Banach spaces in various positions

We formulate a general theory of positions for subspaces of a Banach space: we define equivalent and isomorphic positions, study the automorphy index a(Y,X) that measures how many non-equivalent positions Y admits in X, and obtain estimates of a(Y,X) for X a classical Banach space such as ?p,Lp,L₁,C(ωω) or C[0,1]. Then, we study different aspects of the automorphic space problem posed by Lindenstrauss and Rosenthal; namely, does there exist a separable automorphic space different from c₀ or ?₂? Recall that a Banach space X is said to be automorphic if every subspace Y admits only one position in X; i.e., a(Y,X)=1 for every subspace Y of X. We study the notion of extensible space and uniformly finitely extensible space (UFO), which are relevant since every automorphic space is extensible and every extensible space is UFO. We obtain a dichotomy theorem: Every UFO must be either an L_∞-space or a weak type 2 near-Hilbert space with the Maurey projection property. We show that a Banach space all of whose subspaces are UFO (called hereditarily UFO spaces) must be asymptotically Hilbertian; while a Banach space for which both X and X^∗ are UFO must be weak Hilbert. We then refine the dichotomy theorem for Banach spaces with some additional structure. In particular, we show that an UFO with unconditional basis must be either c₀ or a superreflexive weak type 2 space; that a hereditarily UFO Köthe function space must be Hilbert; and that a rearrangement invariant space UFO must be either L_∞ or a superreflexive type 2 Banach lattice.     

Quantum logics and completeness criteria of inner product spaces

We present the survey of measure-theoretic completeness criteria for inner product spaces using methods and notions important for quantum logics. Moreover, some new criteria and open problems are given.     

Product Effect Algebras

We introduce a product on an effect algebra. We prove that every product effect algebra with the Riesz decomposition property (RDP), is an interval in an Abelian unital interpolation po-ring, and we show that the category of product effect algebras with the RDP is categorically equivalent with the category of unital Abelian interpolation po-rings. In addition, we show that every product effect algebra with the RDP and with 1 as a product unity is a subdirect product of antilattice product effect algebras with the RDP.     

A New Algebraic Criterion for Completeness of Inner Product Spaces

We show that an inner product space S is complete whenever the system E(S) of all splitting subspaces of S, i.e., of all subspaces M of S such that M + M = S holds, satisfies the -Riesz interpolation property. This generalizes the result of H. Gross and H. Keller who required E(S) to be a complete lattice, of G. Cattaneo and G. Marino who required E(S) to be a -complete lattice, and that of the author who required E(S) to be a -orthocomplete OMP.     

Piron's and Bell's Geometric Lemmas and Gleason's Theorem

We study the idea of implantation of Piron's and Bell's geometrical lemmas for proving some results concerning measures on finite as well as infinite-dimensional Hilbert spaces, including also measures with infinite values. In addition, we present parabola based proofs of weak Piron's geometrical and Bell's lemmas. These approaches will not used directly Gleason's theorem, which is a highly non-trivial result.     

When the lexicographic product of two po-groups has the Riesz decomposition property

We study conditions when a certain type of the Riesz Decomposition Property (RDP for short) holds in the lexicographic product of two po-groups. Defining two important properties of po-groups, we extend known situations showing that the lexicographic product satisfies RDP or even \({{\rm RDP}_1}\), a stronger type of RDP. We recall that a very strong type of RDP, \({{\rm RDP}_2}\), entails that the group is lattice ordered. RDP's of the lexicographic products are important for the study of lexicographic pseudo effect algebras, or perfect types of pseudo MV-algebras and pseudo effect algebras, where infinitesimal elements play an important role both for algebras as well as for the first order logic of valid but not provable formulas.     

Atomic Effect Algebras with the Riesz Decomposition Property

We discuss the relationships between effect algebras with the Riesz Decomposition Property and partially ordered groups with interpolation. We show that any σ-orthocomplete atomic effect algebra with the Riesz Decomposition Property is an MV-effect algebras, and we apply this result for pseudo-effect algebras and for states.     

On the structure of linearly ordered pseudo-BCK-algebras

Pseudo-BCK-algebras are a non-commutative generalization of well-known BCK-algebras. The paper describes a situation when a linearly ordered pseudo-BCK-algebra is an ordinal sum of linearly ordered cone algebras. In addition, we present two identities giving such a possibility of the decomposition and axiomatize the residuation subreducts of representable pseudo-hoops and pseudo-BL-algebras.     

Effect of ligand backbone substituents of nickel α‐diimine complexes on livingness/controllability of olefin polymerization: Competition between monomer isomerization and propagation

Four α‐diimine nickel complexes [(Ar? N?C(R)? C(R)?N? Ar)NiBr₂; R?H, CH₃, cyclohexane‐1,2‐diyl, naphthalene‐1,8‐diyl, Ar?2,6‐i‐Pr₂‐C₆H₃‐) were investigated in propene and hex‐1‐ene polymerization to identify the limits of backbone substituent R size needed to provide living/controlled α‐olefins polymerization by the sufficient suppression of β–H elimination transfer. Propagation kinetics measurements, molar mass on monomer conversion dependencies and reinitiation tests were used to evaluate the livingness of hex‐1‐ene polymerization. Interestingly, living/controlled hex‐1‐ene polymerization was observed in the case of all diimine derivatives including the one bearing only hydrogen atom in backbone positions. Unexpectedly, in the case of catalysts bearing H and CH₃ backbone substituents, we observed the unusual isomerization of hex‐1‐ene into internal hexenes in parallel with its polymerization. Nevertheless, by subtracting the amount of monomer consumed in isomerization side reaction, polymerization still keeps the features of living/controlled process. © 2016 Wiley Periodicals, Inc. J. Polym. Sci., Part A: Polym. Chem. 2016 , 54, 3193–3202