149 research outputs found

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

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 $n$ are discussed: (1)$f(0) \leq 0$ and $f$ is
$n$-convex in $[0, \alpha)$. (2)For each matrix $a$ with its spectrum in $[0,
\alpha)$ and a contraction $c$ in the matrix algebra $M_n$, $f(c^*ac) \leq
c^*f(a)c$. (3)The function $f(t)/t$ $(= g(t))$ is $n$-monotone in $(0,
\alpha)$. We know that two conditions $(2)$ and $(3)$ are equivalent and if $f$
with $f(0) \leq 0$ is $n$-convex, then $g$ is $(n -1)$-monotone. In this note
we consider several extra conditions on $g$ to conclude that the implication
from $(3)$ to $(1)$ is true. In particular, we study a class $Q_n([0, \alpha))$
of functions with conditional positive Lowner matrix which contains the class
of matrix $n$-monotone functions and show that if $f \in Q_{n+1}([0, \alpha))$
with $f(0) = 0$ and $g$ is $n$-monotone, then $f$ is $n$-convex. We also
discuss about the local property of $n$-convexity.Comment: 13page

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

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

If $X$ is a compact Hausdorff space and $\sigma$ is a homeomorphism of $X$,
then a Banach algebra $\ell^1(\Sigma)$ of crossed product type is naturally
associated with this topological dynamical system $\Sigma=(X,\sigma)$. If $X$
consists of one point, then $\ell^1(\Sigma)$ is the group algebra of the
integers.
We study the algebraically irreducible representations of $\ell^1(\Sigma)$ 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 $\ell^1(\Sigma)$ is semisimple. All primitive ideals of $\ell^1(\Sigma)$
are selfadjoint, and $\ell^1(\Sigma)$ is Hermitian if there are only periodic
points in $X$. If $X$ is metrisable or all points are periodic, then all
primitive ideals arise as in our construction. A part of the structure space of
$\ell^1(\Sigma)$ is conditionally shown to be homeomorphic to the product of a
space of finite orbits and $\mathbb T$. If $X$ is a finite set, then the
structure space is the topological disjoint union of a number of tori, one for
each orbit in $X$. If all points of $X$ have the same finite period, then it is
the product of the orbit space $X/\mathbb Z$ and $\mathbb T$. For rational
rotations of $\mathbb T$, this implies that the structure space is homeomorphic
to $\mathbb T^2$.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

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

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

- …