*-algebra. (ii) There exists a 2^ℵ₀-universal C*-algebra, but Q(H) is not 2^ℵ₀-universal. (iii) A 2^ℵ₀-universal C*-algebra does not exist. We also prove that it is relatively consistent with ZFC that (iv) there is no ℵ₁-universal nuclear C*-algebra, and that (v) there is no ℵ₁-universal simple nuclear C*-algebra.

Related

• Embedding C^*-algebras into the Calkin algebra
• The Calkin algebra has outer automorphisms
• Calkin algebra, Kazhdan's property (T), strongly self-absorbing C*-algebras
• Universal C*-algebras generated by doubly non-commuting isometries
• The Calkin algebra is not countably homogeneous