The Perfekt theory of $M$-ideals


  • Dirk Werner



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.


Alfsen, E. M., and Effros, E. G., Structure in real Banach spaces. I and II, Ann. Math. (2) 96 (1972), 98–173.

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.

Harmand, P., Werner, D., and Werner, W., $M$-Ideals in Banach Spaces and Banach Algebras, Lecture Notes in Mathematics 1547. Springer-Verlag, Berlin, 1993.

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.

Perfekt, K.-M., Duality and distance formulas in spaces defined by means of oscillation, Ark. Mat. 51 (2013), no. 2, 345–361.

Perfekt, K.-M., On $M$-ideals and $o−O$ type spaces, Math. Scand. 121 (2017), no. 1, 151–160.

Weaver, N., Duality for locally compact Lipschitz spaces, Rocky Mountain J. Math. 26 (1996), no. 1, 337–353.

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.



How to Cite

Werner, D. (2022). The Perfekt theory of $M$-ideals. MATHEMATICA SCANDINAVICA, 128(2).