Spectrally bounded operators on simple 𝐶*-algebras

Mathieu, Martin

doi:10.1090/S0002-9939-03-07215-0

Spectrally bounded operators on simple $C^{*}$-algebras
HTML articles powered by AMS MathViewer

by Martin Mathieu PDF

Proc. Amer. Math. Soc. 132 (2004), 443-446 Request permission

Abstract:

A linear mapping $T$ from a subspace $E$ of a Banach algebra into another Banach algebra is called spectrally bounded if there is a constant $M\geq 0$ such that $r(Tx)\leq M r(x)$ for all $x\in E$, where $r( \cdot )$ denotes the spectral radius. We prove that every spectrally bounded unital operator from a unital purely infinite simple $C^*$-algebra onto a unital semisimple Banach algebra is a Jordan epimorphism.

References

Bernard Aupetit and Martin Mathieu, The continuity of Lie homomorphisms, Studia Math. 138 (2000), no. 2, 193–199. MR 1749080
Lawrence G. Brown and Gert K. Pedersen, $C^*$-algebras of real rank zero, J. Funct. Anal. 99 (1991), no. 1, 131–149. MR 1120918, DOI 10.1016/0022-1236(91)90056-B
Joachim Cuntz, The structure of multiplication and addition in simple $C^*$-algebras, Math. Scand. 40 (1977), no. 2, 215–233. MR 500176, DOI 10.7146/math.scand.a-11691
Kenneth R. Davidson, $C^*$-algebras by example, Fields Institute Monographs, vol. 6, American Mathematical Society, Providence, RI, 1996. MR 1402012, DOI 10.1090/fim/006
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
L. W. Marcoux, On the linear span of the projections in certain simple $C^{*}$-algebras, Indiana Univ. Math. J. 51 (2002), 753–771.
M. Mathieu, Spectrally bounded traces on $C^{*}$-algebras, Bull. Austral. Math. Soc. 68 (2003), 169–173.
Martin Mathieu and Gerhard J. Schick, First results on spectrally bounded operators, Studia Math. 152 (2002), no. 2, 187–199. MR 1916549, DOI 10.4064/sm152-2-6
M. Mathieu and G. J. Schick, Spectrally bounded operators from von Neumann algebras, in press, J. Operator Theory (2003).
M. Mathieu and A. R. Sourour, Hereditary properties of spectral isometries, Archiv Math. (Basel), to appear.
Ciprian Pop, Finite sums of commutators, Proc. Amer. Math. Soc. 130 (2002), no. 10, 3039–3041. MR 1908928, DOI 10.1090/S0002-9939-02-06484-5
M. Rørdam, A simple $C^{*}$-algebra with a finite and an infinite projection, preprint.
Shuang Zhang, A property of purely infinite simple $C^*$-algebras, Proc. Amer. Math. Soc. 109 (1990), no. 3, 717–720. MR 1010004, DOI 10.1090/S0002-9939-1990-1010004-X