We employ the set-theoretic methods yielding the consistency of the existence of nontrivial pairwise disjoint
covers of the unit interval (or equivalently Rn for positive integers n) by less than 2ω
closed sets in the context of covers of infinite dimensional Banach spaces l1(κ)
for infinite κ under geometric conditions arising in tiling theory and in approximation theory.
Specifically, for κ=ω1, ω2,
using side-by-side Sacks forcing we prove the consistency of an arbitrarily large continuum above κ
with the existence of a highly disconnected proximinal set in l1(κ),
built from norm-compact pieces separated by a common positive distance, and the existence of a normal
disjoint tiling of l1(κ) by star-shaped bodies of the form X+B, where X is norm compact and B is the unit ball.
We also prove that if κ is an uncountable cardinal of countable cofinality, then l1(κ)
does not admit any normal disjoint tiling by bodies of the form X+B, where X is closed and norm-separable and B is the unit ball.
These results complement a result of Klee of 1981 obtained for κ satisfying
κω=κ, recent results of De Bernardi, Russo, Sezgek and Somaglia,
and a ZFC a machine-discovered result (included in the appendix) that there are no
nontrivial discrete Chebyshev sets in l1(κ) when κ<2ω.
We show that maximal abelian C*-subalgebras (masas) of the Calkin algebra (the algebra of all bounded operators on the separable Hilbert space modulo compact operators) may consistently have their densities strictly less than continuum and we describe many isomorphism types of such masas. Specifically, we prove that after adding any number of Cohen reals to a model of CH the algebra C(KA) of all complex-valued continuous functions on the Stone space KA of a Boolean algebra A of cardinality ω1 is ∗-isomorphic to a masa of the Calkin algebra if and only if A does not admit a countably generated ultrafilter. Moreover, for every such Boolean algebra we obtain ω2 pairwise unitarily non-equivalent such masas, none of which has a commutative lift. We also show in ZFC that if a C*-algebra of the form C(K) for any compact Hausdorff K is ∗-isomorphic to a masa of the Calkin algebra, then no point of K may have character smaller than p. Therefore, consistently, there may not be any masa of the Calkin algebra of density less than continuum.
We investigate the following general problem, closely related to the problem of isomorphic
classification of Banach spaces of continuous functions:
Let K be a class of compact Hausdorff spaces.
How many isomorphism types of Banach spaces C(K) of real-valued continuous functions on K
with the supremum norm are there, for K in K?
We prove that for any uncountable regular cardinal number κ, there exist exactly
2κ isomorphism types of spaces C(K) for compact spaces K of weight κ.
We show that, for the class Lω1 of separable compact linearly ordered spaces of weight ω1,
the answer to the above question depends on additional set-theoretic axioms. In particular,
assuming the continuum hypothesis, there are 2ω1 isomorphism types of C(L), for L in Lω1,
and assuming a certain axiom proposed by Baumgartner, there is only one type.
We present constructions of previously unknown diverse
maximal abelian self-adjoint subalgebras (masas) of the
Calkin algebra Q(ℓ2) (bounded operators on a separable Hilbert space modulo compact operators)
generated by projections.
First, assuming the continuum hypothesis CH, for every compact totally disconnected Hausdorff space K
of weight not exceeding the continuum and
without Gδ points, we construct a masa of Q(ℓ2) which is *-isomorphic to the algebra C(K)
of complex-valued continuous functions on K. This is sharp in two ways: (1) there cannot be other *-isomorphic types
of masas of Q(ℓ2) generated by projections and so, this result gives a complete *-isomorphic classification
of masas of Q(ℓ2) generated by projections, (2)
some additional
set-theoretic hypothesis, like CH, is necessary to have all these C*-algebras as masas of Q(ℓ2).
This shows that masas of the Calkin algebra could have rather unexpected properties
compared to the previously known three *-isomorphic types of them generated by projections:
ℓ∞/c0, L∞ and ℓ∞/c0⊕L∞.
For example, they may not admit conditional expectations, be factorizable as tensor products of infinite dimensional
C*-algebras,
their Gelfand spaces could be topological groups and admit
nontrivial convergent sequences or could be hyper-Stonean admitting nonseparable category measure,
extremally disconnected but not hyper-Stonean
or could be one of many exotic compact spaces constructed under the continuum hypothesis.
Secondly, without making any additional set-theoretic assumptions we construct a family
of maximal possible cardinality (of the power set of ℝ) of pairwise non-*-isomorphic
masas of Q(ℓ2) generated by projections which (a) are not SAW*-algebras unlike
the liftable masas (Gelfand spaces in this group of our masas are not F-spaces) (b) do not
admit conditional expectations.
This improves
the results which required additional set-theoretic hypotheses to construct
a single masa of Q(ℓ2) generated by projections without a commutative lift.
Answering questions of A. Avilés, F. Cabello Sánchez, J. Castillo,
M. González and Y. Moreno we show that the following statements are independent of the usual axioms ZFC with arbitrarily large continuum: for every
(some) ω < κ < 2ω
- any linear bounded operator T : c0(κ) → ℓ∞/c0 extends to any superspace of c0(κ).
- any isomorphism between any two copies of c0(κ) inside ℓ∞/c0 extends
to an automorphism of ℓ∞/c0.
This contrasts with Boolean, Banach algebraic or isometric levels, where the
objects known as Hausdorff gap and Luzin gap witness the failure in ZFC of
the corresponding properties for the corresponding structures already at the
first uncountable cardinal κ=ω
1.
In particular, consistently, any two pairwise disjoint families in P(ℕ)/Fin
of the same cardinality ω < κ < 2ω can be mapped onto each other by a linear
automorphism of ℓ∞/c0 regardless of their different combinatorial, algebraic
or topological positions in P(ℕ)/ Fin.
Our positive consistency results use a restricted version of Martin’s axiom
for a partial order that adds an infinite block diagonal matrix of an operator on
ℓ∞ which induces an operator on ℓ∞/c0 . The construction of its finite blocks
relies on a lemma of Bourgain and Tzafriri on finite dimensional Banach spaces.
Our negative consistency results rely on an analysis of almost disjoint families of ℕ, the embeddings of
c0(κ) into ℓ∞/c0 they induce and their extensions
to ℓ∞c(κ).
Accepted to Studia Mathematica
