@Article{Werner2018,
author="Werner, Klaus",
title="Complex interpolation and non-commutative integration",
journal="Advances in Operator Theory",
year="2018",
volume="3",
number="1",
pages="1-16",
abstract="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.",
issn="2538-225X",
doi="10.22034/aot.1611-1061",
url="http://www.aot-math.org/article_42356.html"
}
@Article{Brown2018,
author="Brown, Lawrence G.",
title="Semicontinuity and closed faces of C*-algebras",
journal="Advances in Operator Theory",
year="2018",
volume="3",
number="1",
pages="17-41",
abstract="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 $h\geq k$, 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 $k\leq x\leq h$. 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 $k\leq\widetilde x\leq h$. 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.",
issn="2538-225X",
doi="10.22034/aot.1611-1048",
url="http://www.aot-math.org/article_43918.html"
}
@Article{Jeu2018,
author="Jeu, Marcel de
and Tomiyama, Jun",
title="The closure of ideals of $\ell^1(\Sigma)$ in its enveloping $\mathrm{C}^*$-algebra",
journal="Advances in Operator Theory",
year="2018",
volume="3",
number="1",
pages="42-52",
abstract="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 ${\mathrm C}^\ast(\Sigma)$.",
issn="2538-225X",
doi="10.22034/aot.1702-1116",
url="http://www.aot-math.org/article_44047.html"
}
@Article{Ando2018,
author="Ando, Tsuyoshi",
title="Positive map as difference of two completely positive or super-positive maps",
journal="Advances in Operator Theory",
year="2018",
volume="3",
number="1",
pages="53-60",
abstract="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.",
issn="2538-225X",
doi="10.22034/aot.1702-1129",
url="http://www.aot-math.org/article_44116.html"
}
@Article{Godefroy2018,
author="Godefroy, Gilles
and Lerner, Nicolas",
title="Some natural subspaces and quotient spaces of $L^1$",
journal="Advances in Operator Theory",
year="2018",
volume="3",
number="1",
pages="61-74",
abstract="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.",
issn="2538-225X",
doi="10.22034/aot.1702-1124",
url="http://www.aot-math.org/article_44924.html"
}
@Article{Peralta2018,
author="Peralta, Antonio
and Fernandez-Polo, Francisco J",
title="Partial isometries: a survey",
journal="Advances in Operator Theory",
year="2018",
volume="3",
number="1",
pages="75-116",
abstract="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.",
issn="2538-225X",
doi="10.22034/aot.1703-1149",
url="http://www.aot-math.org/article_45165.html"
}
@Article{Spitkovsky2018,
author="Spitkovsky, Ilya M",
title="Operators with compatible ranges in an algebra generated by two orthogonal projections",
journal="Advances in Operator Theory",
year="2018",
volume="3",
number="1",
pages="117-122",
abstract="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.",
issn="2538-225X",
doi="10.22034/aot.1702-1111",
url="http://www.aot-math.org/article_45166.html"
}
@Article{Osaka2018,
author="Osaka, Hiroyuki
and Teruya, Tamotsu",
title="Permanence of nuclear dimension for inclusions of unital $C^*$-algebras with the Rokhlin property",
journal="Advances in Operator Theory",
year="2018",
volume="3",
number="1",
pages="123-136",
abstract="Let $P \subset A$ be an inclusion of unital $C^*$-algebras and $E\colon A \rightarrow P$ 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 $C^*\langle A, e_P\rangle$ are also pure.",
issn="2538-225X",
doi="10.22034/aot.1703-1145",
url="http://www.aot-math.org/article_45177.html"
}
@Article{Banica2018,
author="Banica, Teodor
and Nechita, Ion",
title="Almost Hadamard matrices with complex entries",
journal="Advances in Operator Theory",
year="2018",
volume="3",
number="1",
pages="137-177",
abstract="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.",
issn="2538-225X",
doi="10.22034/aot.1702-1114",
url="http://www.aot-math.org/article_45905.html"
}
@Article{Yin2018,
author="Yin, Sheng",
title="Non-commutative rational functions in strong convergent random variables",
journal="Advances in Operator Theory",
year="2018",
volume="3",
number="1",
pages="178-192",
abstract="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 $(X_{1}^{\left(n\right)},\cdots,X_{m}^{\left(n\right)})$ of independent GUE random matrices, $r(X_{1}^{\left(n\right)},\cdots,X_{m}^{\left(n\right)})$ 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 $r$.",
issn="2538-225X",
doi="10.22034/aot.1702-1126",
url="http://www.aot-math.org/article_46452.html"
}
@Article{Steenstrup2018,
author="Steenstrup, Troels",
title="Fourier multiplier norms of spherical functions on the generalized Lorentz groups",
journal="Advances in Operator Theory",
year="2018",
volume="3",
number="1",
pages="193-230",
abstract="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 $n\geq2$). 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 $n\geq2$) 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.",
issn="2538-225X",
doi="10.22034/aot.1706-1172",
url="http://www.aot-math.org/article_47035.html"
}
@Article{Lau2018,
author="Lau, Anthony To-Ming
and Pham, Hung Le",
title="On a class of Banach algebras associated to harmonic analysis on locally compact groups and semigroups",
journal="Advances in Operator Theory",
year="2018",
volume="3",
number="1",
pages="231-246",
abstract="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.",
issn="2538-225X",
doi="10.22034/aot.1702-1115",
url="http://www.aot-math.org/article_47586.html"
}
@Article{Astengo2018,
author="Astengo, Francesca
and Cowling, Michael G.
and Di Blasio, Bianca",
title="Uniformly bounded representations and completely bounded multipliers of ${\rm SL}(2,\mathbb{R})$",
journal="Advances in Operator Theory",
year="2018",
volume="3",
number="1",
pages="247-270",
abstract="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.",
issn="2538-225X",
doi="10.22034/aot.1707-1207",
url="http://www.aot-math.org/article_49322.html"
}
@Article{Brenken2018,
author="Brenken, Berndt",
title="Completely positive contractive maps and partial isometries",
journal="Advances in Operator Theory",
year="2018",
volume="3",
number="1",
pages="271-294",
abstract="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.",
issn="2538-225X",
doi="10.22034/aot.1703-1131",
url="http://www.aot-math.org/article_49352.html"
}
@Article{Moslehian2018,
author="Moslehian, Mohammad Sal
and Stormer, Erling
and Thorbjoernsen, Steen
and Winslow, Carl",
title="Uffe Haagerup - his life and mathematics",
journal="Advances in Operator Theory",
year="2018",
volume="3",
number="1",
pages="295-325",
abstract="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.",
issn="2538-225X",
doi="10.22034/aot.1708-1213",
url="http://www.aot-math.org/article_50017.html"
}