THE BANACH-LIE GROUP OF LIE TRIPLE AUTOMORPHISMS OF AN H^＊-ALGEBRA^＊

We study the Banach-Lie group Ltaut(A) of Lie triple automorphisms of a complex associative H*-algebra A. Some consequences about its Lie algebra, the algebra of Lie triple derivations of A, Ltder(A), are obtained. For a topologically simple A, in the infinite-dimensional case we have Ltaut(A)0 = Aut(A) implying Ltder(A) = Der(A). In the finite-dimensional case Ltaut(A)0 is a direct product of Aut(A) and a certain subgroup of Lie derivations δ from A to its center, annihilating commutators.     

ON REFLEXIVITY OF HYPONORMAL AND WEIGHTED SHIFT OPERATORS

By an elementary proof, we use a result of Conway and Dudziak to show that if A is a hyponormal operator with spectral radius r(A) such that its spectrum is the closed disc {z:|z| ≤ r(A)} then A is reflexive. Using this result, we give a simple proof of a result of Bercovici, Foias, and Pearcy on reflexivity of shift operators. Also, it is shown that every power of an invertible bilateral weighted shift is reflexive.     

Lattice tensor products. III - Congruences

G. Grätzer and F. Wehrung introduced the lattice tensor product, A?B, of the lattices Aand B. In Part I of this paper, we showed that for any finite lattice A, we can "coordinatize" A?B, that is, represent A?,B as a subset A of B ^A, and provide an effective criteria to recognize the A-tuples of elements of B that occur in this representation. To show the utility of this coordinatization, we prove, for a finite lattice A and a bounded lattice B, the isomorphism Con A ≌ (Con A)B>, which is a special case of a recent result of G. Grätzer and F. Wehrung and a generalization of a 1981 result of G. Grätzer, H. Lakser, and R.W. Quackenbush.     

Automorphism groups of trivial extensions

Given a split basic finite dimensional algebra A over a field, we study the relationship between the groups of categorical automorphisms of A and its trivial extension A?D(A). Our results cover all triangular algebras and all 2-nilpotent algebras whose quiver has no nontrivial oriented cycle of length ?2. In this latter as well as in the hereditary case, we give structure theorem for CAut(A?D(A)) in terms of CAut(A). As a byproduct, we get the precise relationship between the first Hochschild cohomology groups of A and A?D(A).     

A note on vanishing of the functor ext1 for Köthe spaces

In this note we study the relationship between the vanishing of Ext¹(λ(A), λ(A)) and the existence of a regular basis in the Köthe space λ(A). We construct an example of a nuclear Köthe space λ(A) with no regular basis and such that Ext¹(λ(A), λ(A))=0. Then we show that for some classes of Köthe spaces λ(A), the vanishing of Ext¹(λ(A), λ(A)) yields a regular basis for λ(A).     

Algebras generated by inner functions

Let A be a uniform algebra on a compact space X. An inner function is a function in A unimodular on X. For three algebras of type H^∞ we prove A is generated by its inner functions. Whenever A is generated by its inner functions we prove the unit ball of A is the closed convex ball of the inner functions.     

Convex vector lattices and l-algebras

For A an Archimedean Riesz space (=vector lattice) with distinguished positive weak unit e_A, we have the Yosida representation  as a Riesz space in D(X_A), the lattice of extended real valued functions on the space of e_A-maximal ideas. This note is about those A for which  is a convex subset of D(X_A); we call such A “convex”.Convex Riesz spaces arise from the general issue of embedding as a Riesz ideal, from consideration of uniform- and order-completeness, and from some problems involving comparison of maximal ideal spaces (which we won't discuss here; see [10]).The main results here are: (2.4) A is convex iff A is contained as a Riesz ideal in a uniformly complete Φ-algebra B with identity e_A. (3.1) Any A has a convex reflection (i.e., embeds into a convex B with a universal mapping property for Riesz homomorphisms; moreover, the embedding is epic and large).     

Tying up baric algebras

Given two baric algebras (A ₁, ?? ₁) and (A ₂, ?? ₂) we describe a way to define a new baric algebra structure over the vector space A ₁ ?? A ₂, which we shall denote (A ₁ ? A ₂, ?? ₁ ? ?? ₂). We present some easy properties of this construction and we show that in the commutative and unital case it preserves indecomposability. Algebras of the form A ₁ ? A ₂ in the associative, coutable-dimensional, zero-characteristic case are classified.     

Symmetric family of Fredholm operators of indices zero, stability of essential spectra and application to transport operators

In this paper, we prove that, if the product A=A₁?An is a Fredholm operator where the ascent and descent of A are finite, then Aj is a Fredholm operator of index zero for all j, 1?j?n, where A₁,…,An be a symmetric family of bounded operators. Next, we investigate a useful stability result for the Rako?evi?/Schmoeger essential spectra. Moreover, we show that some components of the Fredholm domains of bounded linear operators on a Banach space remain invariant under additive perturbations belonging to broad classes of operators A such as γ(Am)<1 where γ(⋅) is a measure of noncompactness. We also discuss the impact of these results on the behavior of the Rako?evi?/Schmoeger essential spectra. Further, we apply these latter results to investigate the Rako?evi?/Schmoeger essential spectra for singular neutron transport equations in bounded geometries.     

Morita context of weak Hopf algebras

Let H be a finite-dimensional weak Hopf algebra and A a left H-module algebra with its invariant subalgebra A~H.We prove that the smash product A#H is an A-ring with a grouplike character, and give a criterion for A#H to be Frobenius over A. Using the theory of A-rings, we mainly construct a Morita context connecting the smash product A#H and the invariant subalgebra A~H , which generalizes the corresponding results obtained by Cohen, Fischman and Montgomery.     

A semilattice structure for Arens products

Let (A,?) be a Banach algebra. Then for n∈?, A ⁽²ⁿ⁾ has 2 ⁿ Arens products. In this paper we study the relations between the Arens products on A ⁽²ⁿ⁾. Moreover, if P _n (A) denotes the set of all Arens products on A ⁽²ⁿ⁾, for n∈?, we show that $P(A)=\bigcup_{n=1}^{\infty} P_{n}(A)$ is a ∧-semilattice. Also, we study P(A) as an infinite commutative semigroup and P(A)?{?} as a free semigroup generated by two elements. Then we investigate amenability and weak amenability for their semigroup Banach algebras.     

Matrices A such that AA − AA are nonsingular

In this paper we study the class of square matrices A such that AA^† − A^†A is nonsingular, where A^† stands for the Moore-Penrose inverse of A. Among several characterizations we prove that for a matrix A of order n, the difference AA^† − A^†A is nonsingular if and only if R(A)⊕R(A^∗)=C_n,1, where R(·) denotes the range space. Also we study matrices A such that R(A)^⊥=R(A^∗).     

Some new results on subset sums

Let n be a large integer and A be a subset of [n]={1,…,n}. The set SA is the collection of the subset sums of A. In this note, we discuss new results (and proofs) on few well-known problems concerning SA. In particular, we improve an estimate of Alon and Erd?s concerning monochromatic representations.     

Optimal ellipsoids and decomposition of positive definite matrices

Given a positive definite matrix A, we characterize the unique diagonal matrix D, D?A, with the smallest determinant. Equivalently, given an ellipsoid A, we characterize the unique ellipsoid of the largest volume contained in A, with principal axes parallel to the coordinate axes.     

Weak relatively uniform convergences on MV-algebras

Weak relatively uniform convergences (wru-convergences, for short) in lattice ordered groups have been investigated in previous authors’ papers. In the present article, the analogous notion for MV-algebras is studied. The system s(A) of all wru-convergences on an MV-algebra A is considered; this system is partially ordered in a natural way. Assuming that the MV-algebra A is divisible, we prove that s(A) is a Brouwerian lattice and that there exists an isomorphism of s(A) into the system s(G) of all wru-convergences on the lattice ordered group G corresponding to the MV-algebra A. Under the assumption that the MV-algebra A is archimedean and divisible, we investigate atoms and dual atoms in the system s(A).     

Hyper-Tauberian algebras and weak amenability of Figà-Talamanca-Herz algebras

We study certain commutative regular semisimple Banach algebras which we call hyper-Tauberian algebras. We first show that they form a subclass of weakly amenable Tauberian algebras. Then we investigate the basic and hereditary properties of them. Moreover, we show that if A is a hyper-Tauberian algebra, then the linear space of bounded derivations from A into any Banach A-bimodule is reflexive. We apply these results to the Figà-Talamanca-Herz algebra A_p(G) of a locally compact group G for p∈(1,∞). We show that A_p(G) is hyper-Tauberian if the principal component of G is abelian. Finally, by considering the quantization of these results, we show that for any locally compact group G, A_p(G), equipped with an appropriate operator space structure, is a quantized hyper-Tauberian algebra. This, in particular, implies that A_p(G) is operator weakly amenable.     

Module homomorphisms and multipliers on locally compact quantum groups

For a Banach algebra A with a bounded approximate identity, we investigate the A-module homomorphisms of certain introverted subspaces of A^∗, and show that all A-module homomorphisms of A^∗ are normal if and only if A is an ideal of A∗∗. We obtain some characterizations of compactness and discreteness for a locally compact quantum group G. Furthermore, in the co-amenable case we prove that the multiplier algebra of L¹(G) can be identified with M(G). As a consequence, we prove that G is compact if and only if LUC(G)=WAP(G) and M(G)≅Z(LUC^∗(G)); which partially answer a problem raised by Volker Runde.     

Making holes in hyperspaces

Let X be a metric continuum and C(X) the hyperspace of all nonempty subcontinua of X. Let A∈C(X), A is said to make a hole in C(X), if C(X)−{A} is not unicoherent. In this paper we study the following problem.Problem: For which A∈C(X), A makes a hole in C(X).In this paper we present some partial solutions to this problem in the following cases: (1) A is a free arc; (2) A is a one-point set; (3) A is a free simple closed curve; (4) A=X.     

Lattice tensor products. IV

In 1999, for lattices A and B, G. Grätzer and F. Wehrung introduced the lattice tensor product, A?B. In Part I of this paper, we showed that for a finite lattice A and a bounded lattice B, this construction can be "coordinatized,'' that is, represented in B ^A so that the representing elements are easy to recognize. In this note, we show how to extend our method to an arbitrary bounded lattice A to coordinatize A?B.     

Generalizations of ?wierczkowski’s lemma and the arity gap of finite functions

?wierczkowski’s lemma-as it is usually formulated-asserts that if f:Aⁿ→A is an operation on a finite set A, n≥4, and every operation obtained from f by identifying a pair of variables is a projection, then f is a semiprojection. We generalize this lemma in various ways. First, it is extended to B-valued functions on A instead of operations on A and to essentially at most unary functions instead of projections. Then we characterize the arity gap of functions of small arities in terms of quasi-arity, which in turn provides a further generalization of ?wierczkowski’s lemma. Moreover, we explicitly classify all pseudo-Boolean functions according to their arity gap. Finally, we present a general characterization of the arity gaps of B-valued functions on arbitrary finite sets A.