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Type III KMS States on a Class of C*-Algebras Containing O-n and Q(N) and Their Modular Index

Chapter


Abstract


  • We construct a family of purely infinite, simple, separable, nuclear C* -algebras,

    Q\ for,\ E (0, 1). These algebras are also in the class SJlnuc and therefore by results

    of E. Kirchberg and N. C. Phillips they are classified by their K-groups. There is

    an action of the circle 1I' with a unique KMS state 'If; on each Q>.. For ,\ = 1/n,

    Q1fn ~On, with its usual 1I' action and KMS state. For,\= p/q, rational in lowest

    terms, Q>. ~ On (n = q - p + 1) with UHF fixed point algebra of type (pq) 00

    For any n > 0, Q>. ~ On for infinitely many ,\ with distinct KMS states and UHF

    fixed-point algebras. However, none of the Q>. is isomorphic to 0 00 . For,\ irrational

    the fixed point algebras are not AF and the Q>. are usually not Cuntz algebras. For

    ,\transcendental, K1 (Q>.) ~ K0 (Q>.) ~ Z00

    , so that Q>. is Cuntz' QN, [Cul]. If,\ is

    algebraic (and not rational), then K1 (Q>.) and K0 (Q>.) have the same finite rank:

    K1 0 Ql ~ Ko 0 Ql ~ Qlk. If ,\ and >..-1 are both algebraic integers, the only On

    which appear are those for which n = 3(mod4). For each>.., the representation of

    Q>. defined by the KMS state 'If; generates a type III>. factor. These algebras fit into the framework of the modular index theory/twisted cyclic theory of [CPR2, CRT]

    and [CNNR].

Publication Date


  • 2011

Citation


  • Carey, A. L., Phillips, J., Putnam, I. F. & Rennie, A. (2011). Type III KMS States on a Class of C*-Algebras Containing O-n and Q(N) and Their Modular Index. Perspectives on Noncommutative Geometry (pp. 29-40). United States: American Mathematical Society.

International Standard Book Number (isbn) 13


  • 9780821848494

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers1/863

Book Title


  • Perspectives on Noncommutative Geometry

Start Page


  • 29

End Page


  • 40

Place Of Publication


  • United States

Abstract


  • We construct a family of purely infinite, simple, separable, nuclear C* -algebras,

    Q\ for,\ E (0, 1). These algebras are also in the class SJlnuc and therefore by results

    of E. Kirchberg and N. C. Phillips they are classified by their K-groups. There is

    an action of the circle 1I' with a unique KMS state 'If; on each Q>.. For ,\ = 1/n,

    Q1fn ~On, with its usual 1I' action and KMS state. For,\= p/q, rational in lowest

    terms, Q>. ~ On (n = q - p + 1) with UHF fixed point algebra of type (pq) 00

    For any n > 0, Q>. ~ On for infinitely many ,\ with distinct KMS states and UHF

    fixed-point algebras. However, none of the Q>. is isomorphic to 0 00 . For,\ irrational

    the fixed point algebras are not AF and the Q>. are usually not Cuntz algebras. For

    ,\transcendental, K1 (Q>.) ~ K0 (Q>.) ~ Z00

    , so that Q>. is Cuntz' QN, [Cul]. If,\ is

    algebraic (and not rational), then K1 (Q>.) and K0 (Q>.) have the same finite rank:

    K1 0 Ql ~ Ko 0 Ql ~ Qlk. If ,\ and >..-1 are both algebraic integers, the only On

    which appear are those for which n = 3(mod4). For each>.., the representation of

    Q>. defined by the KMS state 'If; generates a type III>. factor. These algebras fit into the framework of the modular index theory/twisted cyclic theory of [CPR2, CRT]

    and [CNNR].

Publication Date


  • 2011

Citation


  • Carey, A. L., Phillips, J., Putnam, I. F. & Rennie, A. (2011). Type III KMS States on a Class of C*-Algebras Containing O-n and Q(N) and Their Modular Index. Perspectives on Noncommutative Geometry (pp. 29-40). United States: American Mathematical Society.

International Standard Book Number (isbn) 13


  • 9780821848494

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers1/863

Book Title


  • Perspectives on Noncommutative Geometry

Start Page


  • 29

End Page


  • 40

Place Of Publication


  • United States