C  * ‐Nuclearity implies CPAP | Zendy

Kirchberg Eberhard | Zendy

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C * ‐Nuclearity implies CPAP

Author(s) -

Kirchberg Eberhard

Publication year - 1977

Publication title -

mathematische nachrichten

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 0.913

H-Index - 50

eISSN - 1522-2616

pISSN - 0025-584X

DOI - 10.1002/mana.19770760115

Subject(s) - mathematics , identity (music) , rank (graph theory) , operator (biology) , norm (philosophy) , algebra over a field , pure mathematics , operator norm , combinatorics , operator theory , chemistry , epistemology , physics , biochemistry , philosophy , repressor , acoustics , transcription factor , gene

In this paper we show: A C * ‐algebra A with identity e is C * ‐nuclear (i.e. “nuclear” in the sense of L ANCE [3, p. 159] or “property T ” in the sense of T AKESAKI [6] if and only if the identity operator id A on A can be approximated in the strong operator topology by completely positive linear operators V from A into A of finite rank with norm one and V(e)=e . From which follows that every C * ‐nuclear C * ‐algebra (possible without identity) possesses the CPAP of L ANCE [3, def. 3.5.]. This answers questions of L ANCE [3, p. 173] and S AKAI [5, p. 64].

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