The Perfekt theory of $M$-ideals

Authors

  • Dirk Werner

DOI:

https://doi.org/10.7146/math.scand.a-132230

Abstract

We revisit some ideas of K.-M. Perfekt who has provided an elegant framework to detect the biduality between function or sequence spaces defined in terms of some $o$- respectively $O$-condition. We present new proofs under somewhat weaker assumptions than before and apply the result to Lipschitz spaces.

References

Alfsen, E. M., and Effros, E. G., Structure in real Banach spaces. I and II, Ann. Math. (2) 96 (1972), 98–173. https://doi.org/10.2307/1970895

Behrends, E., $M$-Structure and the Banach-Stone Theorem, Lecture Notes in Mathematics 736. Springer, Berlin, 1979.

De Leeuw, K., Banach spaces of Lipschitz functions. Studia Math. 21 (1961/62), 55–66. https://doi.org/10.4064/sm-21-1-55-6

Harmand, P., Werner, D., and Werner, W., $M$-Ideals in Banach Spaces and Banach Algebras, Lecture Notes in Mathematics 1547. Springer-Verlag, Berlin, 1993. https://doi.org/10.1007/BFb0084355

Kalton, N. J., Spaces of Lipschitz and Hölder functions and their applications, Collect. Math. 55 (2004), no. 2, 171–217.

D’Onofrio, L., Greco, L., Perfekt, K.-M., Sbordone, C., and Schiattarella, R., Atomic decompositions, two stars theorems, and distances for the Bourgain-Brezis-Mironescu space and other big spaces, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 37 (2020), no. 3, 653–661. https://doi.org/10.1016/j.anihpc.2020.01.004

Perfekt, K.-M., Duality and distance formulas in spaces defined by means of oscillation, Ark. Mat. 51 (2013), no. 2, 345–361. https://doi.org/10.1007/s11512-012-0175-7

Perfekt, K.-M., On $M$-ideals and $o−O$ type spaces, Math. Scand. 121 (2017), no. 1, 151–160. https://doi.org/10.7146/math.scand.a-96626

Weaver, N., Duality for locally compact Lipschitz spaces, Rocky Mountain J. Math. 26 (1996), no. 1, 337–353. https://doi.org/10.1216/rmjm/1181072120

Weaver, N., Lipschitz Algebras, 2nd edition. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2018.

Werner, D., New classes of Banach spaces which are (M)-ideals in their biduals, Math. Proc. Camb. Philos. Soc. 111 (1992), no. 2, 337–354. https://doi.org/10.1017/S0305004100075447

Published

2022-06-11

How to Cite

Werner, D. (2022). The Perfekt theory of $M$-ideals. MATHEMATICA SCANDINAVICA, 128(2). https://doi.org/10.7146/math.scand.a-132230

Issue

Section

Articles