TY - JOUR

T1 - The Calkin algebra is ℵ1-universal

AU - Farah, Ilijas

AU - Hirshberg, Ilan

AU - Vignati, Alessandro

N1 - Funding Information:
I. H. and A. V.'s visit to Toronto were supported by NSERC. I. H. was supported by the Israel Science Foundation, grant no. 476/16. IF's visit to CRM was supported by the Clay Mathematics Institute. A. V. is supported by a PRESTIGE co-fund Scholarship and an FWO scholarship.
Publisher Copyright:
© 2020, The Hebrew University of Jerusalem.

PY - 2020/3/1

Y1 - 2020/3/1

N2 - 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 2ℵ0-universal C*-algebra. (ii) There exists a 2ℵ0-universal C*-algebra, but Q(H) is not 2ℵ0-universal, (iii) A 2ℵ0-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.

AB - 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 2ℵ0-universal C*-algebra. (ii) There exists a 2ℵ0-universal C*-algebra, but Q(H) is not 2ℵ0-universal, (iii) A 2ℵ0-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.

UR - http://www.scopus.com/inward/record.url?scp=85084788474&partnerID=8YFLogxK

U2 - 10.1007/s11856-020-2007-y

DO - 10.1007/s11856-020-2007-y

M3 - Article

AN - SCOPUS:85084788474

VL - 237

SP - 287

EP - 309

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

SN - 0021-2172

IS - 1

ER -