## M(a,B,c)-ideaalid Banachi ruumides

##### Abstract

In their 1973 paper “Structure in real Banach spaces” [AE], E. Alfsen and E. Effros
introduced the notion of an M-ideal. It turned out that a closed subspace of a
Banach space that is an M-ideal enjoys some properties (e.g. uniqueness of a
norm-preserving extension) which do not necessarily occur in arbitrary subspaces.
In [GKS], G. Godefroy, N. Kalton and P. Saphar introduced the notion of an
ideal. It allowed them to make a connection between M-ideals and u-ideals, which
were first introduced in [CK]. They also presented a natural strengthening of the
definition of a u-ideal, an h-ideal. Another important step towards the generalization
of previously studied ideals was made by J. Cabello and E. Nieto in [CN],
where they defined an M(r; s)-ideal.
The idea of studying M(a;B; c)-ideals dates back to a paper [O1] by E. Oja
from 2000. In [OZ], one finds the following definition.
Let a, c > 0 and let B K be a compact set of scalars. We shall say
that a Banach space X satisfies the M(a;B; c)-inequality if
kax + b Xx k + ck Xx k 6 kx k 8b 2 B; 8x 2 X ;
where X : X ! X is the canonical projection.
Based on this definition, we define an M(a;B; c)-ideal for an arbitrary closed
subspace and aim to study some properties of M(a;B; c)-ideals. The pursuit to
studyM(a;B; c)-ideals is motivated by the fact that the definition of anM(a;B; c)-
ideal encompasses all previously studied special cases of ideals and makes it possible
to handle them with a more unified approach.
This bachelor thesis consists of four chapters.
In the first chapter, we give a brief overview of some basic definitions and
results required for further work.
In the second chapter, we introduce the notion of an M(a;B; c)-ideal and study
some basic properties of M(a;B; c)-ideals. We also take a closer look at M-, u-
and h-ideals.
The aim of the third chapter is to study M(a;B; c)-ideals in particular Banach
spaces. First we give necessary and sufficient conditions for a one-dimensional
subspace of a Banach space `2
1 to be an M(a;B; c)-ideal in `2
1. We also provide
a theoretical result which can be used to derive examples of M(a;B; c)-ideals in
L(X).
In the fourth chapter, we study the transitivity of M(a;B; c)-inequality. First
we show that ideal projections are closely connected to Hahn-Banach extension
operators. Using this knowledge, we show as a first main result of this bachelor
thesis that if X is an M(a;B; c)-ideal in Y and Y is an M(d;E; f)-ideal in Z, then
X is an ideal satisfying a certain type of inequality in Z. Relying on this result,
we show as a second main result of this thesis that if X is an M(a;B; c)-ideal in
its second bidual X , then X is an ideal satisfying a certain type of inequality in
X(2n) for every n 2 N.