Advances in Operator TheoryAdvances in Operator Theory
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Sun, 19 Nov 2017 19:37:38 +0100FeedCreatorAdvances in Operator Theory
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Feed provided by Advances in Operator Theory. Click to visit.Complex interpolation and non-commutative integration
http://www.aot-math.org/article_42356_6058.html
We show that under suitable conditions interpolation between a Banach space and its dual yields a Hilbert space at $theta =frac{1}{2}$. By application of this result to the special case of the non-commutative $L^p$-spaces of Leinert [Int. J. Math. textbf{2} (1991), no. 2, 177--182] and Terp [J. Operator Theory textbf{8} (1982), 327--360] we conclude that $L^2$ is a Hilbert space and that $L^p$ is isometrically isomorphic to the dual of $L^q$ without using the isomorphisms of these spaces to $L^p$-spaces of Hilsum [J. Funct. Anal. textbf{40} (1981), 151--169.] and Haagerup [Colloq. Internat. CNRS, 274, CNRS, Paris, 1979].\Haagerup and Pisier [Canad. J. Math. textbf{41} (1989), no. 5, 882--906.], Pisier [Mem. Amer. Math. Soc. textbf{122} (1996), no. 585, viii+103 pp] and Watbled [C. R. Acad. Sci. Paris, t. 321, S'erie I, p. 1437--1440, 1995] gave conditions under which interpolation between a Banach space and its conjugate dual yields a Hilbert space at $frac{1}{2}$. The result mentioned above when put in ``conjugate form'' extends their results.Sun, 31 Dec 2017 20:30:00 +0100On different type of fixed point theorem for multivalued mappings via measure of noncompactness
http://www.aot-math.org/article_48945_0.html
In this paper by using the measure of noncompactness concept, we present new fixed point theorems for multivalued maps. In further we introduce a new class of mappings which are general than Meir--Keeler mappings. Finally, we use these results to investigate the existence of weak solutions to an Evolution differential inclusion with lack of compactness.Sun, 10 Sep 2017 19:30:00 +0100Semicontinuity and closed faces of C*-algebras
http://www.aot-math.org/article_43918_6058.html
C. Akemann and G.K. Pedersen [Duke Math. J. 40 (1973), 785--795.] defined three concepts of semicontinuity for self-adjoint elements of $A^{**}$, the enveloping von Neumann algebra of a $C^*$-algebra $A$. We give the basic properties of the analogous concepts for elements of $pA^{**}p$, where $p$ is a closed projection in $A^{**}$. In other words, in place of affine functionals on $Q$, the quasi--state space of $A$, we consider functionals on $F(p)$, the closed face of $Q$ suppported by $p$. We prove an interpolation theorem: If $hgeq k$, where $h$ is lower semicontinuous on $F(p)$ and $k$ upper semicontinuous, then there is a continuous affine functional $x$ on $F(p)$ such that $kleq xleq h$. We also prove an interpolation--extension theorem: Now $h$ and $k$ are given on $Q$, $x$ is given on $F(p)$ between $h_{|F(p)}$ and $k_{|F(p)}$, and we seek to extend $x$ to $widetilde x$ on $Q$ so that $kleqwidetilde xleq h$. We give a characterization of $pM(A)_{{text{sa}}}p$ in terms of semicontinuity. And we give new characterizations of operator convexity and strong operator convexity in terms of semicontinuity.Sun, 31 Dec 2017 20:30:00 +0100Singular Riesz measures on symmetric cones
http://www.aot-math.org/article_50058_0.html
‎A fondamental theorem due to Gindikin says that the‎ ‎generalized power $Delta_{s}(-theta^{-1})$ defined on a symmetric‎ ‎cone is the Laplace transform of a positive measure $R_{s}$ if and ‎only if $s$ is in a given subset $Xi$ of $Bbb{R}^{r}$‎, ‎where $r$‎ ‎is the rank of the cone‎. ‎When $s$ is in a well defined part of‎ ‎$Xi$‎, ‎the measure $R_{s}$ is absolutely continuous with respect to‎ ‎Lebesgue measure and has a known expression‎. ‎For the other elements‎ ‎$s$ of $Xi$‎, ‎the measure $R_{s}$ is concentrated on the boundary of‎ ‎the cone and it has never been explicitly determined‎. ‎The aim of the‎ ‎present paper is to give an explicit description of the measure‎ ‎$R_{s}$ for all $s$ in $Xi$‎. ‎The work is motivated by the‎ ‎importance of these measures in probability theory and in statistics‎ ‎since they represent a generalization of the class of measures‎ ‎generating the famous Wishart probability distributions‎.Mon, 11 Sep 2017 19:30:00 +0100The closure of ideals of $\ell^1(\Sigma)$ in its enveloping $\mathrm{C}^*$-algebra
http://www.aot-math.org/article_44047_6058.html
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 ${mathrm C}(X)rtimes_sigmamathbb Z$ 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 ${mathrm C}^ast(Sigma)$.Sun, 31 Dec 2017 20:30:00 +0100Cover topologies, subspaces, and quotients for some spaces of vector-valued functions
http://www.aot-math.org/article_51020_0.html
‎Let $X$ be a completely regular Hausdorff space‎, ‎and let $mathcal{D}$ be a‎ ‎cover of $X$ by $C_{b}$-embedded sets‎. ‎Let $pi‎ :‎mathcal{E}$ $rightarrow X$‎ ‎be a bundle of Banach spaces (algebras)‎, ‎and let $Gamma(pi)$ be the‎ ‎section space of the bundle $pi‎ .‎$ Denote by $Gamma _{b}(pi‎,‎mathcal{D})$‎ ‎the subspace of $Gamma (pi )$ consisting of sections which are bounded on‎ ‎each $Din mathcal{D}$. We construct a bundle $rho ^{prime }:mathcal{F}‎^{prime}rightarrow beta X$ such that $Gamma _{b}(pi‎ ,‎mathcal{D}) ‎$ is topologically and algebraically isomorphic to $Gamma(rho^prime‎‎)‎$, ‎and use this to study the subspaces (ideals) and quotients resulting‎ ‎from endowing $Gamma _{b}(pi‎,‎mathcal{D})$ with the cover topology‎ ‎determined by $mathcal{D}$‎.Sun, 15 Oct 2017 20:30:00 +0100Positive map as difference of two completely positive or super-positive maps
http://www.aot-math.org/article_44116_6058.html
For a linear map from ${mathbb M}_m$ to ${mathbb M}_n$, besides the usual positivity, there are two stronger notions, complete positivity and super positivity. Given a positive linear map $varphi$ we study a decomposition $varphi = varphi^{(1)} - varphi^{(2)}$ with completely positive linear maps $varphi^{(j)} (j = 1,2)$. Here $varphi^{(1)} + varphi^{(2)}$ is of simple form with norm small as possible. The same problem is discussed with super-positivity in place of complete positivity.Sun, 31 Dec 2017 20:30:00 +0100Integral representations and asymptotic behaviour of a Mittag-Leffler type function of two variables
http://www.aot-math.org/article_51110_0.html
Integral representations play a prominent role in the analysis of entire functions. The representations of generalized Mittag-Leffler type functions and their asymptotics have been (and still are) investigated by plenty of authors in various conditions and cases.The present paper explores the integral representations of a special function extending to two variables the two-parametric Mittag-Leffler type function. Integral representations of this functions within different variation ranges of its arguments for certain values of the parameters are thus obtained. Asymptotic expansion formulas and asymptotic properties of this function are also established for large values of the variables. This yields corresponding theorems providing integral representations as well as expansion formulas.Tue, 17 Oct 2017 20:30:00 +0100Some natural subspaces and quotient spaces of $L^1$
http://www.aot-math.org/article_44924_6058.html
We show that the space $text{Lip}_0(mathbb R^n)$ is the dual space of $L^{1}({mathbb R}^{n}; {mathbb R}^{n})/N$ where $N$ is the subspace of $L^{1}({mathbb R}^{n}; {mathbb R}^{n})$ consisting of vector fields whose divergence vanishes identically. We prove that although the quotient space $L^{1}({mathbb R}^{n}; {mathbb R}^{n})/N$ is weakly sequentially complete, the subspace $N$ is not nicely placed - in other words, its unit ball is not closed for the topology $tau_m$ of local convergence in measure. We prove that if $Omega$ is a bounded open star-shaped subset of $mathbb {R}^n$ and $X$ is a dilation-stable closed subspace of $L^1(Omega)$ consisting of continuous functions, then the unit ball of $X$ is compact for the compact-open topology on $Omega$. It follows in particular that such spaces $X$, when they have Grothendieck's approximation property, have unconditional finite-dimensional decompositions and are isomorphic to weak*-closed subspaces of $l^1$. Numerous examples are provided where such results apply.Sun, 31 Dec 2017 20:30:00 +0100Operator algebras associated to modules over an integral domain
http://www.aot-math.org/article_51119_0.html
We use the Fock semicrossed product to define an operator algebra associated to a module over an integral domain. We consider the $C^*$-envelope of the semicrossed product, and then consider properties of these algebras as models for studying general semicrossed products.Thu, 19 Oct 2017 20:30:00 +0100Partial isometries: a survey
http://www.aot-math.org/article_45165_6058.html
We survey the main results characterizing partial isometries in C$^*$-algebras and tripotents in JB$^*$-triples obtained in terms of regularity, conorm, quadratic-conorm, and the geometric structure of the underlying Banach spaces.Sun, 31 Dec 2017 20:30:00 +0100On the truncated two-dimensional moment problem
http://www.aot-math.org/article_51181_0.html
We study the truncated two-dimensional moment problem (with rectangular data) to find a non-negative measure $mu(delta)$, $deltainmathfrak{B}(mathbb{R}^2)$, such that $int_{mathbb{R}^2} x_1^m x_2^n dmu = s_{m,n}$, $0leq mleq M,quad 0leq nleq N$, where ${ s_{m,n} }_{0leq mleq M, 0leq nleq N}$ is a prescribed sequence of real numbers; $M,Ninmathbb{Z}_+$. For the cases $M=N=1$ and $M=1, N=2$ explicit numerical necessary and sufficient conditions for the solvability of the moment problem are given. In the cases $M=N=2$; $M=2, N=3$; $M=3, N=2$; $M=3, N=3$ some explicit numerical sufficient conditions for the solvability are obtained. In all the cases some solutions (not necessarily atomic) of the moment problem can be constructed.Sat, 21 Oct 2017 20:30:00 +0100Operators with compatible ranges in an algebra generated by two orthogonal projections
http://www.aot-math.org/article_45166_6058.html
The criterion is obtained for operators A from the algebra generated by two orthogonal projections P,Q to have a compatible range, i.e., coincide with the hermitian conjugate of A on the orthogonal complement to the sum of their kernels. In the particular case of A being a polynomial in P,Q, some easily verifiable conditions are derived.Sun, 31 Dec 2017 20:30:00 +0100The compactness of a class of radial operators on weighted Bergman spaces
http://www.aot-math.org/article_51302_0.html
In this paper, we study some connection between the compactness of radial operators and the boundary behavior of the corresponding Berezin transform on weighted Bergman spaces. More precisely, we prove that, under some mild condition, the vanishing of the Berezin transform on the unit circle is equivalent to the compactness of a class of radial operators on weighted Bergman spaces. Moreover, we also study the radial essential commutant of the Toeplitz operator $T_z$.Wed, 25 Oct 2017 20:30:00 +0100Permanence of nuclear dimension for inclusions of unital $C^*$-algebras with the Rokhlin property
http://www.aot-math.org/article_45177_6058.html
Let $P subset A$ be an inclusion of unital $C^*$-algebras and $Ecolon A rightarrow P$ be a faithful conditional expectation of index finite type. Suppose that $E$ has the Rokhlin property. Then $dr(P) leq dr(A)$ and $dim_{nuc}(P) leq dim_{nuc}(A)$. This can be applied to Rokhlin actions of finite groups. We also show that under the same above assumption if $A$ is exact and pure, that is, the Cuntz semigroups $W(A)$ has strict comparison and is almost divisible, then $P$ and the basic contruction $C^*langle A, e_Prangle$ are also pure.Sun, 31 Dec 2017 20:30:00 +0100Extensions of theory of regular and weak regular splittings to singular matrices
http://www.aot-math.org/article_51467_0.html
Matrix splittings are useful in finding a solution of linear systems of equations, iteratively. In this note, we present some more convergence and comparison results for recently introduced matrix splittings called index-proper regular and index-proper weak regular splittings. We then apply to theory of double index-proper splittings.Sat, 28 Oct 2017 20:30:00 +0100Almost Hadamard matrices with complex entries
http://www.aot-math.org/article_45905_6058.html
We discuss an extension of the almost Hadamard matrix formalism, to the case of complex matrices. Quite surprisingly, the situation here is very different from the one in the real case, and our conjectural conclusion is that there should be no such matrices, besides the usual Hadamard ones. We verify this conjecture in a number of situations, and notably for most of the known examples of real almost Hadamard matrices, and for some of their complex extensions. We discuss as well some potential applications of our conjecture, to the general study of complex Hadamard matrices.Sun, 31 Dec 2017 20:30:00 +0100On linear maps preserving certain pseudospectrum and condition spectrum subsets
http://www.aot-math.org/article_51460_0.html
We define two new types of spectrum, called the $varepsilon$-left (or right) pseudospectrum and the $varepsilon$-left (or right) condition spectrum, of an element $a$ in a complex unital Banach algebra $A$. We prove some basic properties among them the property that the $varepsilon$-left (or right) condition spectrum is a particular case of Ransford spectrum. We study also the linear preserver problem for our defined functions and we establish the following: (1) Let $A$ and $B$ be complex unital Banach algebras and $varepsilon>0$. Let $phi : Alongrightarrow B $ be an $varepsilon$-left (or right) pseudospectrum preserving onto linear map. Then $phi$ preserves certain standart spectral functions.(2) Let $A$ and $B$ be complex unital Banach algebras and $0< varepsilon<1$. Let $phi : Alongrightarrow B $ be unital linear map. Then(a) If $phi $ is $varepsilon$-almost multiplicative map, then $sigma^{l}(phi(a))subseteq sigma^{l}_varepsilon(a)$ and $sigma^{r}(phi(a))subseteq sigma^{r}_varepsilon(a)$, for all $a in A$.(b) If $phi$ is an $varepsilon$-left (or right) condition spectrum preserving, then (i) if $A$ is semi-simple, then $phi$ is injective; (ii) if B is spectrally normed, then $phi$ is continuous.Sat, 28 Oct 2017 20:30:00 +0100Non-commutative rational functions in strong convergent random variables
http://www.aot-math.org/article_46452_6058.html
Random matrices like GUE, GOE and GSE have been shown that they possess a lot of nice properties. In 2005, a new property of independent GUE random matrices is discovered by Haagerup and ThorbjÃ¸rnsen in their paper in 2005, it is called strong convergence property and then more random matrices with this property are followed. In general, the definition can be stated for a sequence of tuples over some $text{C}^{ast}$-algebras. In this paper, we want to show that, for a sequence of strongly convergent random variables, non-commutative polynomials can be extended to non-commutative rational functions under certain assumptions. As a direct corollary, we can conclude that for a tuple $(X_{1}^{left(nright)},cdots,X_{m}^{left(nright)})$ of independent GUE random matrices, $r(X_{1}^{left(nright)},cdots,X_{m}^{left(nright)})$ converges in trace and in norm to $r(s_{1},cdots,s_{m})$ almost surely, where $r$ is a rational function and $(s_{1},cdots,s_{m})$ is a tuple of freely independent semi-circular elements which lies in the domain of $r$.Sun, 31 Dec 2017 20:30:00 +0100Fourier multiplier norms of spherical functions on the generalized Lorentz groups
http://www.aot-math.org/article_47035_6058.html
Our main result provides a closed expression for the completely bounded Fourier multiplier norm of the spherical functions on the generalized Lorentz groups $SO_0(1,n)$ (for $ngeq2$). As a corollary, we find that there is no uniform bound on the completely bounded Fourier multiplier norm of the spherical functions on the generalized Lorentz groups. We extend the latter result to the groups $SU(1,n)$, $Sp(1,n)$ (for $ngeq2$) and the exceptional group $F_{4(-20)}$, and as an application we obtain that each of the above mentioned groups has a completely bounded Fourier multiplier, which is not the coefficient of a uniformly bounded representation of the group on a Hilbert space.Sun, 31 Dec 2017 20:30:00 +0100On a class of Banach algebras associated to harmonic analysis on locally compact groups and ...
http://www.aot-math.org/article_47586_6058.html
The purpose of this paper is to present some old and recent results for the class of $F$-algebras which include most classes of Banach algebras that are important in abstract harmonic analysis. We also introduce a subclass of the class of $F$-algebras, called normal $F$-algebras, that captures better the measure algebras and the (reduced) Fourier--Stieltjes algebras, and use this to give new characterisations the reduced Fourier--Stieltjes algebras of discrete groups.Sun, 31 Dec 2017 20:30:00 +0100Uniformly bounded representations and completely bounded multipliers of ${\rm SL}(2,\mathbb{R})$
http://www.aot-math.org/article_49322_6058.html
We estimate the norms of many matrix coefficients of irreducible uniformly bounded representations of ${rm SL}(2,mathbb{R})$ as completely bounded multipliers of the Fourier algebra.Our results suggest that the known inequality relating the uniformly bounded norm of a representation and the completely bounded norm of its coefficients may not be optimal.Sun, 31 Dec 2017 20:30:00 +0100Completely positive contractive maps and partial isometries
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Associated with a completely positive contractive map $varphi$ of a $C^*$-algebra $A$ is a universal $C^*$-algebra generated by the $C^*$-algebra $A$ along with a contraction implementing $varphi$. We prove a dilation theorem: the map $varphi$ may be extended to a completely positive contractive map of an augmentation of $A.$ The associated $C^*$-algebra of the augmented system contains the original universal $C^*$-algebra as a corner, and the extended completely positive contractive map is implemented by a partial isometry.Sun, 31 Dec 2017 20:30:00 +0100Uffe Haagerup - his life and mathematics
http://www.aot-math.org/article_50017_6058.html
In remembrance of Professor Uffe Valentin Haagerup (1949--2015), as a brilliant mathematician, we review some aspects of his life, and his outstanding mathematical accomplishments.Sun, 31 Dec 2017 20:30:00 +0100