149 research outputs found

    Double piling structure of matrix monotone functions and of matrix convex functions II

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    We continue the analysis in [H. Osaka and J. Tomiyama, Double piling structure of matrix monotone functions and of matrix convex functions, Linear and its Applications 431(2009), 1825 - 1832] in which the followings three assertions at each label nn are discussed: (1)f(0)0f(0) \leq 0 and ff is nn-convex in [0,α)[0, \alpha). (2)For each matrix aa with its spectrum in [0,α)[0, \alpha) and a contraction cc in the matrix algebra MnM_n, f(cac)cf(a)cf(c^*ac) \leq c^*f(a)c. (3)The function f(t)/tf(t)/t (=g(t))(= g(t)) is nn-monotone in (0,α)(0, \alpha). We know that two conditions (2)(2) and (3)(3) are equivalent and if ff with f(0)0f(0) \leq 0 is nn-convex, then gg is (n1)(n -1)-monotone. In this note we consider several extra conditions on gg to conclude that the implication from (3)(3) to (1)(1) is true. In particular, we study a class Qn([0,α))Q_n([0, \alpha)) of functions with conditional positive Lowner matrix which contains the class of matrix nn-monotone functions and show that if fQn+1([0,α))f \in Q_{n+1}([0, \alpha)) with f(0)=0f(0) = 0 and gg is nn-monotone, then ff is nn-convex. We also discuss about the local property of nn-convexity.Comment: 13page

    Noncommutative spectral synthesis for the involutive Banach algebra associated with a topological dynamical system

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    If X is a compact Hausdorff space, supplied with a homeomorphism, then a crossed product involutive Banach algebra is naturally associated with these data. If X consists of one point, then this algebra is the group algebra of the integers. In this paper, we study spectral synthesis for the closed ideals of this associated algebra in two versions, one modeled after C(X), and one modeled after the group algebra of the integers. We identify the closed ideals which are equal to (what is the analogue of) the kernel of their hull, and determine when this holds for all closed ideals, i.e., when spectral synthesis holds. In both models, this is the case precisely when the homeomorphism has no periodic points.Comment: 28 page

    Algebraically irreducible representations and structure space of the Banach algebra associated with a topological dynamical system

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    If XX is a compact Hausdorff space and σ\sigma is a homeomorphism of XX, then a Banach algebra 1(Σ)\ell^1(\Sigma) of crossed product type is naturally associated with this topological dynamical system Σ=(X,σ)\Sigma=(X,\sigma). If XX consists of one point, then 1(Σ)\ell^1(\Sigma) is the group algebra of the integers. We study the algebraically irreducible representations of 1(Σ)\ell^1(\Sigma) on complex vector spaces, its primitive ideals and its structure space. The finite dimensional algebraically irreducible representations are determined up to algebraic equivalence, and a sufficiently rich family of infinite dimensional algebraically irreducible representations is constructed to be able to conclude that 1(Σ)\ell^1(\Sigma) is semisimple. All primitive ideals of 1(Σ)\ell^1(\Sigma) are selfadjoint, and 1(Σ)\ell^1(\Sigma) is Hermitian if there are only periodic points in XX. If XX is metrisable or all points are periodic, then all primitive ideals arise as in our construction. A part of the structure space of 1(Σ)\ell^1(\Sigma) is conditionally shown to be homeomorphic to the product of a space of finite orbits and T\mathbb T. If XX is a finite set, then the structure space is the topological disjoint union of a number of tori, one for each orbit in XX. If all points of XX have the same finite period, then it is the product of the orbit space X/ZX/\mathbb Z and T\mathbb T. For rational rotations of T\mathbb T, this implies that the structure space is homeomorphic to T2\mathbb T^2.Comment: 32 pages. Editorial improvements from the first version, and a few remarks added. Final version, to appear in Advances in Mathematic

    Gaps between classes of matrix monotone functions

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    We prove the existence of gaps between all the different classes of matrix monotone functions defined on an interval, provided the interval is non trivial and different from the whole real line. We then show how matrix monotone functions may be used in the characterization of certain C*-algebras as an alternative to the study of the matricial structure by positive linear maps

    Fish fauna off sandy beaches, in an estuary, and in a seagrass bed in Hiroshima Bay, Seto Inland Sea

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    From February 2015 to January 2016, we collected fish monthly using a beach seine net at two sandy beaches (B1 and B2), in a muddy sand estuary (MS), and in a seagrass bed (SG) in Hiroshima Bay, western Japan. A total of 2920 fish in 50 species were collected. The number of species, individuals, and biomass (total weight) were greater at SG and MS than at B1 and B2. The numerically most dominant species were Favonigobius gymnauchen and Tridentiger trigonocephalus at B1 and B2, F. gymnauchen and Acentrogobius sp. 2 at MS, and Plotosus japonicus and Rudarius ercodes at SG. Fish diversity also was higher at MS and SG than at B1 and B2 throughout the year. Fish assemblages and their patterns varied between sites, indicating that each habitat plays an important role as the nursery ground for different fishes