Uniqueness of the trace and simplicity

Murphy, Gerard

doi:10.1090/S0002-9939-00-05605-7

Uniqueness of the trace and simplicity
HTML articles powered by AMS MathViewer

by Gerard J. Murphy PDF

Proc. Amer. Math. Soc. 128 (2000), 3563-3570 Request permission

Abstract:

It is proved that for certain large classes of unital C*-algebras, the existence of a unique, faithful tracial state implies simplicity. An example is given to show that this implication does not hold for all unital C*-algebras.

References

William Arveson, Improper filtrations for $C^*$-algebras: spectra of unilateral tridiagonal operators, Acta Sci. Math. (Szeged) 57 (1993), no. 1-4, 11–24. MR 1243265
Erik Bédos, Notes on hypertraces and $C^*$-algebras, J. Operator Theory 34 (1995), no. 2, 285–306. MR 1373325
Erik Bédos, Simple $C^*$-crossed products with a unique trace, Ergodic Theory Dynam. Systems 16 (1996), no. 3, 415–429. MR 1395046, DOI 10.1017/S0143385700008907
A. R. Collar, On the reciprocation of certain matrices, Proc. Roy. Soc. Edinburgh 59 (1939), 195–206. MR 8, DOI 10.1017/S0370164600012281
Joachim Cuntz and Gert Kjaergȧrd Pedersen, Equivalence and traces on $C^{\ast }$-algebras, J. Functional Analysis 33 (1979), no. 2, 135–164. MR 546503, DOI 10.1016/0022-1236(79)90108-3
George A. Elliott, Derivations of matroid $C^{\ast }$-algebras. II, Ann. of Math. (2) 100 (1974), 407–422. MR 352999, DOI 10.2307/1971079
George A. Elliott and David E. Evans, The structure of the irrational rotation $C^*$-algebra, Ann. of Math. (2) 138 (1993), no. 3, 477–501. MR 1247990, DOI 10.2307/2946553
Pratibha G. Ghatage and William J. Phillips, $C^{\ast }$-algebras generated by weighted shifts. II, Indiana Univ. Math. J. 30 (1981), no. 4, 539–546. MR 620266, DOI 10.1512/iumj.1981.30.30044
U. Haagerup and S. Thorbjornsen, Random matrices and K-theory for exact C*-algebras, Doc. Math. 4 (1999), 341–450 (electronic).
David Handelman, Homomorphisms of $C^{\ast }$ algebras to finite $AW^{\ast }$ algebras, Michigan Math. J. 28 (1981), no. 2, 229–240. MR 616272
Shinz\B{o} Kawamura, Hideo Takemoto, and Jun Tomiyama, State extensions in transformation group $C^*$-algebras, Acta Sci. Math. (Szeged) 54 (1990), no. 1-2, 191–200. MR 1073433
P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
Marc A. Rieffel, Dimension and stable rank in the $K$-theory of $C^{\ast }$-algebras, Proc. London Math. Soc. (3) 46 (1983), no. 2, 301–333. MR 693043, DOI 10.1112/plms/s3-46.2.301
Jun Tomiyama, The interplay between topological dynamics and theory of $C^*$-algebras, Lecture Notes Series, vol. 2, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1992. MR 1160781
Simon Wassermann, Exact $C^*$-algebras and related topics, Lecture Notes Series, vol. 19, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1994. MR 1271145