Related papers: A note on the diamond operator

Assuming Jensen's diamond principle ($\diamondsuit$) 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…

It is well known that a Lipschitz function on the real line does not have to be operator Lipschitz. We show that the situation changes dramatically if we pass to H\"older classes. Namely, we prove that if $f$ belongs to the H\"older class…

We give an explicit description of the lattice $\Semistar(D)$ of all semistar operations on any Dedekind domain $D$ from its set $\Max(D)$ of maximal ideals. This descpription is constructive if $\Max(D)$ is finite. As a corollary we show…

In this paper we obtain the asymptotic formulas of arbitrary order for the Bloch eigenvalues and Bloch functions of the multidimensional polyharmonic operator with periodic, with respect to arbitrary lattice, potential. Then we prove that…

Any Lipschitz map $f : M \to N$ between two pointed metric spaces may be extended in a unique way to a bounded linear operator $\widehat{f} : \mathcal F(M) \to \mathcal F(N)$ between their corresponding Lipschitz-free spaces. In this paper,…

A modular or distributive lattice is `diamond-colored' if its order diagram edges are colored in such a way that, within any diamond of edges, parallel edges have the same color. Such lattices arise naturally in combinatorial representation…

Suppose that $\Omega$ is a complex lattice that is closed under complex conjugation and that $I$ is a small real interval, and that $D$ is a disc in $ \mathbb{C}$. Then the restriction $\wp|_D$ is definable in the structure…

The diamond product is the poset operation that when applied to the face lattices of two polytopes results in the face lattice of the Cartesian product of the polytopes. Application of the diamond product to two Eulerian posets is a…

We show that the centre of a Dedekind complete complex Banach lattice is a commutative $\mathrm{C}^\ast$-algebra in the order unit norm. This implies that the order unit norm and the operator norm coincide. As an application of the latter,…

Let $D$ be an integral domain with quotient field $K$. A star-operation $\star$ on $D$ is a closure operation $A \longmapsto A^\star$ on the set of nonzero fractional ideals, $F(D)$, of $D$ satisfying the properties: $(xD)^\star = xD$ and…

Various authors have investigated properties of the star order (introduced by M.P. Drazin in 1978) on algebras of matrices and of bounded linear operators on a Hilbert space. Rickart involution rings (*-rings) are a certain algebraic…

The present paper deals with complemented lattices where, however, a unary operation of complementation is not explicitly assumed. This means that an element can have several complements. The mapping $^+$ assigning to each element $a$ the…

We prove that the Weihrauch lattice can be transformed into a Brouwer algebra by the consecutive application of two closure operators in the appropriate order: first completion and then parallelization. The closure operator of completion is…

We prove that if $0<\a<1$ and $f$ is in the H\"older class $\L_\a(\R)$, then for arbitrary self-adjoint operators $A$ and $B$ with bounded $A-B$, the operator $f(A)-f(B)$ is bounded and $\|f(A)-f(B)\|\le\const\|A-B\|^\a$. We prove a similar…

We prove that, consistently, there exists a weakly but not strongly inaccessible cardinal $\lambda$ for which the sequence $\langle 2^\theta:\theta<\lambda\rangle$ is not eventually constant and the weak diamond fails at $\lambda$. We also…

We construct a C*-algebra that has only one irreducible representation up to unitary equivalence but is not isomorphic to the algebra of compact operators on any Hilbert space. This answers an old question of Naimark. Our construction uses…

In this paper, we introduce and study new concepts of order L-weakly and order M-weakly compact operators. As consequences, we obtain some characterizations of Banach lattices with order continuous norms or whose topological duals have…

We consider composition operators $\mathscr{C}_\varphi$ on the Hardy space of Dirichlet series $\mathscr{H}^2$, generated by Dirichlet series symbols $\varphi$. We prove two different subordination principles for such operators. One…

We answer an open question in the theory of transducer degrees initially posed in [1] on the existence of a diamond structure in the transducer hierarchy. Transducer degrees are the equivalence classes formed by word transformations which…

In the theory of partially-ordered sets, the two-dimensional Boolean lattice is known as the diamond. In this paper, we show that, if $\mathcal{F}$ is a family in the $n$-dimensional Boolean lattice that has no diamond as a subposet, then…