Primitive Limit Algebras and C*-Envelopes

✍ Scribed by Kenneth R Davidson; Elias Katsoulis

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 252 KB
• Volume: 170
• Category: Article
• ISSN: 0001-8708
• DOI: 10.1006/aima.2001.2068

✦ Synopsis

In this paper, we study irreducible representations of regular limit subalgebras of AF-algebras. The main result is twofold: every closed prime ideal of a limit of direct sums of nest algebras (NSAF) is primitive, and every prime regular limit algebra is primitive. A key step is that the quotient of an NSAF algebra by any closed ideal has an AF C n -envelope, and this algebra is exhibited as a quotient of a concretely represented AF-algebra. When the ideal is prime, the C n -envelope is primitive. The GNS construction is used to produce algebraically irreducible (in fact n-transitive for all n51) representations for quotients of NSAF algebras by closed prime ideals. Thus the closed prime ideals of NSAF algebras coincide with the primitive ideals. Moreover, these representations extend to * -representations of the C n -envelope of the quotient, so that a fortiori these algebras are also operator primitive. The same holds true for arbitrary limit algebras and the f0g ideal. # 2002 Elsevier Science (USA)