We introduce a weaker notion of central subspace called almost central subspace, and we study Banach spaces that belong to the class (GC), introduced by Veselý (1997). In particular, we prove that if

In [

For a Banach space

We now recall the definition of the class (GC) from [

Let

For a Banach space

If

One shall denote by (GC) the class of all Banach spaces

Next we recall the definition of a central subspace which generalizes the notion of (GC).

Let

Clearly

An infinite version of central subspace called almost constrained subspace was investigated in [

A subspace

We recall that a subspace

A subspace

Clearly

An important concept in the

A projection

For

In Section

In Section

We begin this section with the definition of an “almost central subspace” of a Banach space which is the generalization of the concept central subspace, defined in [

A subspace

Clearly central subspaces of Banach spaces are almost central. As in the case of central subspace, it is easy to observe that

Let

Let

Since every Banach space is an ideal in its bidual, we have the following result.

Every Banach space is almost central in its bidual.

Since every

Let

In [

Let

Let

Now let

Our next result gives a sufficient condition for an almost central subspace to be an AC-subspace.

Let

Let

We now give a class of Banach spaces where almost central subspaces are central. We recall that a Banach space

Let

Let

Following the same line of argument as in the proof of [

Since every ideal is almost central, our next result generalizes Proposition 14 of [

Let

If

Conversely suppose

By a similar transitivity argument used in the proof of the Proposition

Let

It is well known that a semi

Let

Since

We now give a sufficient condition for a semi

Let

Suppose

Our next theorem gives another sufficient condition for a semi

Let

Suppose

We now recall the following theorem of Bandyopadhyay and Dutta that characterizes an AC-subspace of finite codimension in the space

Let

There exist measures

In our next proposition, we observe a simple proof for the implication

We recall that a compact Hausdorff space

For any infinite discrete set

The following lemma is the uncountable version of the main theorem of [

Let

Our next result shows that the space

Let

Suppose

Let

Let

Let

Now define

Now let

Let

Let

In an

Coming to quotient spaces, one can easily observe that if

For a subspace

Our next result solves the above problem for AC-subspaces.

Let

Let

We now prove the stability of ideals in quotient spaces.

Let

Since

Our next result proves the stability of almost central subspaces in quotient spaces.

Let

Let

Now, for Banach spaces

Combining Proposition

Let

As a consequence of the above corollary, we have the following result.

Let

By Corollary

We recall that a subspace

Let

We now prove the converse of Proposition

Let

Let

The following corollary is the converse of Proposition

Let

Since

We recall that, for any collection

It is easy to observe that, for any family

We now prove the stability of almost central subspaces in vector-valued continuous function spaces. For a compact Hausdorff space

Let

Let

Let

For a central subspace

Let

We now discuss the stability problem in injective tensor product spaces.

Let

Since

Let

By Proposition

Let

Since

We now answer a question raised in [

Let

Let

Define

if

if

The class

If

In [

A norm

We say that a Banach space

Our next theorem proves that the property of being a central subspace is stable under polyhedral direct sums.

For

Let

Suppose

Conversely suppose

An argument similar to the one used to prove Theorem

Let

The author would like to thank Professor T. S. S. R. K. Rao for many helpful discussions and valuable suggestions. The author also thanks the referees for their extensive comments that lead to an improved version of the paper.