Diamond Principle Research Articles

A CONSTRUCTION VIA FORCING OF A HEREDITARILY WEAKLY KOSZMIDER SPACE

We present an alternative construction – using forcing, instead of diamond principle – of a Hausdorff compactum K such that C(L) has few operators, for every closed L ⊂ K. Moreover, we prove a new result about the density of C(K) with few operators and present some open problems.

Approximating diamond principles on products at an inaccessible cardinal

We isolate the approximating diamond principles, which are consequences of the diamond principle at an inaccessible cardinal. We use these principles to find new methods for negating the diamond principle at large cardinals. Most notably, we demonstrate, using Gitik’s overlapping extenders forcing, a new method to get the consistency of the failure of the diamond principle at a large cardinal θ \theta without changing cofinalities or adding fast clubs to θ \theta . In addition, we show that the approximating diamond principles necessarily hold at a weakly compact cardinal. This result, combined with the fact that in all known models where the diamond principle fails the approximating diamond principles also fail at an inaccessible cardinal, exhibits essential combinatorial obstacles to make the diamond principle fail at a weakly compact cardinal.

Can you take Akemann–Weaver's [formula omitted] away?

By Glimm's dichotomy, a separable, simple C⁎-algebra has continuum many unitarily inequivalent irreducible representations if, and only if, it is non-type I while all of its irreducible representations are unitarily equivalent if, and only if, it is type I. Naimark asked whether the latter equivalence holds for all C⁎-algebras.In 2004, Akemann and Weaver gave a negative answer to Naimark's problem using Jensen's Diamond Principle ⋄ℵ1, a powerful diagonalization principle that implies the Continuum Hypothesis (CH). By a result of Rosenberg, a separably represented, simple C⁎-algebra with a unique irreducible representation is necessarily of type I. We show that this result is sharp by constructing an example of a separably represented, simple C⁎-algebra that has exactly two inequivalent irreducible representations, and therefore does not satisfy the conclusion of Glimm's dichotomy. Our construction uses a weakening of Jensen's ⋄ℵ1, denoted ⋄Cohen, that holds in the original Cohen's model for the negation of CH. We also prove that ⋄Cohen suffices to give a negative answer to Naimark's problem. Our main technical tool is a forcing notion that generically adds an automorphism of a given C⁎-algebra with a prescribed action on its space of pure states.

A Banach space C(K) reading the dimension of K

Assuming Jensen's diamond principle (⋄) we construct for every natural number n>0 a compact Hausdorff space K such that whenever the Banach spaces C(K) and C(L) are isomorphic for some compact Hausdorff L, then the covering dimension of L is equal to n. The constructed space K is separable and connected, and the Banach space C(K) has few operators i.e. every bounded linear operator T:C(K)→C(K) is of the form T(f)=fg+S(f), where g∈C(K) and S is weakly compact.

‘We don’t openly discuss these things’: Cultural complexities in teaching about sexuality and HIV in South Africa

It is well documented that rates of HIV infection in South Africa are alarmingly high, with approximately 7.5 million people living with HIV in 2021. This study aimed to explore how culture in the form of the values, practices, norms and beliefs prevalent in society influences teaching about sexuality and HIV in South Africa. The study adopted a qualitative, narrative approach and drew on findings from a purposive sample of six further education and training life orientation teachers from six schools in the province of KwaZulu-Natal, South Africa. Data were analysed using thematic analysis and cultural diamond principles. Socio-cultural complexities were found to shape discussion of sexuality and HIV. Five key themes were developed from an analysis of participants’ responses: school guidelines, culture of silence, personal experiences, cultural taboos, and language as a barrier. Findings signal the value of an integrated whole-school approach to the design and delivery of the curriculum involving key stakeholders and the perspectives of parents and religious leaders about the teaching of sexuality and HIV. The national departments of education and health in South Africa should also provide resources and guidelines detailing best practices to assist life orientation teachers.

Compactness and guessing principles in the Radin extensions

We investigate the interaction between compactness principles and guessing principles in the Radin forcing extensions. In particular, we show that in any Radin forcing extension with respect to a measure sequence on [Formula: see text], if [Formula: see text] is weakly compact, then [Formula: see text] holds. This provides contrast with a well-known theorem of Woodin, who showed that in a certain Radin extension over a suitably prepared ground model relative to the existence of large cardinals, the diamond principle fails at a strongly inaccessible Mahlo cardinal. Refining the analysis of the Radin extensions, we consistently demonstrate a scenario where a compactness principle, stronger than the diagonal stationary reflection principle, holds yet the diamond principle fails at a strongly inaccessible cardinal, improving a result from [O. B. -Neria, Diamonds, compactness, and measure sequences, J. Math. Log. 19(1) (2019) 1950002].

Minimally generated Boolean algebras and the Nikodym property

A Boolean algebra A has the Nikodym property if every pointwise bounded sequence of bounded finitely additive measures on A is uniformly bounded. Assuming the Diamond Principle ◇, we will construct an example of a minimally generated Boolean algebra A with the Nikodym property. The Stone space of such an algebra must necessarily be an Efimov space. The converse is, however, not true—again under ◇ we will provide an example of a minimally generated Boolean algebra whose Stone space is Efimov but which does not have the Nikodym property. The results have interesting measure-theoretic and topological consequences.

The ultrafilter number and

AbstractThe cardinal invariant $\mathfrak {hm}$ is defined as the minimum size of a family of $\mathsf {c}_{\mathsf {min}}$ -monochromatic sets that cover $2^{\omega }$ (where $\mathsf {c}_{\mathsf {min}}( x,y) $ is the parity of the biggest initial segment both x and y have in common). We prove that $\mathfrak {hm}=\omega _{1}$ holds in Shelah’s model of $\mathfrak {i<u},$ so the inequality $\mathfrak {hm<u}$ is consistent with the axioms of $\mathsf {ZFC}$ . This answers a question of Thilo Weinert. We prove that the diamond principle $\mathfrak {\Diamond }_{\mathfrak {d}}$ also holds in that model.

A forcing axiom for a non-special Aronszajn tree

Suppose that T⁎ is an ω1-Aronszajn tree with no stationary antichain. We introduce a forcing axiom PFA(T⁎) for proper forcings which preserve these properties of T⁎. We prove that PFA(T⁎) implies many of the strong consequences of PFA, such as the failure of very weak club guessing, that all of the cardinal characteristics of the continuum are greater than ω1, and the P-ideal dichotomy. On the other hand, PFA(T⁎) implies some of the consequences of diamond principles, such as the existence of Knaster forcings which are not stationarily Knaster.

Characterizations of the weakly compact ideal on Pκλ

Hellsten [15] gave a characterization of Πn1-indescribable subsets of a Πn1-indescribable cardinal in terms of a natural filter base: when κ is a Πn1-indescribable cardinal, a set S⊆κ is Πn1-indescribable if and only if S∩C≠∅ for every n-club C⊆κ. We generalize Hellsten's characterization to Πn1-indescribable subsets of Pκλ, which were first defined by Baumgartner. First we show that under reasonable assumptions the Π01-indescribability ideal on Pκλ equals the minimal strongly normal ideal NSSκ,λ on Pκλ, which is different from NSκ,λ. We then formulate a notion of n-club subset of Pκλ and prove that a set S⊆Pκλ is Πn1-indescribable if and only if S∩C≠∅ for every n-club C⊆Pκλ. We also prove that elementary embeddings considered by Schanker [26] witnessing near supercompactness lead to the definition of a normal ideal on Pκλ, and indeed, this ideal is equal to Baumgartner's ideal of non–Π11-indescribable subsets of Pκλ. Additionally, as applications of these results we answer a question of Cox-Lücke [10] about F-layered posets, provide a characterization of Πnm-indescribable subsets of Pκλ in terms of generic elementary embeddings and prove several results involving a two-cardinal weakly compact diamond principle.

Non homeomorphic hereditarily weakly Koszmider spaces

It was claimed in a previous work of the second author that, assuming Jensen's diamond principle (⋄), there exists a hereditarily Koszmider space. We notice that there is a gap in that proof. It is still true, under ⋄, that there exists a hereditarily weakly Koszmider space and we prove that there is even a family (Kξ)ξ<2c of weakly Koszmider spaces such that C(Kξ) and C(Kη) are essentially incomparable, for ξ≠η. In particular, Axiom ⋄ implies the existence of 2c non-homeomorphic Efimov's spaces.

Diamonds, compactness, and measure sequences

We establish the consistency of the failure of the diamond principle on a cardinal [Formula: see text] which satisfies a strong simultaneous reflection property. The result is based on an analysis of Radin forcing, and further leads to a characterization of weak compactness of [Formula: see text] in a Radin generic extension.

Construction with opposition: cardinal invariants and games

We consider several game versions of the cardinal invariants {mathfrak {t}}, {mathfrak {u}} and {mathfrak {a}}. We show that the standard proof that parametrized diamond principles prove that the cardinal invariants are small actually shows that their game counterparts are small. On the other hand we show that {mathfrak {t}}<{mathfrak {t}}_{Builder} and {mathfrak {u}}<{mathfrak {u}}_{Builder} are both relatively consistent with ZFC, where {mathfrak {t}}_{Builder} and {mathfrak {u}}_{Builder} are the principal game versions of {mathfrak {t}} and {mathfrak {u}}, respectively. The corresponding question for {mathfrak {a}} remains open.

Chains of P-points

AbstractIt is proved that the Continuum Hypothesis implies that any sequence of rapid P-points of length ${&lt;}\mathfrak{c}^{+}$ that is increasing with respect to the Rudin–Keisler ordering is bounded above by a rapid P-point. This is an improvement of a result from B. Kuzeljevic and D. Raghavan. It is also proved that Jensen’s diamond principle implies the existence of an unbounded strictly increasing sequence of P-points of length $\unicode[STIX]{x1D714}_{1}$ in the Rudin–Keisler ordering. This shows that restricting to the class of rapid P-points is essential for the first result.

On the existence of skinny stationary subsets

Matsubara–Usuba [13] introduced the notion of skinniness and its variants for subsets of Pκλ and showed that the existence of skinny stationary subsets of Pκλ is related to large cardinal properties of ideals over Pκλ and to Jensen's diamond principle on λ. In this paper, we study the existence of skinny stationary sets in more detail.

Trace spaces of counterexamples to Naimark's problem

A counterexample to Naimark's problem is a C⁎-algebra that is not isomorphic to the algebra of compact operators on some Hilbert space, yet still has only one irreducible representation up to unitary equivalence. It is well-known that such algebras must be nonseparable, and in 2004 Akemann and Weaver used the diamond principle (a set theoretic axiom independent from ZFC) to give the first counterexamples. For any such counterexample A, the unitary group U(A) acts transitively on the pure states, which are the extreme points of the state space S(A). It is conceivable that this implies (as happens for finite-dimensional simplexes) that the action of U(A) on S(A) has at most one fixed point, i.e. A has at most one trace. We give a strong negative answer here assuming diamond. In particular, we adapt the Akemann–Weaver construction to show that the trace space of a counterexample to Naimark's problem can be affinely homeomorphic to any metrizable Choquet simplex, and can also be nonseparable.

On Some Properties of Shelah Cardinals

We present several results concerning Shelah cardinals including the fact that small and fast function forcings preserve Shelah and $$(^{\kappa }\kappa \cap V)$$ -Shelah cardinals, respectively. Furthermore, we prove that the Laver Diamond Principle holds for Shelah cardinals and use this fact to show that Shelah cardinals can be made indestructible under $$\le \kappa $$ -directed closed forcings of size $$<wt(\kappa )$$ .

A parametrized diamond principle and union ultrafilters

We consider a cardinal invariant closely related to Hindman's theorem. We prove that this cardinal invariant is small in the iterated Sacks perfect set forcing model, and that its corresponding parametrized diamond principle implies the existence of union ultrafilters. As a corollary, this establishes the existence of union ultrafilters in the iterated Sacks model of Set Theory.

SQUARE WITH BUILT-IN DIAMOND-PLUS

AbstractWe formulate combinatorial principles that combine the square principle with various strong forms of the diamond principle, and prove that the strongest amongst them holds in L for every infinite cardinal.As an application, we prove that the following two hold in L:1.For every infinite regular cardinal λ, there exists a special λ+-Aronszajn tree whose projection is almost Souslin;2.For every infinite cardinal λ, there exists a respecting λ+-Kurepa tree; Roughly speaking, this means that this λ+-Kurepa tree looks very much like the λ+-Souslin trees that Jensen constructed in L.

A Diamond Principle Consistent with AD

We present a diamond principle ◊R concerning all subsets of Θ, the supremum of the ordinals that are the surjective image of R. We prove that ◊R holds in Steel’s core model K(R), a canonical inner model for determinacy.