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If $X$ is a compact Hausdorff space and $\sigma$ is a homeomorphism of $X$, then an involutive Banach algebra $\ell^1(\Sigma)$ of crossed product type is naturally associated with the topological dynamical system $\Sigma=(X,\sigma)$. We initiate the study of the relation between two-sided ideals of $\ell^1(\Sigma)$ and ${\mathrm C}^\ast(\Sigma)$, the enveloping $\mathrm{C}^\ast$-algebra ${\mathrm C}(X)\rtimes_\sigma \mathbb Z$ of $\ell^1(\Sigma)$. Among others, we prove that the closure of a proper two-sided ideal of $\ell^1(\Sigma)$ in ${\mathrm C}^\ast(\Sigma)$ is again a proper two-sided ideal of ${\mathrm C}^\ast(\Sigma)$., Comment: 9 pages. Minor changes in presentation from the original. Final version; to appear in Adv. Oper. Theory

We describe (infinite-dimensional) irreducible representations of the crossed product C$^*$-algebra associated with a topological dynamical system (based on $Z$) and we show that their restrictions to the underling $\ell^1$-Banach $*$-algebra are not algebraically irreducible under mild conditions on the dynamical system. The above description of irreducible representations has two ingredients, ergodic measures on the space and ergodic extensions for the tensor product with type I factors; the latter which may not have been explicitly taken up before will be explored by examples. A new class of ergodic measures is also constructed for irrational rotations on the circle.

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 Mathematics

Let $\varphi$ be a normal state on the algebra $B(H)$ of all bounded operators on a Hilbert space $H$, $f$ a strictly positive, continuous function on $(0, \infty)$, and let $g$ be a function on $(0, \infty)$ defined by $g(t) = \frac{t}{f(t)}$. We will give characterizations of matrix and operator monotonicity by the following generalized Powers-St\ormer inequality: $$ \varphi(A + B) - \varphi(|A - B|) \leq 2\varphi(f(A)^1/2g(B)f(A)^1/2), $$ whenever $A, B$ are positive invertible operators in $B(H).$, Comment: 10 pages

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 pages

We show that a unital involutive Banach algebra, with identity of norm one and continuous involution, is a C*-algebra, with the given involution and norm, if every continuous linear functional attaining its norm at the identity is positive., Comment: 3 pages. Some typos corrected. Final version, to appear in Ann. Funct. Anal

If $\Sigma=(X,\sigma)$ is a topological dynamical system, where $X$ is a compact Hausdorff space and $\sigma$ is a homeomorphism of $X$, then a crossed product Banach $\sp{*}$-algebra $\ell^1(\Sigma)$ is naturally associated with these data. If $X$ consists of one point, then $\ell^1(\Sigma)$ is the group algebra of the integers. The commutant $C(X)'_1$ of $C(X)$ in $\ell^1(\Sigma)$ is known to be a maximal abelian subalgebra which has non-zero intersection with each non-zero closed ideal, and the same holds for the commutant $C(X)'_*$ of $C(X)$ in $C^*(\Sigma)$, the enveloping $C^*$-algebra of $\ell^1(\Sigma)$. This intersection property has proven to be a valuable tool in investigating these algebras. Motivated by this pivotal role, we study $C(X)'_1$ and $C(X)'_*$ in detail in the present paper. The maximal ideal space of $C(X)'_1$ is described explicitly, and is seen to coincide with its pure state space and to be a topological quotient of $X\times\mathbb{T}$. We show that $C(X)'_1$ is hermitian and semisimple, and that its enveloping $C^*$-algebra is $C(X)'_*$. Furthermore, we establish necessary and sufficient conditions for projections onto $C(X)'_1$ and $C(X)'_*$ to exist, and give explicit formulas for such projections, which we show to be unique. In the appendix, topological results for the periodic points of a homeomorphism of a locally compact Hausdorff space are given., Comment: Some typos corrected. Final version, 23 pages, to appear in Studia Mathematica

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: 13pages

Given a topological dynamical system $\Sigma = (X, \sigma)$, where $X$ is a compact Hausdorff space and $\sigma$ a homeomorphism of $X$, we introduce the associated Banach $^*$-algebra crossed product $\ell^1 (\Sigma)$ and analyse its ideal structure. This algebra is the Banach algebra most naturally associated with the dynamical system, and it has a richer structure than its well studied $C^*$-envelope, as becomes evident from the possible existence of non-self-adjoint closed ideals. This paper initiates the study of these algebras and links their ideal structure to the topological dynamics. It is determined when exactly the algebra is simple, or prime, and when there exists a non-self-adjoint closed ideal. In addition, a structure theorem is obtained for the case when $X$ consists of one finite orbit, and the algebra is shown to be Hermitian if $X$ is finite. The key to these results lies in analysing the commutant of $C(X)$ in the algebra, which can be shown to be a maximal abelian subalgebra with non-zero intersection with each non-zero closed ideal., Comment: 19 pages. Some typos corrected and minor changes in presentation. Final version, to appear in Journal of Functional Analysis

For the $C^*$-crossed product $C^*(\Sigma)$ associated with an arbitrary topological dynamical system $\Sigma = (X, \sigma)$, we provide a detailed analysis of the commutant, in $C^* (\Sigma)$, of $C(X)$ and the commutant of the image of $C(X)$ under an arbitrary Hilbert space representation $\tilde{\pi}$ of $C^* (\Sigma)$. In particular, we give a concrete description of these commutants, and also determine their spectra. We show that, regardless of the system $\Sigma$, the commutant of $C(X)$ has non-zero intersection with every non-zero, not necessarily closed or self-adjoint, ideal of $C^* (\Sigma)$. We also show that the corresponding statement holds true for the commutant of $\tilde{\pi}(C(X))$ under the assumption that a certain family of pure states of $\tilde{\pi}(C^* (\Sigma))$ is total. Furthermore we establish that, if $C(X) \subsetneq C(X)'$, there exist both a $C^*$-subalgebra properly between $C(X)$ and $C(X)'$ which has the aforementioned intersection property, and such a $C^*$-subalgebra which does not have this property. We also discuss existence of a projection of norm one from $C^*(\Sigma)$ onto the commutant of $C(X)$., Comment: 19 pages. Minor clarifications and improvements of exposition made. Some typos corrected

Let $n \in \N$ and $M_n$ be the algebra of $n \times n$ matrices. We call a function $f$ matrix monotone of order $n$ or $n$-monotone in short whenever the inequality $f(a) \leq f(b)$ holds for every pair of selfadjoint matrices $a, b \in M_n$ such that $a \leq b$ and all eigenvalues of $a$ and $b$ are contained in $I$. Matrix convex (concave) functions on $I$ are similarily defined. The spaces for $n$-monotone functions and $n$-convex functions are written as $P_n(I)$ and $K_n(I)$. In this note we discuss several assertions at each leven $n$ for which we regard themas the problems of double piling structure of those sequences $\{P_n(I)\}_{n\in\N}$ and $\{K_n(I)\}_{n\in\N}$. In order to see clear insight of the aspect of the problems, however, we choose the following three main assertions among them and discuss their mutual dependence: \begin{enumerate} \item[(i)] $f(0)\leq 0$ and $f$ is $n$-convex in $[0,\alpha)$, \item[(ii)] For each matrix $a$ with its spectrum in $[0,\alpha)$ and a contraction $c$ in the matrix algebra $M_n$, \[ f(c^{\star}a c)\leq c^{\star}f(a)c, \] \item[(iii)] The functon $g(t)/t$ is $n$-monotone in $(0,\alpha)$. \end{enumerate} In particular, we show that for any $n \in \N$ two conditions $(ii)$ and $(iii)$ are equivalent.

We continue the analysis in [3] of matrix convex functions of a fixed order defined in a real interval by differential methods as opposed to the characterization in terms of divided differences given by Kraus [5]. We amend and improve some points in the previously given presentation, and we give a number of simple but important consequences of matrix convexity of low orders.

The article is devoted to investigation of the classes of functions belonging to the gaps between classes $P_{n+1}(I)$ and $P_{n}(I)$ of matrix monotone functions for full matrix algebras of successive dimensions. In this paper we address the problem of characterizing polynomials belonging to the gaps $P_{n}(I) \setminus P_{n+1}(I)$ for bounded intervals $I$. We show that solution of this problem is closely linked to solution of truncated moment problems, Hankel matrices and Hankel extensions. Namely, we show that using the solutions to truncated moment problems we can construct continuum many polynomials in the gaps. We also provide via several examples some first insights into the further problem of description of polynomials in the gaps that are not coming from the truncated moment problem. Also, in this article, we deepen further in another way into the structure of the classes of matrix monotone functions and of the gaps between them by considering the problem of position in the gaps of certain interesting subclasses of matrix monotone functions that appeared in connection to interpolation of spaces and in a prove of the L{\"o}wner theorem on integral representation of operator monotone functions., Comment: This paper is to appear in Math. Scand. ;39 pages, 22 figures

We analyze matrix convex functions of a fixed order defined on a real interval by differential methods as opposed to the characterization in terms of divided differences given by Kraus. We obtain for each order conditions for matrix convexity which are necessary and locally sufficient, and they allow us to prove the existence of gaps between classes of matrix convex functions of successive orders, and to give explicit examples of the type of functions contained in each of these gaps. The given conditions are shown to be also globally sufficient for matrix convexity of order two. We finally introduce a fractional transformation which connects the set of matrix monotone functions of each order n with the set of matrix convex functions of order n+1.

The article is devoted to investigation of classes of functions monotone as functions on general $C^*$-algebras that are not necessarily the $C^*$-algebras of all bounded linear operators on a Hilbert space as it is in classical case of matrix and operator monotone functions. We show that for general $C^*$-algebras the classes of monotone functions coincide with the standard classes of matrix and operator monotone functions. For every class we give exact characterization of $C^*$-algebras that have this class of monotone functions, providing at the same time a monotonicity characterization of subhomogeneous $C^*$-algebras. We use this characterization to generalize one function based monotonicity conditions for commutativity of a $C^*$-algebra, to one function based monotonicity conditions for subhomogeneity. As a $C^*$-algebraic counterpart of standard matrix and operator monotone scaling, we investigate, by means of projective $C^*$-algebras and relation lifting, the existence of $C^*$-subalgebras of a given monotonicity class., Comment: 16 pages, LaTex

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.