Derivations,Local and 2-Local Derivations on Some Algebras of Operators on Hilbert C*-Modules

For a commutative C*-algebra \({\mathcal {A}}\) with unit e and a Hilbert \({\mathcal {A}}\)-module \({\mathcal {M}}\), denote by End\(_{{\mathcal {A}}}({\mathcal {M}})\) the algebra of all bounded \({\mathcal {A}}\)-linear mappings on \({\mathcal {M}}\), and by End\(^*_{{\mathcal {A}}}({\mathcal {M}})\) the algebra of all adjointable mappings on \({\mathcal {M}}\). We prove that if \({\mathcal {M}}\) is full, then each derivation on End\(_{{\mathcal {A}}}({\mathcal {M}})\) is \({\mathcal {A}}\)-linear, continuous, and inner, and each 2-local derivation on End\(_{{\mathcal {A}}}({\mathcal {M}})\) or End\(^{*}_{{\mathcal {A}}}({\mathcal {M}})\) is a derivation. If there exist \(x_0\) in \({\mathcal {M}}\) and \(f_0\) in \({\mathcal {M}}^{'}\), such that \(f_0(x_0)=e\), where \({\mathcal {M}}^{'}\) denotes the set of all bounded \({\mathcal {A}}\)-linear mappings from \({\mathcal {M}}\) to \({\mathcal {A}}\), then each \({\mathcal {A}}\)-linear local derivation on End\(_{{\mathcal {A}}}({\mathcal {M}})\) is a derivation.     

Joins and subdirect products of varieties

We generalise in three different directions two well-known results in universal algebra. Grätzer, Lakser and P?onka proved that independent subvarieties \({\mathcal{V}_{1}, \mathcal{V}_{2}}\) of a variety \({\mathcal{V}}\) are disjoint and such that their join \({\mathcal{V}_{1} \vee \mathcal{V}_{2}}\) (in the lattice of subvarieties of \({\mathcal{V}}\)) is their direct product \({\mathcal{V}_{1} \times \mathcal{V}_{2}}\) . Jónsson and Tsinakis provided a partial converse to this result: if \({\mathcal{V}}\) is congruence permutable and \({\mathcal{V}_{1}, \mathcal{V}_{2}}\) are disjoint, then they are independent (and so \({\mathcal{V}_{1} \vee \mathcal{V}_{2} = \mathcal{V}_{1} \times \mathcal{V}_{2}}\)). We show that (i) if \({\mathcal{V}}\) is subtractive, then Jónsson’s and Tsinakis’ result holds under some minimal assumptions; (ii) if \({\mathcal{V}}\) satisfies some weakened permutability conditions, then disjointness implies a generalised notion of independence and \({\mathcal{V}_{1} \vee \mathcal{V}_{2}}\) is the subdirect product of \({\mathcal{V}_{1}}\) and \({\mathcal{V}_2}\) ; (iii) the same holds if \({\mathcal{V}}\) is congruence 3-permutable.     

Multipliers of perturbed Cartesian products with an application to BSE-property

Given semisimple commutative Banach algebras \({\mathcal{A}}\) and \({\mathcal{B}}\) and a norm decreasing homomorphism \({\mathcal{T} : \mathcal{B} \rightarrow \mathcal{B}}\), we characterize the multipliers of the perturbed product Banach algebra \({\mathcal{A}\times_T \mathcal{B}}\). As an application it is shown that \({\mathcal{A}\times_T \mathcal{B}}\) has the Bochner–Schoenberg–Eberlein property if and only if both \({\mathcal{A}}\) and \({\mathcal{B}}\) have this property.     

Spectral properties and stability of perturbed Cartesian product

Let \({\mathcal {A}}\) and \({\mathcal {B}}\) be commutative Banach algebras, and let \(T:{\mathcal {B}} \rightarrow {\mathcal {A}}\) be an algebra homomorphism with \({\Vert T\Vert }\le 1\). Then T induces a Banach algebra product \(\times _T\) perturbing the coordinatewise product on the Cartesian product space \({\mathcal {A}} \times {\mathcal {B}}\). We show that the spectral properties like spectral extension property, unique uniform norm property, regularity, weak regularity as well as Ditkin’s condition are stable with respect to this product.     

Integer Decomposition Property of Free Sums of Convex Polytopes

Let \({\mathcal{P} \subset \mathbb{R}^{d}}\) and \({\mathcal{Q} \subset \mathbb{R}^{e}}\) be integral convex polytopes of dimension d and e which contain the origin of \({\mathbb{R}^{d}}\) and \({\mathbb{R}^{e}}\), respectively. We say that an integral convex polytope \({\mathcal{P}\subset \mathbb{R}^{d}}\) possesses the integer decomposition property if, for each \({n\geq1}\) and for each \({\gamma \in n\mathcal{P}\cap\mathbb{Z}^{d}}\), there exist \({\gamma^{(1)}, . . . , \gamma^{(n)}}\) belonging to \({\mathcal{P}\cap\mathbb{Z}^{d}}\) such that \({\gamma = \gamma^{(1)} +. . .+\gamma^{(n)}}\). In the present paper, under some assumptions, the necessary and sufficient condition for the free sum of \({\mathcal{P}}\) and \({\mathcal{Q}}\) to possess the integer decomposition property will be presented.     

Dirac–Krein Systems on Star Graphs

Suppose that \({\mathcal {M}}\) is a countably decomposable type II\({_1}\) von Neumann algebra and \({\mathcal {A}}\) is a separable, non-nuclear, unital C\({^*}\)-algebra. We show that, if \({\mathcal {M}}\) has Property \({\Gamma}\), then the similarity degree of \({\mathcal {M}}\) is less than or equal to 5. If \({\mathcal {A}}\) has Property c\({^*}\)-\({\Gamma}\), then the similarity degree of \({\mathcal {A}}\) is equal to 3. In particular, the similarity degree of a \({\mathcal {Z}}\)-stable, separable, non-nuclear, unital C\({^*}\)-algebra is equal to 3.     

Centralizing traces with automorphisms on triangular algebras

Let \({\mathcal{T}}\) be a triangular algebra over a commutative ring \({\mathcal{R}}\), \({\xi}\) be an automorphism of \({\mathcal{T}}\) and \({\mathcal{Z}_{\xi}(\mathcal{T})}\) be the \({\xi}\)-center of \({\mathcal{T}}\). Suppose that \({\mathfrak{q}\colon \mathcal{T}\times \mathcal{T}\longrightarrow \mathcal{T}}\) is an \({\mathcal{R}}\)-bilinear mapping and that \({\mathfrak{T}_{\mathfrak{q}}\colon \mathcal{T}\longrightarrow \mathcal{T}}\) is a trace of \({\mathfrak{q}}\). The aim of this article is to describe the form of \({\mathfrak{T}_{\mathfrak{q}}}\) satisfying the commuting condition \({[\mathfrak{T}_{\mathfrak{q}}(x), x]_{\xi}=0}\) (resp. the centralizing condition \({[\mathfrak{T}_{\mathfrak{q}}(x), x]_{\xi}\in \mathcal{Z}_\xi(\mathcal{T})}\)) for all \({x\in \mathcal{T}}\). More precisely, we will consider the question of when \({\mathfrak{T}_{\mathfrak{q}}}\) satisfying the previous condition has the so-called proper form.     

Gauge functions,Eikonal equations and Bôcher’s theorem on stratified Lie groups

Let \({\mathcal{L} = \sum_{i=1}^m X_i^2}\) be a real sub-Laplacian on a Carnot group \({\mathbb{G}}\) and denote by \({\nabla_\mathcal{L} = (X_1,\ldots,X_m)}\) the intrinsic gradient related to \({\mathcal{L}}\). Our aim in this present paper is to analyze some features of the \({\mathcal{L}}\)-gauge functions on \({\mathbb{G}}\), i.e., the homogeneous functions d such that \({\mathcal{L}(d^\gamma) = 0}\) in \({\mathbb{G} \setminus \{0\}}\) , for some \({\gamma \in \mathbb{R} \setminus \{0\}}\). We consider the relation of \({\mathcal{L}}\)-gauge functions with: the \({\mathcal{L}}\)-Eikonal equation \({|\nabla_\mathcal{L} u| = 1}\) in \({\mathbb{G}}\); the Mean Value Formulas for the \({\mathcal{L}}\)-harmonic functions; the fundamental solution for \({\mathcal{L}}\); the Bôcher-type theorems for nonnegative \({\mathcal{L}}\)-harmonic functions in “punctured” open sets \({\dot \Omega:= \Omega \setminus \{x_0\}}\).     

Tempered representations of <Emphasis Type="Italic">p</Emphasis>-adic groups: special idempotents and topology

Let \(G=G(k)\) be a connected reductive group over a p-adic field k. The smooth (and tempered) complex representations of G can be considered as the nondegenerate modules over the Hecke algebra \({\mathcal {H}}={\mathcal {H}}(G)\) and the Schwartz algebra \({\mathcal {S}}={\mathcal {S}}(G)\) forming abelian categories \({\mathcal {M}}(G)\) and \({\mathcal {M}}^t(G)\), respectively. Idempotents \(e\in {\mathcal {H}}\) or \({\mathcal {S}}\) define full subcategories \({\mathcal {M}}_e(G)= \{V : {\mathcal {H}}eV=V\}\) and \({\mathcal {M}}_e^t(G)= \{V : {\mathcal {S}}eV=V\}\). Such an e is said to be special (in \({\mathcal {H}}\) or \({\mathcal {S}}\)) if the corresponding subcategory is abelian. Parallel to Bernstein’s result for \(e\in {\mathcal {H}}\) we will prove that, for special \(e \in {\mathcal {S}}\), \({\mathcal {M}}_e^t(G) = \prod _{\Theta \in \theta _e} {\mathcal {M}}^t(\Theta )\) is a finite direct product of component categories \({\mathcal {M}}^t(\Theta )\), now referring to connected components of the center of \({\mathcal {S}}\). A special \(e\in {\mathcal {H}}\) will be also special in \({\mathcal {S}}\), but idempotents \(e\in {\mathcal {H}}\) not being special can become special in \({\mathcal {S}}\). To obtain conditions we consider the sets \(\mathrm{Irr}^t(G) \subset \mathrm{Irr}(G)\) of (tempered) smooth irreducible representations of G, and we view \(\mathrm{Irr}(G)\) as a topological space for the Jacobson topology defined by the algebra \({\mathcal {H}}\). We use this topology to introduce a preorder on the connected components of \(\mathrm{Irr}^t(G)\). Then we prove that, for an idempotent \(e \in {\mathcal {H}}\) which becomes special in \({\mathcal {S}}\), its support \(\theta _e\) must be saturated with respect to that preorder. We further analyze the above decomposition of \({\mathcal {M}}_e^t(G)\) in the case where G is k-split with connected center and where \(e = e_J \in {\mathcal {H}}\) is the Iwahori idempotent. Here we can use work of Kazhdan and Lusztig to relate our preorder on the support \(\theta _{e_J}\) to the reverse of the natural partial order on the unipotent classes in G. We finish by explicitly computing the case \(G=GL_n\), where \(\theta _{e_J}\) identifies with the set of partitions of n. Surprisingly our preorder (which is a partial order now) is strictly coarser than the reverse of the dominance order on partitions.     

A decomposition theorem for Lie ideals in nest algebras

Let \({\mathcal {N}}\) be a nest and let \({\mathcal {L}}\) be a weakly closed Lie ideal of the nest algebra \({\mathcal {T} (\mathcal {N})}\) . We explicitly construct the greatest weakly closed associative ideal \({\mathcal {J} (\mathcal {L})}\) contained in \({\mathcal {L}}\) and show that \({\mathcal {J} (\mathcal {L}) \subseteq \mathcal {L} \subseteq \mathcal {J} (\mathcal {L})\oplus {\breve{\mathcal{D}}} (\mathcal {L})}\) , where \({{\breve{\mathcal{D}}}} (\mathcal {L})\) is an appropriate subalgebra of the diagonal \({\mathcal {D} (\mathcal {N})}\) of the nest algebra \({\mathcal {T} (\mathcal {N})}\) . We show that norm-preserving linear extensions of elements of the dual of \({\mathcal {L}}\) , satisfying a certain condition, are uniquely determined on the diagonal of the nest algebra by the ideal \({\mathcal {J} (\mathcal {L})}\) .     

Some remarks on energy inequalities for harmonic maps with potential

We call the \({\delta}\)-vector of an integral convex polytope of dimension d flat if the \({\delta}\)-vector is of the form \({(1,0,\ldots,0,a,\ldots,a,0,\ldots,0)}\), where \({a \geq 1}\). In this paper, we give the complete characterization of possible flat \({\delta}\)-vectors. Moreover, for an integral convex polytope \({\mathcal{P}\subset \mathbb{R}^N}\) of dimension d, we let \({i(\mathcal{P},n)=|n\mathcal{P}\cap \mathbb{Z}^N|}\) and \({i^*(\mathcal{P},n)=|n(\mathcal{P} {\setminus}\partial \mathcal{P})\cap \mathbb{Z}^N|}\). By this characterization, we show that for any \({d \geq 1}\) and for any \({k,\ell \geq 0}\) with \({k+\ell \leq d-1}\), there exist integral convex polytopes \({\mathcal{P}}\) and \({\mathcal{Q}}\) of dimension d such that (i) For \({t=1,\ldots,k}\), we have \({i(\mathcal{P},t)=i(\mathcal{Q},t),}\) (ii) For \({t=1,\ldots,\ell}\), we have \({i^*(\mathcal{P},t)=i^*(\mathcal{Q},t)}\), and (iii) \({i(\mathcal{P},k+1) \neq i(\mathcal{Q},k+1)}\) and \({i^*(\mathcal{P},\ell+1)\neq i^*(\mathcal{Q},\ell+1)}\).     

Homomorphisms of $$\ell ^1$$-Munn algebras and Rees matrix semigroup algebras

Let \({\mathcal {LM}}\left( {\mathcal {A}}, P\right) \) be an \(\ell ^1\)-Munn algebra over an arbitrary unital Banach algebra \({\mathcal {A}}\). We characterize homomorphisms from \({\mathcal {LM}}\left( {\mathcal {A}}, P\right) \) into an arbitrary Banach algebra \({\mathcal {B}}\) in terms of homomorphisms from \({\mathcal {A}}\) into \({\mathcal {B}}\). Then we discuss homomorphisms from arbitrary Banach algebras into \({\mathcal {LM}}\left( {\mathcal {A}}, P\right) \). Existence and uniqueness of homomorphisms under certain conditions are also discussed. We apply these results to the concrete case of \(\ell ^1(S)\) where S is a Rees matrix semigroup, to identify characters of \(\ell ^1(S)\) in both cases where S is with or without zero. As a consequence if the sandwich matrix of S has a zero entry, then \(\ell ^1(S)\) is character amenable.     

Relative Igusa-Todorov Functions and Relative Homological Dimensions

We develope the theory of \({\mathcal {E}}\)-relative Igusa-Todorov functions in an exact I T-context \(({\mathcal {C}},{\mathcal {E}})\) (see Definition 2.1). In the case when \({\mathcal {C}}={\text {mod}}\, ({\Lambda })\) is the category of finitely generated left Λ-modules, for an artin algebra Λ, and \({\mathcal {E}}\) is the class of all exact sequences in \({\mathcal {C}},\) we recover the usual Igusa-Todorov functions, Igusa K. and Todorov G. (2005). We use the setting of the exact structures and the Auslander-Solberg relative homological theory to generalise the original Igusa-Todorov’s results. Furthermore, we introduce the \({\mathcal {E}}\)-relative Igusa-Todorov dimension and also we obtain relationships with the relative global and relative finitistic dimensions and the Gorenstein homological dimensions.     

Global determinism of semigroups having regular globals

The aim of this paper is to study the global determinism of the class \({\mathcal {A}}\) of all semigroups having regular globals. It is known from PeliKán (Periodica Math Hungarica 4:103–106, 1973) and Pondělí?ek (On semigroups having regular globals, 1976) that \({\mathcal {A}}\) can be divided into two subclasses: the class \({\mathcal {A}}_{2}\) of all semigroups having idempotent globals and the class \({\mathcal {A}}_{3}\) of all semigroups having regular but non-idempotent globals. We prove that \({\mathcal {A}}_{2}\) is globally determined and that \({\mathcal {A}}_{3}\) satisfies the strong isomorphism property. This shows that \({\mathcal {A}}\) is globally determined.     

Exponential decay towards equilibrium and global classical solutions for nonlinear reaction–diffusion systems

We consider a system of reaction–diffusion equations describing the reversible reaction of two species \({\mathcal{U}}\), \({\mathcal{V}}\) forming a third species \({\mathcal{W}}\) and vice versa according to mass action law kinetics with arbitrary stoichiometric coefficients (equal or larger than one). Firstly, we prove existence of global classical solutions via improved duality estimates under the assumption that one of the diffusion coefficients of \({\mathcal{U}}\) or \({\mathcal{V}}\) is sufficiently close to the diffusion coefficient of \({\mathcal{W}}\). Secondly, we derive an entropy entropy-dissipation estimate, that is a functional inequality, which applied to global solutions of these reaction–diffusion systems proves exponential convergence to equilibrium with explicit rates and constants.     

A Beurling-Blecher-Labuschagne Theorem for Noncommutative Hardy Spaces Associated with Semifinite von Neumann Algebras

We prove a Beurling-Blecher-Labuschagne theorem for \({H^\infty}\)-invariant spaces of \({L^p(\mathcal{M},\tau)}\) when \({0 < p \leq\infty}\), using Arveson’s non-commutative Hardy space \({H^\infty}\) in relation to a von Neumann algebra \({\mathcal{M}}\) with a semifinite, faithful, normal tracial weight \({\tau}\). Using the main result, we are able to completely characterize all \({H^\infty}\)-invariant subspaces of \({L^p(\mathcal{M} \rtimes_\alpha \mathbb{Z},\tau)}\), where \({\mathcal{M} \rtimes_\alpha \mathbb{Z} }\) is a crossed product of a semifinite von Neumann algebra \({\mathcal{M}}\) by the integer group \({\mathbb{Z}}\), and \({H^\infty}\) is a non-selfadjoint crossed product of \({\mathcal{M}}\) by \({\mathbb{Z}^+}\). As an example, we characterize all \({H^\infty}\)-invariant subspaces of the Schatten p-class \({S^p(\mathcal{H})}\), where \({H^\infty}\) is the lower triangular subalgebra of \({B(\mathcal{H})}\), for each \({0 < p \leq\infty}\).     

Semantical conditions for the definability of functions and relations

Let \({\mathcal{L}\subseteq \mathcal{L}^\prime}\) be first order languages, let \({R \in \mathcal{L}^\prime- \mathcal{L}}\) be a relation symbol, and let \({\mathcal{K}}\) be a class of \({\mathcal{L}^\prime}\)-structures. In this paper, we present semantical conditions equivalent to the existence of an \({\mathcal{L}}\)-formula \({\varphi(\vec{x})}\) such that \({\mathcal{K}\vDash \varphi(\vec{x}) \leftrightarrow R(\vec{x})}\), where \({\varphi}\) has a specific syntactical form (e.g., quantifier free, positive and quantifier free, existential Horn, etc.). For each of these definability results for relations, we also present an analogous version for the definability of functions. Several applications to natural definability questions in universal algebra have been included; most notably definability of principal congruences. The paper concludes with a look at term-interpolation in classes of structures with the same techniques used for definability. Here we obtain generalizations of two classical term-interpolation results: Pixley’s theorem for quasiprimal algebras, and the Baker–Pixley Theorem for finite algebras with a majority term.     

M-$${\mathcal{}}$$-Injective (Flat) and Strongly $${\mathcal{B}}$$B-Injective (Flat) Modules

Let M be a left R-module, \({\mathcal{A}}\)be a family of some submodules of M and \({\mathcal{B}}\)be a family of some left R-modules. In this article, we introduce and characterize \({\mathcal{A}}\)-coherent, \({P\mathcal{A}}\), \({F\mathcal{A}}\), M-\({\mathcal{A}}\)-injective (flat) and strongly \({\mathcal{B}}\)-injective (flat) modules, which are generalizations of coherent, PS, FS, M-injective (flat) and strongly M-injective modules, respectively. We extend some known results to this general structure.     

Local derivations of full matrix rings

Let \({\mathcal{R}}\) be a unital commutative ring and \({\mathcal{M}}\) be a 2-torsion free central \({\mathcal{R}}\) -bimodule. In this paper, for \({n \geqq 3}\), we show that every local derivation from M _n (\({\mathcal{R}}\)) into M _n (\({\mathcal{M}}\)) is a derivation.