On subalgebras of the CAR-algebra. (English) Zbl 0912.46059

Kirchberg, Eberhard

J. Funct. Anal. 129, No. 1, 35-63 (1995).

Based on a construction of Voiculescu, B. Blackadar [J. Oper. Theory 14, 347-350 (1985; Zbl 0598.46037)] has shown that the Cuntz algebra \({\mathcal O}_2\) is a subquotient of the CAR-algebra. Because Choi found a \(C^*\)-algebra monomorphism of the regular \(C^*\)-group algebra of \(\text{PSL}_2(Z)\) into \({\mathcal O}_2\), by a theorem of Glimm, every \(C^*\)-algebra not of type \(I\) contains a non-nuclear \(C^*\)-subalgebra.
In the present paper, we improve the result of Blackadar and a theorem of Glimm, cf. Corollary 1.4. Moreover, we characterize the quotient \(C^*\)-algebras of the \(C^*\)-subalgebras of the CAR-algebra by a fairly simple functional analytic criterion, cf. Corollary 1.3. For that purpose, we need some investigations concerning general nuclear operator spaces in the sense of E. G. Effros and M.-D. Choi [Ann. Math., II. Ser. 104, 585-609 (1976; Zbl 0361.46067)] as we partly have done in [the author, this J. 129, 1-34 (1995; Zbl 0828.46052)].

MSC:

46L05

General theory of \(C^*\)-algebras

Keywords:

Cuntz algebra; subquotient of the CAR-algebra; nuclear operator spaces

Citations:

Zbl 0598.46037; Zbl 0361.46067; Zbl 0828.46052