Local automorphisms

Crist, Randall

doi:10.1090/S0002-9939-99-05282-X

Local automorphisms
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Proc. Amer. Math. Soc. 128 (2000), 1409-1414 Request permission

Abstract:

We show that any linear map on a finite dimensional CSL algebra $\mathcal {A}$ which at each point is equal to the value of some automorphism of $\mathcal {A}$ is either an automorphism or can be factored as an automorphism and the transpose of a self-adjoint summand of $\mathcal {A}$. New examples of local mappings are constructed.

References

Matej Brešar, Characterizations of derivations on some normed algebras with involution, J. Algebra 152 (1992), no. 2, 454–462. MR 1194314, DOI 10.1016/0021-8693(92)90043-L
Matej Brešar and Peter emrl, On local automorphisms and mappings that preserve idempotents, Studia Math. 113 (1995), no. 2, 101–108. MR 1318418, DOI 10.4064/sm-113-2-101-108
Matej Brešar and Peter emrl, Mappings which preserve idempotents, local automorphisms, and local derivations, Canad. J. Math. 45 (1993), no. 3, 483–496. MR 1222512, DOI 10.4153/CJM-1993-025-4
Randall L. Crist, Local derivations on operator algebras, J. Funct. Anal. 135 (1996), no. 1, 76–92. MR 1367625, DOI 10.1006/jfan.1996.0004
Kenneth R. Davidson, When locally contractive representations are completely contractive, J. Funct. Anal. 128 (1995), no. 1, 186–225. MR 1317715, DOI 10.1006/jfan.1995.1028
K. R. Davidson and S. C. Power, Isometric automorphisms and homology for nonselfadjoint operator algebras, Quart. J. Math. Oxford Ser. (2) 42 (1991), no. 167, 271–292. MR 1120989, DOI 10.1093/qmath/42.1.271
G. Frobenius, Über die Darstellung der einlichen Gruppen durch linear Substitutionen: I, Sitzungsberichte der Preuss, Acad. Wiss. zu Berlin (1897), 994-1015.
Frank Gilfeather and Robert L. Moore, Isomorphisms of certain CSL algebras, J. Funct. Anal. 67 (1986), no. 2, 264–291. MR 845200, DOI 10.1016/0022-1236(86)90039-X
De Guang Han and Shu Yun Wei, Local derivations of nest algebras, Proc. Amer. Math. Soc. 123 (1995), no. 10, 3095–3100. MR 1246521, DOI 10.1090/S0002-9939-1995-1246521-1
Ali A. Jafarian and A. R. Sourour, Spectrum-preserving linear maps, J. Funct. Anal. 66 (1986), no. 2, 255–261. MR 832991, DOI 10.1016/0022-1236(86)90073-X
Richard V. Kadison, Local derivations, J. Algebra 130 (1990), no. 2, 494–509. MR 1051316, DOI 10.1016/0021-8693(90)90095-6
David R. Larson, Reflexivity, algebraic reflexivity and linear interpolation, Amer. J. Math. 110 (1988), no. 2, 283–299. MR 935008, DOI 10.2307/2374503
David R. Larson and Ahmed R. Sourour, Local derivations and local automorphisms of ${\scr B}(X)$, Operator theory: operator algebras and applications, Part 2 (Durham, NH, 1988) Proc. Sympos. Pure Math., vol. 51, Amer. Math. Soc., Providence, RI, 1990, pp. 187–194. MR 1077437, DOI 10.1090/pspum/051.2/1077437
Stephen C. Power, Limit algebras: an introduction to subalgebras of $C^*$-algebras, Pitman Research Notes in Mathematics Series, vol. 278, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1992. MR 1204657
Peter emrl, Isomorphisms of standard operator algebras, Proc. Amer. Math. Soc. 123 (1995), no. 6, 1851–1855. MR 1242104, DOI 10.1090/S0002-9939-1995-1242104-8
Peter emrl, Local automorphisms and derivations on ${\scr B}(H)$, Proc. Amer. Math. Soc. 125 (1997), no. 9, 2677–2680. MR 1415338, DOI 10.1090/S0002-9939-97-04073-2
V. S. Shul′man, Operators preserving ideals in $C^*$-algebras, Studia Math. 109 (1994), no. 1, 67–72. MR 1267712, DOI 10.4064/sm-109-1-67-72