Arens regularity of projective tensor products

For completely contractive Banach algebras A and B (respectively operator algebras A and B), the necessary and sufficient conditions for the operator space projective tensor product \({A\widehat{\otimes}B}\) (respectively the Haagerup tensor product \({A\otimes^{h}B}\)) to be Arens regular are obtained. Using the non-commutative Grothendieck inequality, we show that, for C*-algebras A and B, \({A\otimes^{\gamma} B}\) is Arens regular if \({A\widehat{\otimes}B}\) and \({A\widehat{\otimes}B^{op}}\) are Arens regular whereas \({A\widehat{\otimes}B}\) is Arens regular if and only if \({A\otimes^{h}B}\) and \({B\otimes^{h}A}\) are, where \({\otimes^h}\), \({\otimes^{\gamma}}\), and \({\widehat{\otimes}}\) are the Haagerup, the Banach space projective tensor norm, and the operator space projective tensor norm, respectively.     

Combinatorial Yang–Baxter maps arising from the tetrahedron equation

We survey the matrix product solutions of the Yang–Baxter equation recently obtained from the tetrahedron equation. They form a family of quantum R-matrices of generalized quantum groups interpolating the symmetric tensor representations of U_q(A_n?1⁽¹⁾) and the antisymmetric tensor representations of \({U_{ - {q^{ - 1}}}}\left( {A_{n - 1}^{\left( 1 \right)}} \right)\). We show that at q = 0, they all reduce to the Yang–Baxter maps called combinatorial R-matrices and describe the latter by an explicit algorithm.     

On the structure of a mutually permutable product of finite groups

Let a finite group \({G = AB}\) be the product of the mutually permutable subgroups A and B. We investigate the structure of G given by conditions on conjugacy class sizes of elements in \({A \cup B}\) . Some recent results are extended.     

On semiconjugate rational functions

We investigate semiconjugate rational functions, that is rational functions A, B related by the functional equation \({A \circ X = X \circ B}\), where X is a rational function. We show that if A and B is a pair of such functions, then either A can be obtained from B by a certain iterative process, or A and B can be described in terms of orbifolds of non-negative Euler characteristic on the Riemann sphere.     

Completely representable lattices

It is known that a lattice is representable as a ring of sets iff the lattice is distributive. CRL is the class of bounded distributive lattices (DLs) which have representations preserving arbitrary joins and meets. jCRL is the class of DLs which have representations preserving arbitrary joins, mCRL is the class of DLs which have representations preserving arbitrary meets, and biCRL is defined to be \({{\bf jCRL} \cap {\bf mCRL}}\) . We prove

${\bf CRL} \subset {\bf biCRL} = {\bf mCRL} \cap {\bf jCRL} \subset {\bf mCRL} \neq {\bf jCRL} \subset {\bf DL}$

where the marked inclusions are proper.

Let L be a DL. Then \({L \in {\bf mCRL}}\) iff L has a distinguishing set of complete, prime filters. Similarly, \({L \in {\bf jCRL}}\) iff L has a distinguishing set of completely prime filters, and \({L \in {\bf CRL}}\) iff L has a distinguishing set of complete, completely prime filters.Each of the classes above is shown to be pseudo-elementary, hence closed under ultraproducts. The class CRL is not closed under elementary equivalence, hence it is not elementary.     

Tensor products of tilting modules

We consider whether the tilting properties of a tilting A-module T and a tilting B-module T′ can convey to their tensor product T ? T′: The main result is that T ? T′ turns out to be an (n + m)-tilting A ? B-module, where T is an m-tilting A-module and T′ is an n-tilting B-module.     

A note on generating <Emphasis Type="Italic">P</Emphasis>-matrices

We prove that for any \({A,B\in\mathbb{R}^{n\times n}}\) such that each matrix S satisfying min(A, B) ≤ S ≤ max(A, B) is nonsingular, all four matrices A ^?1 B, AB ^?1, B ^?1 A and BA ^?1 are P-matrices. A practical method for generating P-matrices is drawn from this result.     

Symmetrization of Cuntz’ Picture for the Kasparov <Emphasis Type="Italic">KK</Emphasis>-Bifunctor

Given C^*-algebras A and B, we generalize the notion of a quasi-homomorphism from A to B in the sense of Cuntz by considering quasi-homomorphisms from some C^*-algebra C to B such that C surjects onto A and the two maps forming the quasi-homomorphism agree on the kernel of this surjection. Under an additional assumption, the group of homotopy classes of such generalized quasi-homomorphisms coincides with KK(A, B). This makes the definition of the Kasparov bifunctor slightly more symmetric and provides more flexibility in constructing elements of KK-groups. These generalized quasi-homomorphisms can be viewed as pairs of maps directly from A (instead of various C’s), but these maps need not be *-homomorphisms.     

<Emphasis Type="Italic">θ</Emphasis>-Pairs for 2-maximal subgroups of finite groups

Let H, A and B be subgroups of a group G. We call the pair (A, B) a θ-pair for H in G if: (i) \({\langle H, A\rangle=G}\) and B = (A ∩ H) _G ; (ii) if A ₁/B is a proper subgroup of A/B and \({{A_1/B \vartriangleleft G/B}}\), then \({G\neq \langle H, A_1\rangle}\). In this paper, we study the θ-pairs for 2-maximal subgroups of a group, which imply a group to be solvable or supersolvable.     

Maps preserving common zeros between subspaces of vector-valued continuous functions

For metric spaces X and Y, normed spaces E and F, and certain subspaces A(X, E) and A(Y, F) of vector-valued continuous functions, we obtain a complete characterization of linear and bijective maps \({T:A(X,E)\rightarrow A(Y,F)}\) preserving common zeros, that is, maps satisfying the property

$Z(f) \cap Z(g) \neq \emptyset \Longleftrightarrow Z(Tf) \cap Z(Tg) \neq \emptyset \quad\quad\quad{\rm (P)}$

for any \({f, g \in A(X, E)}\), where \({Z(f) = \{x \in X: f(x) = 0\}}\). Moreover, we provide some examples of subspaces for which the automatic continuity of linear bijections having the property (P) is derived.

     

Rational approximation to surfaces defined by polynomials in one variable

For a Tychonoff space X, we denote by C _p (X) the space of all real-valued continuous functions on X with the topology of pointwise convergence.

In this paper we prove that:

• If every finite power of X is Lindelöf then C _p (X) is strongly sequentially separable iff X is \({\gamma}\)-set.
• \({B_{\alpha}(X)}\) (= functions of Baire class \({\alpha}\) (\({1 < \alpha \leq \omega_1}\)) on a Tychonoff space X with the pointwise topology) is sequentially separable iff there exists a Baire isomorphism class \({\alpha}\) from a space X onto a \({\sigma}\)-set.
• \({B_{\alpha}(X)}\) is strongly sequentially separable iff \({iw(X)=\aleph_0}\) and X is a \({Z^{\alpha}}\)-cover \({\gamma}\)-set for \({0 < \alpha \leq \omega_1}\).
• There is a consistent example of a set of reals X such that C _p (X) is strongly sequentially separable but B₁(X) is not strongly sequentially separable.
• B(X) is sequentially separable but is not strongly sequentially separable for a \({\mathfrak{b}}\)-Sierpiński set X.

     

Twisted Frobenius Extensions of Graded Superrings

We define twisted Frobenius extensions of graded superrings. We develop equivalent definitions in terms of bimodule isomorphisms, trace maps, bilinear forms, and dual sets of generators. The motivation for our study comes from categorification, where one is often interested in the adjointness properties of induction and restriction functors. We show that A is a twisted Frobenius extension of B if and only if induction of B-modules to A-modules is twisted shifted right adjoint to restriction of A-modules to B-modules. A large (non-exhaustive) class of examples is given by the fact that any time A is a Frobenius graded superalgebra, B is a graded subalgebra that is also a Frobenius graded superalgebra, and A is projective as a left B-module, then A is a twisted Frobenius extension of B.     

Representing some families of monotone maps by principal lattice congruences

For a lattice L with 0 and 1, let Princ(L) denote the set of principal congruences of L. Ordered by set inclusion, it is a bounded ordered set. In 2013, G. Grätzer proved that every bounded ordered set is representable as Princ(L); in fact, he constructed L as a lattice of length 5. For {0, 1}-sublattices \({A \subseteq B}\) of L, congruence generation defines a natural map Princ(A) \({\longrightarrow}\) Princ(B). In this way, every family of {0, 1}-sublattices of L yields a small category of bounded ordered sets as objects and certain 0-separating {0, 1}-preserving monotone maps as morphisms such that every hom-set consists of at most one morphism. We prove the converse: every small category of bounded ordered sets with these properties is representable by principal congruences of selfdual lattices of length 5 in the above sense. As a corollary, we can construct a selfdual lattice L in G. Grätzer's above-mentioned result.     

Secondary Hochschild Cohomology

For a B-algebra A we introduce a Hochschild-like cohomology and use it to describe simultaneous deformations of the product and of the B-algebra structure on A[[t]]. These deformations have the property that the natural projection A[[t]]→A is a morphism of B-algebras.     

Ranks of matrices with few distinct entries

It is consistent that P(ω ₁) is the union of less than \({2^{{\aleph _1}}}\) parts such that if A ₀,..., A _n?1, B ₀,..., B _m?1 are distinct elements of the same part, then |A ₀ ∩ · · · ∩ A _n?1 ∩ (ω ₁ ? B ₀) ∩ · · ·∩ (ω ₁ ? B _m?1)| = N₁.     

Lattices of topologies of unary algebras of the variety $$ \mathcal{A}_{1,1} $$

We consider the variety of unary algebras 〈A, f, g〉 defined by the identities f(g(x)) = g(f(x)) = x. We describe algebras of this variety, whose topology lattices are modular, distributive, linearly ordered, complemented, or pseudocomplemented.     

Common best proximity point theorems: Global minimization of some real-valued multi-objective functions

Let us deliberate the question of computing a solution to the problems that can be articulated as the simultaneous equations \({Sx = x}\) and \({Tx = x}\) in the framework of metric spaces. However, when the mappings in context are not necessarily self-mappings, then it may be consequential that the equations do not have a common solution. At this juncture, one contemplates to compute a common approximate solution of such a system with the least possible error. Indeed, for a common approximate solution \({x^*}\) of the equations, the real numbers \({d(x^*, Sx^*)}\) and \({d(x^*,Tx^*)}\) measure the errors due to approximation. Eventually, it is imperative that one pulls off the global minimization of the multiobjective functions \({x \rightarrow d(x, Sx)}\) and \({x \rightarrow d(x, Tx)}\). When S and T are mappings from A to B, it follows that \({d(x, Sx) \geq d(A, B)}\) and \({d(x, Tx) \geq d(A, B)}\) for every \({x \in A}\). As a result, the global minimum of the aforesaid problem shall be actualized if it is ascertained that the functions \({x \rightarrow d(x, Sx)}\) and \({x \rightarrow d(x, Tx)}\) attain the lowest possible value d(A, B). The target of this paper is to resolve the preceding multiobjective global minimization problem when S is a T-cyclic contraction or a generalized cyclic contraction, thereby enabling one to determine a common optimal approximate solution to the aforesaid simultaneous equations.     

Effect Algebras as Presheaves on Finite Boolean Algebras

For an effect algebra A, we examine the category of all morphisms from finite Boolean algebras into A. This category can be described as a category of elements of a presheaf R(A) on the category of finite Boolean algebras. We prove that some properties (being an orthoalgebra, the Riesz decomposition property, being a Boolean algebra) of an effect algebra A can be characterized in terms of some properties of the category of elements of the presheaf R(A). We prove that the tensor product of effect algebras arises as a left Kan extension of the free product of finite Boolean algebras along the inclusion functor. The tensor product of effect algebras can be expressed by means of the Day convolution of presheaves on finite Boolean algebras.     

On 2-blocks with {k(B) - l(B) = 1}

We investigate some properties of Cartan matrices of symmetric algebras. In particular, we study the Cartan matrices of p-blocks B of finite groups for the cases that \({k(B) - l(B) = 1}\) and that \({k(B) = 3}\) where k(B) and l(B) are the numbers of irreducible ordinary and Brauer characters associated to B, respectively.     

Best proximity points: global optimal approximate solutions

Let A and B be non-empty subsets of a metric space. As a non-self mapping \({T:A\longrightarrow B}\) does not necessarily have a fixed point, it is of considerable interest to find an element x in A that is as close to Tx in B as possible. In other words, if the fixed point equation Tx = x has no exact solution, then it is contemplated to find an approximate solution x in A such that the error d(x, Tx) is minimum, where d is the distance function. Indeed, best proximity point theorems investigate the existence of such optimal approximate solutions, called best proximity points, to the fixed point equation Tx = x when there is no exact solution. As the distance between any element x in A and its image Tx in B is at least the distance between the sets A and B, a best proximity pair theorem achieves global minimum of d(x, Tx) by stipulating an approximate solution x of the fixed point equation Tx = x to satisfy the condition that d(x, Tx) = d(A, B). The purpose of this article is to establish best proximity point theorems for contractive non-self mappings, yielding global optimal approximate solutions of certain fixed point equations. Besides establishing the existence of best proximity points, iterative algorithms are also furnished to determine such optimal approximate solutions.