A reflexivity problem concerning the 𝐶*-algebra 𝐶(𝑋)⊗ℬ(ℋ)

Molnár, Lajos

doi:10.1090/S0002-9939-00-05604-5

A reflexivity problem concerning the $C^*$-algebra $C(X)\otimes \mathscr {B}(\mathscr {H})$
HTML articles powered by AMS MathViewer

by Lajos Molnár PDF

Proc. Amer. Math. Soc. 129 (2001), 531-537 Request permission

Abstract:

Let $X$ be a compact Hausdorff space and let $\mathscr {H}$ be a separable Hilbert space. We prove that the group of all order automorphisms of the $C^*$-algebra $C(X)\otimes \mathscr {B}(\mathscr {H})$ is algebraically reflexive.

References

Charles J. K. Batty and Lajos Molnár, On topological reflexivity of the groups of $\ast$-automorphisms and surjective isometries of $\scr B(H)$, Arch. Math. (Basel) 67 (1996), no. 5, 415–421. MR 1411996, DOI 10.1007/BF01189101
P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
Richard V. Kadison, Local derivations, J. Algebra 130 (1990), no. 2, 494–509. MR 1051316, DOI 10.1016/0021-8693(90)90095-6
Richard V. Kadison and John R. Ringrose, Derivations and automorphisms of operator algebras, Comm. Math. Phys. 4 (1967), 32–63. MR 206735, DOI 10.1007/BF01645176
E. C. Lance, Automorphisms of certain operator algebras, Amer. J. Math. 91 (1969), 160–174. MR 241989, DOI 10.2307/2373275
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
Lajos Molnár, The set of automorphisms of $B(H)$ is topologically reflexive in $B(B(H))$, Studia Math. 122 (1997), no. 2, 183–193. MR 1432168, DOI 10.4064/sm-122-2-183-193
Lajos Molnár and Peter emrl, Order isomorphisms and triple isomorphisms of operator ideals and their reflexivity, Arch. Math. (Basel) 69 (1997), no. 6, 497–506. MR 1480517, DOI 10.1007/s000130050152
L. Molnár and M. Győry, Reflexivity of the automorphism and isometry groups of the suspension of $\mathcal {B}(\mathcal {H})$, J. Funct. Anal. 159 (1998), 568–586.
L. Molnár and B. Zalar, Reflexivity of the group of surjective isometries on some Banach spaces, Proc. Edinb. Math. Soc. 42 (1999), 17–36.
L. Molnár and B. Zalar, On local automorphisms of group algebras of compact groups, Proc. Amer. Math. Soc. 128 (2000), 93–99.
L. Molnár, Reflexivity of the automorphism and isometry groups of $C^*$-algebras in BDF theory, Arch. Math. (to appear)
Heydar Radjavi and Peter Rosenthal, On invariant subspaces and reflexive algebras, Amer. J. Math. 91 (1969), 683–692. MR 251569, DOI 10.2307/2373347
Russell C. Walker, The Stone-Čech compactification, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 83, Springer-Verlag, New York-Berlin, 1974. MR 0380698, DOI 10.1007/978-3-642-61935-9