A general view of reflexivity

Hadwin, Don

doi:10.1090/S0002-9947-1994-1239639-4

A general view of reflexivity
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by Don Hadwin

Trans. Amer. Math. Soc. 344 (1994), 325-360

DOI: https://doi.org/10.1090/S0002-9947-1994-1239639-4

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Abstract:

Various concepts of reflexivity for an algebra or linear space of operators have been studied by operator theorists and algebraists. This paper contains a very general version of reflexivity based on dual pairs of vector spaces over a Hausdorff field. The special cases include topological, algebraic and approximate reflexivity. In addition general versions of hyperreflexivity and direct integrals are introduced. We prove general versions of many known (and some new) theorems, often with simpler proofs.

References

Banach

modules and operators preserving disjointness

Constantin Apostol, Ciprian Foiaş, and Dan Voiculescu, Strongly reductive operators are normal, Acta Sci. Math. (Szeged) 38 (1976), no. 3-4, 261–263. MR 433241
Constantin Apostol, Ciprian Foiaş, and Dan Voiculescu, On strongly reductive algebras, Rev. Roumaine Math. Pures Appl. 21 (1976), no. 6, 633–641. MR 417804
William Arveson, Notes on extensions of $C^{^*}$-algebras, Duke Math. J. 44 (1977), no. 2, 329–355. MR 438137
William Arveson, Ten lectures on operator algebras, CBMS Regional Conference Series in Mathematics, vol. 55, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1984. MR 762819, DOI 10.1090/cbms/055

An invitation to

algebras

Edward A. Azoff, On finite rank operators and preannihilators, Mem. Amer. Math. Soc. 64 (1986), no. 357, vi+85. MR 858467, DOI 10.1090/memo/0357
E. A. Azoff, C. K. Fong, and F. Gilfeather, A reduction theory for non-self-adjoint operator algebras, Trans. Amer. Math. Soc. 224 (1976), no. 2, 351–366 (1977). MR 448109, DOI 10.1090/S0002-9947-1976-0448109-1
E. A. Azoff and H. A. Shehada, On separation by families of linear functionals, J. Funct. Anal. 96 (1991), no. 1, 96–116. MR 1093508, DOI 10.1016/0022-1236(91)90074-F
E. A. Azoff and H. A. Shehada, From algebras of normal operators to intersecting hyperplanes, Operator theory: operator algebras and applications, Part 2 (Durham, NH, 1988) Proc. Sympos. Pure Math., vol. 51, Amer. Math. Soc., Providence, RI, 1990, pp. 11–16. MR 1077415, DOI 10.1090/pspum/051.2/1077415
Kenneth R. Davidson, On operators commuting with Toeplitz operators modulo the compact operators, J. Functional Analysis 24 (1977), no. 3, 291–302. MR 0454715, DOI 10.1016/0022-1236(77)90060-x
Kenneth R. Davidson, The distance to the analytic Toeplitz operators, Illinois J. Math. 31 (1987), no. 2, 265–273. MR 882114
J. A. Deddens and P. A. Fillmore, Reflexive linear transformations, Linear Algebra Appl. 10 (1975), 89–93. MR 358390, DOI 10.1016/0024-3795(75)90099-3
J. Diestel and J. J. Uhl Jr., Vector measures, Mathematical Surveys, No. 15, American Mathematical Society, Providence, R.I., 1977. With a foreword by B. J. Pettis. MR 0453964
Jacques Dixmier, Les algèbres d’opérateurs dans l’espace hilbertien (Algèbres de von Neumann), Cahiers Scientifiques, Fasc. XXV, Gauthier-Villars, Paris, 1957 (French). MR 0094722
R. G. Douglas and C. Foiaş, Infinite dimensional versions of a theorem of Brickman-Fillmore, Indiana Univ. Math. J. 25 (1976), no. 4, 315–320. MR 407622, DOI 10.1512/iumj.1976.25.25027
John Ernest, Charting the operator terrain, Mem. Amer. Math. Soc. 6 (1976), no. 171, iii+207. MR 463941, DOI 10.1090/memo/0171
P. A. Fillmore, On invariant linear manifolds, Proc. Amer. Math. Soc. 41 (1973), 501–505. MR 338804, DOI 10.1090/S0002-9939-1973-0338804-X
James Glimm, A Stone-Weierstrass theorem for $C^{\ast }$-algebras, Ann. of Math. (2) 72 (1960), 216–244. MR 116210, DOI 10.2307/1970133
T. A. Gillespie, Boolean algebras of projections and reflexive algebras of operators, Proc. London Math. Soc. (3) 37 (1978), no. 1, 56–74. MR 482360, DOI 10.1112/plms/s3-37.1.56
Donald W. Hadwin, An asymptotic double commutant theorem for $C^{\ast }$-algebras, Trans. Amer. Math. Soc. 244 (1978), 273–297. MR 506620, DOI 10.1090/S0002-9947-1978-0506620-0
Don Hadwin, Algebraically reflexive linear transformations, Linear and Multilinear Algebra 14 (1983), no. 3, 225–233. MR 718951, DOI 10.1080/03081088308817559
Don Hadwin, Approximately hyperreflexive algebras, J. Operator Theory 28 (1992), no. 1, 51–64. MR 1259915
Don Hadwin, A reflexivity theorem for subspaces of Calkin algebras, J. Funct. Anal. 123 (1994), no. 1, 1–11. MR 1279293, DOI 10.1006/jfan.1994.1080
D. W. Hadwin and E. A. Nordgren, Subalgebras of reflexive algebras, J. Operator Theory 7 (1982), no. 1, 3–23. MR 650190
D. W. Hadwin and E. A. Nordgren, Erratum: “Subalgebras of reflexive algebras” [J. Operator Theory 7 (1982), no. 1, 3–23; MR0650190 (83f:47033)], J. Operator Theory 15 (1986), no. 1, 203–204. MR 816239
Don Hadwin and Eric A. Nordgren, Reflexivity and direct sums, Acta Sci. Math. (Szeged) 55 (1991), no. 1-2, 181–197. MR 1124956
Don Hadwin, Eric Nordgren, Heydar Radjavi, and Peter Rosenthal, Orbit-reflexive operators, J. London Math. Soc. (2) 34 (1986), no. 1, 111–119. MR 859152, DOI 10.1112/jlms/s2-34.1.111
D. W. Hadwin and S.-C. Ong, On algebraic and para-reflexivity, J. Operator Theory 17 (1987), no. 1, 23–31. MR 873461
Don Hadwin and Mehmet Orhon, Reflexivity and approximate reflexivity for bounded Boolean algebras of projections, J. Funct. Anal. 87 (1989), no. 2, 348–358. MR 1026857, DOI 10.1016/0022-1236(89)90014-1
P. R. Halmos, Ten problems in Hilbert space, Bull. Amer. Math. Soc. 76 (1970), 887–933. MR 270173, DOI 10.1090/S0002-9904-1970-12502-2
C. J. Himmelberg, Measurable relations, Fund. Math. 87 (1975), 53–72. MR 367142, DOI 10.4064/fm-87-1-53-72
Jon Kraus and David R. Larson, Reflexivity and distance formulae, Proc. London Math. Soc. (3) 53 (1986), no. 2, 340–356. MR 850224, DOI 10.1112/plms/s3-53.2.340
Alan Lambert, Strictly cyclic operator algebras, Pacific J. Math. 39 (1971), 717–726. MR 310664
David R. Larson, Annihilators of operator algebras, Invariant subspaces and other topics (Timişoara/Herculane, 1981), Operator Theory: Advances and Applications, vol. 6, Birkhäuser, Basel-Boston, Mass., 1982, pp. 119–130. MR 685459
David R. Larson, Hyperreflexivity and a dual product construction, Trans. Amer. Math. Soc. 294 (1986), no. 1, 79–88. MR 819936, DOI 10.1090/S0002-9947-1986-0819936-X
David R. Larson, Reflexivity, algebraic reflexivity and linear interpolation, Amer. J. Math. 110 (1988), no. 2, 283–299. MR 935008, DOI 10.2307/2374503

On hereditary and intermediate reflexivity of

algebras

396

9

W. Mlak, Operator valued representations of function algebras, Linear operators and approximation, II (Proc. Conf., Math. Res. Inst., Oberwolfach, 1974) Internat. Ser. Numer. Math., Vol. 25, Birkhäuser, Basel, 1974, pp. 49–79. MR 0394220
Robert F. Olin and James E. Thomson, Algebras of subnormal operators, J. Functional Analysis 37 (1980), no. 3, 271–301. MR 581424, DOI 10.1016/0022-1236(80)90045-2

Locally cyclic representations of

Vern I. Paulsen, Completely bounded maps and dilations, Pitman Research Notes in Mathematics Series, vol. 146, Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York, 1986. MR 868472
A. Pełczyński, Projections in certain Banach spaces, Studia Math. 19 (1960), 209–228. MR 126145, DOI 10.4064/sm-19-2-209-228
Heydar Radjavi and Peter Rosenthal, Invariant subspaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 77, Springer-Verlag, New York-Heidelberg, 1973. MR 0367682
C. J. Read, The invariant subspace problem for a class of Banach spaces. II. Hypercyclic operators, Israel J. Math. 63 (1988), no. 1, 1–40. MR 959046, DOI 10.1007/BF02765019

Topological vector spaces

Shlomo Rosenoer, Distance estimates for von Neumann algebras, Proc. Amer. Math. Soc. 86 (1982), no. 2, 248–252. MR 667283, DOI 10.1090/S0002-9939-1982-0667283-3
M.-F. Sainte-Beuve, On the extension of von Neumann-Aumann’s theorem, J. Functional Analysis 17 (1974), 112–129. MR 0374364, DOI 10.1016/0022-1236(74)90008-1
Dan Voiculescu, A non-commutative Weyl-von Neumann theorem, Rev. Roumaine Math. Pures Appl. 21 (1976), no. 1, 97–113. MR 415338