Unitarily-invariant linear spaces in C*-algebras

Marcoux, L.; Murphy, G.

doi:10.1090/S0002-9939-98-04934-X

Unitarily-invariant linear spaces in C*-algebras
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by L. W. Marcoux and G. J. Murphy PDF

Proc. Amer. Math. Soc. 126 (1998), 3597-3605 Request permission

Abstract:

Characterisations and containment results are given for linear subspaces of a unital C*-algebra that are invariant under conjugation by sets of unitary elements of the algebra. The (unitarily-invariant) linear span of the projections in a simple, unital C*-algebra having non-scalar projections is shown to contain all additive commutators of the algebra and, in certain cases, to be equal to the algebra.

References

Bruce E. Blackadar, A simple unital projectionless $C^{\ast }$-algebra, J. Operator Theory 5 (1981), no. 1, 63–71. MR 613047
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 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
Ryszard Engelking, Teoria wymiaru, Biblioteka Matematyczna, Tom 51. [Mathematics Library, Vol. 51], Państwowe Wydawnictwo Naukowe, Warsaw, 1977 (Polish). MR 0482696
Thierry Fack, Finite sums of commutators in $C^{\ast }$-algebras, Ann. Inst. Fourier (Grenoble) 32 (1982), no. 1, vii, 129–137 (English, with French summary). MR 658946
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
I.N. Herstein, On the Lie and Jordan rings of a simple associative ring, Amer. J. Math. 77 (1955), 279—285.
I. N. Herstein, Topics in ring theory, University of Chicago Press, Chicago, Ill.-London, 1969. MR 0271135
Laurent Marcoux, On the closed Lie ideals of certain $C^*$-algebras, Integral Equations Operator Theory 22 (1995), no. 4, 463–475. MR 1343340, DOI 10.1007/BF01203386
Gerard J. Murphy, $C^*$-algebras and operator theory, Academic Press, Inc., Boston, MA, 1990. MR 1074574
Gert K. Pedersen, The linear span of projections in simple $C^{\ast }$-algebras, J. Operator Theory 4 (1980), no. 2, 289–296. MR 595417
Klaus Thomsen, Finite sums and products of commutators in inductive limit $C^\ast$-algebras, Ann. Inst. Fourier (Grenoble) 43 (1993), no. 1, 225–249 (English, with English and French summaries). MR 1209702, DOI 10.5802/aif.1328