135 research outputs found
A C(K) Banach space which does not have the Schroeder-Bernstein property
We construct a totally disconnected compact Hausdorff space N which has
clopen subsets M included in L included in N such that N is homeomorphic to M
and hence C(N) is isometric as a Banach space to C(M) but C(N) is not
isomorphic to C(L). This gives two nonisomorphic Banach spaces of the form C(K)
which are isomorphic to complemented subspaces of each other (even in the above
strong isometric sense), providing a solution to the Schroeder-Bernstein
problem for Banach spaces of the form C(K). N is obtained as a particular
compactification of the pairwise disjoint union of a sequence of Ks for which
C(K)s have few operators
On constructions with -cardinals
We propose developing the theory of consequences of morasses relevant in
mathematical applications in the language alternative to the usual one,
replacing commonly used structures by families of sets originating with
Velleman's neat simplified morasses called -cardinals. The theory of related
trees, gaps, colorings of pairs and forcing notions is reformulated and
sketched from a unifying point of view with the focus on the applicability to
constructions of mathematical structures like Boolean algebras, Banach spaces
or compact spaces.
A new result which we obtain as a side product is the consistency of the
existence of a function
with the
appropriate -version of property for regular
satisfying .Comment: Minor correction
Some topological invariants and biorthogonal systems in Banach spaces
We consider topological invariants on compact spaces related to the sizes of
discrete subspaces (spread), densities of subspaces, Lindelof degree of
subspaces, irredundant families of clopen sets and others and look at the
following associations between compact topological spaces and Banach spaces: a
compact K induces a Banach space C(K) of real valued continuous functions on K
with the supremum norm; a Banach space X induces a compact space - the dual
ball with the weak* topology. We inquire on how topological invariants on K and
the dual ball are linked to the sizes of biorthogonal systems and their
versions in C(K) and X respectively. We gather folkloric facts and survey
recent results like that of Lopez-Abad and Todorcevic that it is consistent
that there is a Banach space X without uncountable biorthogonal systems such
that the spread of the dual ball is uncountable or that of Brech and Koszmider
that it is consistent that there is a compact space where spread of the square
of K ic countable but C(K) has uncountable biorthogonal systems
On large indecomposable Banach spaces
Hereditarily indecomposable Banach spaces may have density at most continuum
(Plichko-Yost, Argyros-Tolias). In this paper we show that this cannot be
proved for indecomposable Banach spaces. We provide the first example of an
indecomposable Banach space of density two to continuum. The space exists
consistently, is of the form C(K) and it has few operators in the sense that
any bounded linear operator T on C(K) satisfies T(f)=gf+S(f) for every f in
C(K), where g is in C(K) and S is weakly compact (strictly singular)
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