Local and 2-Local Lie Derivations of Operator Algebras on Banach Spaces

Let X be a Banach space of dimension > 2. We show that every local Lie derivation of B(X) is a Lie derivation, and that a map of B(X) is a 2-local Lie derivation if and only if it has the form ${A \mapsto AT - TA + \psi(A)}$ A ? A T - T A + ψ ( A ) , where ${T \in B(X)}$ T ∈ B ( X ) and ψ is a homogeneous map from B(X) into ${\mathbb{F}I}$ F I satisfying ${\psi(A + B) = \psi(A)}$ ψ ( A + B ) = ψ ( A ) for ${A, B \in B(X)}$ A , B ∈ B ( X ) with B being a sum of commutators.     

New estimates of continuity in $$M/GI/1/ \infty$$ queues

The paper deals with an estimation of the total variation distance between stationary distributions of waiting time in two queueing systems with equal Poisson inputs and different distributions B and $\widetilde B$ of service time. Assuming equality of two first moments of B and $\widetilde B$ the continuity inequalities are derived in terms of difference pseudomoments of B and $\widetilde B$ . When in addition the third moments of B and $\widetilde B$ coincide then the constant involved in the corresponding inequality has the asymptotics ${\text{O}}\left[ {\left( {1 - \rho } \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}} } \right]$ in the heavy traffic limit $\rho \to 1$ .     

Additivity of homological dimensions for tensor products of some Banach algebras

It is proved that if A = C(Ω), where Ω is an infinite metrizable compact space such that some finite-order iterated derived set of Ω is empty, then for every unital Banach algebra B the global dimensions and the bidimensions of the Banach algebras A \(\hat \otimes \) B and B are related as dg A \(\hat \otimes \) B = 2 + dg B and db A \(\hat \otimes \) B = 2 + db B. Thus, a partial extension of Selivanov’s result is obtained.     

A Sequence to Compute the Brauer Group of Certain Quasi-Triangular Hopf Algebras

A deeper understanding of recent computations of the Brauer group of Hopf algebras is attained by explaining why a direct product decomposition for this group holds and describing the non-interpreted factor occurring in it. For a Hopf algebra B in a braided monoidal category ${{\mathcal C}}$ , and under certain assumptions on the braiding (fulfilled if ${{\mathcal C}}$ is symmetric), we construct a sequence for the Brauer group ${{\rm{BM}}}({{\mathcal C}};B)$ of B-module algebras, generalizing Beattie’s one. It allows one to prove that ${{\rm{BM}}}({{\mathcal C}};B) \cong {{\rm{Br}}}({{\mathcal C}}) \times {\operatorname{Gal}}({{\mathcal C}};B)$ , where ${{\rm{Br}}}({{\mathcal C}})$ is the Brauer group of ${{\mathcal C}}$ and ${\operatorname{Gal}}({{\mathcal C}};B)$ the group of B-Galois objects. We also show that ${{\rm{BM}}}({{\mathcal C}};B)$ contains a subgroup isomorphic to ${{\rm{Br}}}({{\mathcal C}}) \times {\operatorname{H^2}}({{\mathcal C}};B,I),$ where ${\operatorname{H^2}}({{\mathcal C}};B,I)$ is the second Sweedler cohomology group of B with values in the unit object I of ${{\mathcal C}}$ . These results are applied to the Brauer group ${{\rm{BM}}}(K,B \times H,{{\mathcal R}})$ of a quasi-triangular Hopf algebra that is a Radford biproduct B × H, where H is a usual Hopf algebra over a field K, the Hopf subalgebra generated by the quasi-triangular structure ${{\mathcal R}}$ is contained in H and B is a Hopf algebra in the category ${}_H{{\mathcal M}}$ of left H-modules. The Hopf algebras whose Brauer group was recently computed fit this framework. We finally show that ${{\rm{BM}}}(K,H,{{\mathcal R}}) \times {\operatorname{H^2}}({}_H{{\mathcal M}};B,K)$ is a subgroup of ${{\rm{BM}}}(K,B \times H,{{\mathcal R}})$ , confirming the suspicion that a certain cohomology group of B × H (second lazy cohomology group was conjectured) embeds into it. New examples of Brauer groups of quasi-triangular Hopf algebras are computed using this sequence.     

Proof of the BMV conjecture

We prove the BMV (Bessis, Moussa, Villani, [1]) conjecture, which states that the function ${t \mapsto \mathop{\rm Tr}\exp(A-tB)}$ , ${t \geqslant 0}$ , is the Laplace transform of a positive measure on [0,∞) if A and B are ${n \times n}$ Hermitian matrices and B is positive semidefinite. A semi-explicit representation for this measure is given.     

Walker groups

A reformulation of Walker’s theorem on the cancellation of \(\mathbf {Z}\) says that any two homomorphisms from an abelian group W onto \(\mathbf {Z}\) have isomorphic kernels. It does not have a constructive proof, even for W a subgroup of \(\mathbf {Z}^{3}.\) In this paper we give a constructive proof of Walker’s theorem for W a direct sum, over any discrete index set, of groups of the following two kinds: Butler groups with weakly computable heights, and finite-rank torsion-free groups B with computable relative heights (that is, all quotients of B by finite-rank pure subgroups have computable heights). Throughout, “group” means abelian group. The infinite cyclic group, and the ring of integers, is denoted by \(\mathbf {Z}.\) The nonnegative integers are denoted by \(\mathbf {N},\) the positive integers by \(\mathbf {Z}^{+},\) and the rational numbers by \(\mathbf {Q}.\) We say that a set is discrete if any two elements are either equal or different. A subset A of a set B is detachable (from B) if for each \(b\in B,\) either \(b\in A\) or \(b\notin A.\) A group is discrete if and only if its subset \(\{0\}\) is detachable. Walker, in his dissertation [7], and Cohn in [2], showed that \(\mathbf {Z}\) is cancellable in the sense that if \(\mathbf {Z}\oplus B\cong \mathbf {Z}\oplus B^{\prime },\) then \(B\cong B^{\prime }.\) It is somewhat of an oddity that \(\mathbf {Z}\) is cancellable. A rank-one torsion-free group A is cancellable if and only if \(A\cong \mathbf {Z}\) or the endomorphism ring of A has stable range one [3], [1, Theorem 8.12]. (A ring R has stable range one if whenever \(aR+bR=R,\) then \(a+bR\) contains a unit of R.) In fact, any object in an abelian category whose endomorphism ring has stable range one is cancellable. The endomorphism ring of \(\mathbf {Z}\) does not have stable range one, so \(\mathbf {Z}\) is the unique rank-one torsion-free group that is cancellable for some reason other than its endomorphism ring. Walker’s theorem can be reformulated to say that any two maps from an abelian group W onto \(\mathbf {Z}\) have isomorphic kernels. Accordingly, we define a Walker group to be such a group W. Of course, Walker’s theorem says that every abelian group is a Walker group. However, a counterexample in the (abelian) category of diagrams \(\cdot \rightarrow \cdot \rightarrow \cdot \) of abelian groups provides a Kripke model which shows that there is no constructive proof that even every subgroup of \(\mathbf {Z}^{3}\) is a Walker group [4]. Thus, from a constructive point of view, it is of interest to explore the class of Walker groups. We will say that a group is a cZ-group if every homomorphism from it into \(\mathbf {Z}\) has a cyclic image. An easy classical argument shows that every group is a cZ-group. It is an immediate (constructive) consequence of [4, Theorem 1] that every cZ-group is a Walker group. This is not a complete triviality because it provides a classical proof of Walker’s theorem! The question remains as to how extensive the class of cZ-groups is. That question motivated the current paper. Our main results along these lines are Theorem 1.3, which says that Butler groups with weakly computable heights are cZ-groups, and Theorem 1.6, which says that a finite-rank torsion-free group with computable relative heights is a cZ-group (Butler groups with computable heights have computable relative heights). The relevance of the ability to compute heights to the study of Walker groups was suggested by the fact that this was not possible for the group corresponding to the counterexample. The notions of weakly computable heights and computable heights already appeared in [5, 6], papers that are over 20 years old. The notion of computable relative heights is stronger than these and originates in the current paper, just after Theorem 1.3. Note that B is a cZ-group if and only if \(\mathbf {Z}\oplus B\) is a cZ-group. Finitely generated groups are clearly cZ-groups. Finite direct sums of cZ-groups, and quotients of cZ-groups, are cZ-groups. As any map into \(\mathbf {Z}\) kills all torsion elements, we will focus on torsion-free groups B. However, not even nonzero subgroups of \(\mathbf {Z}\) need be cZ-groups: for example, \(\{x\in \mathbf {Z}:\,x\,\mathrm{is\,even,\, or}\,P\}.\) In Sect. 2 we show that a direct sum of cZ-groups over a discrete index set is a Walker group (Corollary 2.3). This gives essentially the largest class of Walker groups that we currently know (Corollary 2.4), although in [4, Theorem 5] it was shown that if B is a torsion-free group such that every nonzero map from B into \(\mathbf {Z}\) is one-to-one, then \(\mathbf {Z}\oplus B\) is a Walker group. Rank-one torsion-free groups B have that property, as do subgroups of \(\mathbf {Z},\) and any group with no nontrivial maps into \(\mathbf {Z}.\) The question regarding a group B that is a finite direct sum of such groups was left open, and is still open as far as I know. Section 3 deals with the idea of the height of a subgroup. This idea arose in an effort to formulate a strong height condition that would imply that a group was a cZ-group. That approach failed and was replaced by the notion of computable relative heights. However, I still feel that the idea is interesting and may prove fruitful for some other purpose.     

The Two-Weighted Inequalities for Sublinear Operators Generated by B Singular Integrals in Weighted Lebesgue Spaces

In this paper, the authors establish several general theorems for the boundedness of sublinear operators (B sublinear operators) satisfies the condition (1.2), generated by B singular integrals on a weighted Lebesgue spaces $L_{p,\omega,\gamma}(\mathbb{R}_{k,+}^{n})$ , where $B=\sum_{i=1}^{k} (\frac{\partial^{2}}{\partial x_{k}^{2}} + \frac{\gamma_{i}}{x_{i}}\frac{\partial}{\partial x_{i}} )$ . The condition (1.2) are satisfied by many important operators in analysis, including B maximal operator and B singular integral operators. Sufficient conditions on weighted functions ω and ω ₁ are given so that B sublinear operators satisfies the condition (1.2) are bounded from $L_{p,\omega,\gamma}(\mathbb{R}_{k,+}^{n})$ to $L_{p,\omega_{1},\gamma}(\mathbb{R}_{k,+}^{n})$ .     

Additivity of homological dimensions for a class of Banach algebras

Let Ω be a metrizable compact space. Suppose that its derived set of some finite order is empty. Let B be a unital Banach algebra, and let $\widehat \otimes $ stand for the projective tensor product. We prove the additivity formulas dg C(Ω)B $\widehat \otimes $ =dgB and db C(Ω) $\widehat \otimes $ B=dbC(Ω)+dbB for the global homological dimension and the homological bidimension. Thus these formulas are true for a new class of commutative Banach algebras in addition to those considered earlier by Selivanov.     

Expansion of open sets and decomposition of continuous mappings

The notion of expansionA of open sets is introduced. ThenA-expansion continuous mappingf:X→Y is defined. The main result of this note is that a mappingf is continuous if and only if it is bothA-expansion continuous andB-expansion continuous, whereA-expansion,B-expansion are two mutually dual expansions.     

Hall-type representations for generalised orthogroups

An orthogroup is a completely regular orthodox semigroup. The main purpose of this paper is to find a representation of a (generalised) orthogroup with band of idempotents B in terms of a fundamental (generalised) orthogroup. The latter is a subsemigroup of the Hall semigroup W _B (or of its generalisations V _B ,U _B and S _B ). We proceed in the regular case by constructing a fundamental completely regular subsemigroup \(\overline{W_{B}}\) of W _B , using two different methods. Our subsemigroup plays the role for orthogroups that W _B plays for orthodox semigroups, in that it contains a representation of every orthogroup with band of idempotents B, with kernel of the representation being μ, the greatest congruence contained in \(\mathcal{H}\) . To develop an analogous theory for classes of generalised orthogroups, that is, to extend beyond the regular case, we replace \(\mathcal{H}\) by \(\widetilde{\mathcal{H}}_{B}\) . Generalised orthogroups are then classes of weakly B-superabundant semigroups with (C). We first consider those satisfying an idempotent connected condition (IC) or (WIC). We construct fundamental weakly B-superabundant subsemigroups \(\overline{V_{B}}\) (respectively, \(\overline{U_{B}}\) ) of V _B (respectively, U _B ) with (C) and (IC) (respectively, with (C) and (WIC)) such that any weakly B-superabundant semigroup with (C) and (IC) (respectively, with (C) and (WIC)) admits a representation to \(\overline{V_{B}}\) (respectively, \(\overline{U_{B}}\) ), with kernel of the respresentation being μ _B , the greatest congruence contained in \(\widetilde{\mathcal{H}}_{B}\) . Finally, we remove the idempotent connected condition and find a representation for an arbitrary weakly B-superabundant semigroup with (C), making use of fresh technology, constructing a fundamental weakly B-superabundant subsemigroup \(\overline{S_{B}}\) of S _B , with the appropriate universal properties. We note that our results are needed in a parallel paper to complete the representation of arbitrary weakly B-superabundant semigroups with (C) as spined products of superabundant Ehresmann semigroups and subsemigroups of S _B .     

Reduced Load Equivalence under Subexponentiality

The stationary workload W _A+B ^φ of a queue with capacity φ loaded by two independent processes A and B is investigated. When the probability of load deviation in process A decays slower than both in B and $e^{ - \sqrt x } $ , we show that W _A+B ^φ is asymptotically equal to the reduced load queue W _A ^φ?b , where b is the mean rate of B. Given that this property does not hold when both processes have lighter than $e^{ - \sqrt x } $ deviation decay rates, our result establishes the criticality of $e^{ - \sqrt x } $ in the functional behavior of the workload distribution. Furthermore, using the same methodology, we show that under an equivalent set of conditions the results on sampling at subexponential times hold.     

Endomorphism algebras and Igusa–Todorov algebras

Let A be an Artin algebra. If $V\in \operatorname{mod} A$ such that the global dimension of  $\operatorname{End}_{A}V$ is at most 3, then for any ${M\in \operatorname{add}_{A}V}$ , both B and B ^op are 2-Igusa–Todorov algebras, where ${B=\operatorname{End}_{A}M}$ . Let ${P\in \operatorname{mod} A}$ be projective and ${B=\operatorname{End}_{A}P}$ such that the projective dimension of P as a right B-module is at most n(<∞). If A is an m-syzygy-finite algebra (resp. an m-Igusa–Todorov algebra), then B is an (m+n)-syzygy-finite algebra (resp. an (m+n)-Igusa–Todorov algebra); in particular, the finitistic dimension of B is finite in both cases. Some applications of these results are given.     

Maxima for the Expectation of the Lifetime of a Brownian Motion in the Ball

Let G _B (x, y) be the Green’s function of the unit ball B in ${\mathbb{R}^n, n \ge 3,}$ and ${\Gamma_B (x,y)=\int_BG_B(x, z)G_B(z, y)dz}$ the iterated Green’s function. The function $$E_x^y(\tau_B) = \frac{\Gamma_B(x, y)}{G_B(x, y)}$$ is the expectation of the lifetime of a Brownian motion starting at ${x \in \overline{B}}$ , killed on exiting B and conditioned to converge to and to be stopped at ${y \in \overline{B}}$ . The aim of the paper is to prove that $$\sup_{x \in \partial B,y \in B} E_x^y(\tau_B) = \sup_{x,y \in \partial B} E_x^y(\tau_B) = E_{x_0}^{-x_0}(\tau_B), x_0 \in\partial B$$ and that the maximum value of ${E_x^y(\tau_B)}$ occurs if and only if x, y are diametrically opposite points on the boundary of B.     

Projective Hausdorff gaps

Todor?evi? (Fund Math 150(1):55–66, 1996) shows that there is no Hausdorff gap (A, B) if A is analytic. In this note we extend the result by showing that the assertion “there is no Hausdorff gap (A, B) if A is coanalytic” is equivalent to “there is no Hausdorff gap (A, B) if A is ${{\bf \it{\Sigma}}^{1}_{2}}$ ”, and equivalent to ${\forall r \; (\aleph_1^{L[r]}\,< \aleph_1)}$ . We also consider real-valued games corresponding to Hausdorff gaps, and show that ${\mathsf{AD}_\mathbb{R}}$ for pointclasses Γ implies that there are no Hausdorff gaps (A, B) if ${{\it{A}} \in {\bf \it{\Gamma}}}$ .     

Uniforme Räume mit einer linear geordneten Basis

A. K. Steiner undE. F. Steiner described the socalled natural topology κ on spacesA ^B of transfinite sequences (a _β), β∈B,a _β∈A [J. Math. Anal. Appl.19, 174–178 (1967)]. These spaces generalize Baire's zerodimensional sequence-spaces. Using these spaces (A ^B, κ), we generalize two well known theorems of F. Hausdorff, W. Hurewicz, C. Kuratowski and K. Morita on metric spaces and their Lebesgue-dimension respectively, both involving Baire's sequence spaces. Thus we obtain a topological characterization of uniform spaces \((X,\mathfrak{U})\) with a linearly ordered base \(\mathfrak{B}\) of \(\mathfrak{U}\) .     

Automorphisms of {{\rm M_{n}}(\mathbb{D})}, Partially Ordered by Rank Subtractivity Ordering

Let \({\mathbb{D}}\) be an arbitrary division ring and \({{\rm M_{n}}(\mathbb{D})}\) be the set of all n × n matrices over \({\mathbb{D}}\) . We define the rank subtractivity or minus partial order on \({{\rm M_{n}}(\mathbb{D})}\) as defined on \({{\rm M_{n}}(\mathbb{C})}\) , i.e., \({A \leqslant B}\) iff rank(B) = rank(A) + rank(B?A). We describe the structure of maps Φ on \({{\rm M_{n}}(\mathbb{D})}\) such that \({A\leqslant B}\) iff \({\Phi(A)\leqslant \Phi(B) (A, B\in {\rm M_{n}}(\mathbb{D}) )}\) .