<Emphasis Type="Italic">C</Emphasis>-distribution semigroups and <Emphasis Type="Italic">C</Emphasis>-ultradistribution semigroups in locally convex spaces

The main purpose of this paper is to study C-distribution semigroups and C-ultradistribution semigroups in the setting of sequentially complete locally convex spaces. We provide a few important theoretical novelties in this field and some interesting examples. Under consideration are stationary dense operators in a sequentially complete locally convex space.     

Numerical Ranges of <Emphasis Type="Italic">C</Emphasis><Subscript>0</Subscript>(<Emphasis Type="Italic">N</Emphasis>) Contractions

A conjecture of Halmos proved by Choi and Li states that the closure of the numerical range of a contraction on a Hilbert space is the intersection of the closure of the numerical ranges of all its unitary dilations. We show that for C ₀(N) contractions one can restrict the intersection to a smaller family of dilations. This generalizes a finite dimensional result of Gau and Wu.     

Subgenerators of <Emphasis Type="Italic">C</Emphasis>-semigroups on <Emphasis Type="Italic">l</Emphasis><Superscript>2</Superscript>

This paper gives some results about the subgenerators of special C-semigroups on l2, partly answers the problem mentioned in [1].     

On the product formulas for <Emphasis Type="Italic">C</Emphasis>-semigroups

In this paper we obtain a Chernoff-type approximation theorem for C-semigroups, from which we derive the Trotter product formula and the Post-Widder inversion formula for C-semigroups. The first author was supported by the NSF of China (Grant No. 10501032) and the NSFC-RFBR Programm (Grant No. 108011120015), and the second author by the NSF of China (Grant No. 10671079).     

On a new property of <Emphasis Type="Italic">n</Emphasis>-poised and <Emphasis Type="Italic">G</Emphasis><Emphasis Type="Italic">C</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript> sets

In this paper we consider n-poised planar node sets, as well as more special ones, called G C _n sets. For the latter sets each n-fundamental polynomial is a product of n linear factors as it always holds in the univariate case. A line ? is called k-node line for a node set \(\mathcal X\) if it passes through exactly k nodes. An (n + 1)-node line is called maximal line. In 1982 M. Gasca and J. I. Maeztu conjectured that every G C _n set possesses necessarily a maximal line. Till now the conjecture is confirmed to be true for n ≤ 5. It is well-known that any maximal line M of \(\mathcal X\) is used by each node in \(\mathcal X\setminus M, \)meaning that it is a factor of the fundamental polynomial. In this paper we prove, in particular, that if the Gasca-Maeztu conjecture is true then any n-node line of G C _n set \(\mathcal {X}\) is used either by exactly \(\binom {n}{2}\) nodes or by exactly \(\binom {n-1}{2}\) nodes. We prove also similar statements concerning n-node or (n ? 1)-node lines in more general n-poised sets. This is a new phenomenon in n-poised and G C _n sets. At the end we present a conjecture concerning any k-node line.     

The -conjecture and equivariant <Emphasis Type="Italic">e</Emphasis><Subscript><Emphasis Type="Italic">C</Emphasis></Subscript>-invariants

Let X be a smooth closed oriented non-spin 4-manifold with even intersection form kE₈nH (n1). The -conjecture states that n is greater than or equal to |k|. In this paper we give a proof of the -conjecture. The strategy of this paper is to use the finite dimensional approximation of the map induced from the Seiberg-Witten equations and equivariant e_C-invariants as in the paper of M. Furuta and Y. Kametani.Mathematics Subject Classification (1991): 57R55This work was supported by Korea Research Foundation Grant (KRF–2002–003–C00011).     

A space <Emphasis Type="Italic">C</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>(<Emphasis Type="Italic">X</Emphasis>) is dominated by irrationals if and only if it is<Emphasis Type="Italic"> K</Emphasis>-analytic

Summary We prove that, for any Tychonoff X, the space C_p(X) is K-analytic if and only if it has a compact cover {K_p: p } such that K_p subset K_q whenever p,q and p q. Applying this result we show that if C_p(X) is K-analytic then C_p(X) is K-analytic as well. We also establish that a space C_p(X) is K-analytic and Baire if and only if X is countable and discrete.     

A generalized Kiepert formula for <Emphasis Type="Italic">C</Emphasis><Subscript><Emphasis Type="Italic">ab</Emphasis></Subscript> curves

Kiepert (1873) and Brioschi (1864) published algebraic equations for the n-division points of an elliptic curve, in terms of the Weierstrass ℘-function and its derivatives with respect to a uniformizing parameter, or another elliptic function, respectively. We generalize both types of formulas for a compact Riemann surface which, outside from one point, has a smooth polynomial equation in the plane, in the sense that we characterize the points whose n-th multiple in the Jacobian belongs to the theta divisor.     

On (<Emphasis Type="Italic">n</Emphasis>,<Emphasis Type="Italic">m</Emphasis>)-semigroups

We generalize Green’s lemma and Green’s theorem for usual binary semigroups to (n,m)-semigroups, define and describe the regularity for an element of an (n,m)-semigroup, give some criteria for an element of an (n,m)-semigroup to be invertible, and further apply the invertibility for (n,m)-semigroups to (n,m)-groups and give some equivalent characterizations for (n,m)-groups. We establish Hosszú-Gluskin theorems for (n,m)-semigroups in two cases, as generalizations of the corresponding theorems for n-groups.     

Novikov superalgebras with <Emphasis Type="Italic">A</Emphasis><Subscript>0</Subscript> = <Emphasis Type="Italic">A</Emphasis><Subscript>1</Subscript><Emphasis Type="Italic">A</Emphasis><Subscript>1</Subscript>

Novikov superalgebras are related to quadratic conformal superalgebras which correspond to the Hamiltonian pairs and play a fundamental role in completely integrable systems. In this note we show that the Novikov superalgebras with A ₀ = A ₁ A ₁ and dim A ₁ = 2 are of type N and give a class of Novikov superalgebras of type S with A ₀ = A ₁ A ₁.     

On the Theory of <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>(<Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">q</Emphasis></Subscript>)-Banach Lattices

Given 1≤ p,q < ∞, let BL_pL_q be the class of all Banach lattices X such that X is isometrically lattice isomorphic to a band in some L_p(L_q)-Banach lattice. We show that the range of a positive contractive projection on any BL_pL_q-Banach lattice is itself in BL_pL_q. It is a consequence of this theorem and previous results that BL_pL_q is first-order axiomatizable in the language of Banach lattices. By studying the pavings of arbitrary BL_pL_q-Banach lattices by finite dimensional sublattices that are themselves in this class, we give an explicit set of axioms for BL_pL_q. We also consider the class of all sublattices of L_p(L_q)-Banach lattices; for this class (when p/q is not an integer) we give a set of axioms that are similar to Krivine’s well-known axioms for the subspaces of L_p-Banach spaces (when p/2 is not an integer). We also extend this result to the limiting case q = ∞.     

<Emphasis Type="Italic">A</Emphasis><Subscript>1</Subscript> weights and <Emphasis Type="Italic">d</Emphasis>-homogeneous measures on ahlfors <Emphasis Type="Italic">d</Emphasis>-regular space

Let X be an Ahlfors d-regular space and mad-regular measure on X . We prove that a measure μ on X is d-homogeneous if and only if μ is mutually absolutely continuous with respect to m and the derivative Dmμ(x) is an A1 weight. Also, we show by an example that every Ahlfors d-regular space carries a measure which is d-homogeneous but not d-regular.     

Similarity Results for Operators of Class <Emphasis Type="Italic">C</Emphasis><Subscript>0</Subscript>

If T is a multiplicity-free contraction of class C ₀ with minimal function m _T , then it is quasisimilar to the Jordan block S(m _T ). In case m _T is a Blaschke product with simple roots forming a Carleson sequence, we show that the relation between T and S(m _T ) can be strengthened to similarity. Under the additional assumption that u(T) has closed range for every inner divisor \({u\in H^\infty}\) of m _T , the result also holds in the more general setting where the roots have bounded multiplicities.     

<Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-Boundedness of general index transforms

We establish the boundedness properties in L _p for a class of integral transformations with respect to an index of hypergeometric functions. In particular, by using the Riesz-Thorin interpolation theorem, we get the corresponding results in L _p (R ₊), 1 p 2, for the Kontorovich-Lebedev, Mehler-Fock, and Olevskii index transforms. An inversion theorem is proved for a general index transformation. The case p=2 is known as the Plancherel-type theory for this class of transformations.__________Published in Lietuvos Matematikos Rinkinys, Vol. 45, No. 1, pp. 127–147, January–March, 2005.     

New results on <Emphasis Type="Italic">C</Emphasis><Subscript>11</Subscript> and <Emphasis Type="Italic">C</Emphasis><Subscript>12</Subscript> lattices with applications to Grothendieck categories and torsion theories

In this paper, which is a continuation of our previous paper [T. Albu, M. Iosif, A. Tercan, The conditions (C _i ) in modular lattices, and applications, J. Algebra Appl. 15 (2016), http: dx.doi.org/10.1142/S0219498816500018], we investigate the latticial counterparts of some results about modules satisfying the conditions (C ₁₁) or (C ₁₂). Applications are given to Grothendieck categories and module categories equipped with hereditary torsion theories.     

Uniquely <Emphasis Type="Italic">C</Emphasis><Subscript>4</Subscript>-Saturated Graphs

For a fixed graph H, a graph G is uniquely H-saturated if G does not contain H, but the addition of any edge from [`(G)]{\overline{G}} to G completes exactly one copy of H. Using a combination of algebraic methods and counting arguments, we determine all the uniquely C ₄-saturated graphs; there are only ten of them.     

Some Notes on <Emphasis Type="Italic">C</Emphasis>- and <Emphasis Type="Italic">C</Emphasis>*-quotients of Frames

We characterise C*-quotients and C-quotients of completely regular frames in terms of ?ech-Stone compactifications and Lindelöfications, respectively. The latter is a frame-theoretic result with no spatial counterpart. We also characterise C*-quotients and dense C-quotients of completely regular frames in terms of normal covers.     

A Class of <Emphasis Type="Italic">C</Emphasis>*-algebras with non-stable <Emphasis Type="Italic">K</Emphasis><Subscript>1</Subscript>-group property

In this paper, we give a class of C*-algebras with non-stable K ₁-group property which include the example non-simple tracial topological rank zero and stable rank two C*-algebra given by Lin and Osaka.     

<Emphasis Type="Italic">C</Emphasis>*-Homomorphisms and duality of <Emphasis Type="Italic">C</Emphasis>*-discrete quantum groups

Let

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be a C*-discrete quantum group and let

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be the discrete quantum group associated with

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. Suppose that there exists a continuous action of

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on a unital C*-algebra

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so that

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becomes a

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-algebra. If there is a faithful irreducible vacuum representation π of

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on a Hilbert space H =

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with a vacuum vector Ω, which gives rise to a

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-invariant state, then there is a unique C*-representation (θ, H) of

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supplemented by the action. The fixed point subspace of

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under the action of

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is exactly the commutant of θ(

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).