2018-01-23T17:14:56Z
http://www.aot-math.org/?_action=export&rf=summon&issue=6059
Advances in Operator Theory
Adv. Operator Theory (AOT)
2018
3
1
Complex interpolation and non-commutative integration
Klaus
Werner
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.
Hilbert space
interpolation
Banach space
2018
01
01
1
16
http://www.aot-math.org/article_42356_71777b147346763bfff24fc7d39d965f.pdf
Advances in Operator Theory
Adv. Operator Theory (AOT)
2018
3
1
Semicontinuity and closed faces of C*-algebras
Lawrence
Brown
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.
operator algebras
Semicontinuity
Closed projection
Operator convex
2018
01
01
17
41
http://www.aot-math.org/article_43918_bf8da69fd044f09da9c3e4f4db9277c1.pdf
Advances in Operator Theory
Adv. Operator Theory (AOT)
2018
3
1
The closure of ideals of $ell^1(Sigma)$ in its enveloping $mathrm{C}^*$-algebra
Marcel
Jeu
Jun
Tomiyama
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 ${mathrm C}^ast(Sigma)$.
Primary 46K99
Secondary 46H10, 47L65, 54H20
2018
01
01
42
52
http://www.aot-math.org/article_44047_f65a8f1062ea283744db5848a9363ba9.pdf
Advances in Operator Theory
Adv. Operator Theory (AOT)
2018
3
1
Positive map as difference of two completely positive or super-positive maps
Tsuyoshi
Ando
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.
Positive map
completely positive map
super-positive map
Norm
tensor product
2018
01
01
53
60
http://www.aot-math.org/article_44116_531ea9ae7786a407c42b2866cb0dd368.pdf
Advances in Operator Theory
Adv. Operator Theory (AOT)
2018
3
1
Some natural subspaces and quotient spaces of $L^1$
Gilles
Godefroy
Nicolas
Lerner
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.
nicely placed subspaces of $L^1$
Lipschitz-free spaces over $mathbb{R}^n$
subspaces of $l^1$
2018
01
01
61
74
http://www.aot-math.org/article_44924_ba9c3c5c2f3766d6635df15b74db8914.pdf
Advances in Operator Theory
Adv. Operator Theory (AOT)
2018
3
1
Partial isometries: a survey
Antonio
Peralta
Francisco
Fernandez-Polo
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.
partial isometry
von Neumann regularity
Moore-Penrose invertibility
tripotent
reduced minimum modulus
conorm
quadratic-conorm, extreme points
2018
01
01
75
116
http://www.aot-math.org/article_45165_94afcb414af03a75e4a512f171c4db10.pdf
Advances in Operator Theory
Adv. Operator Theory (AOT)
2018
3
1
Operators with compatible ranges in an algebra generated by two orthogonal projections
Ilya
Spitkovsky
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.
hermitian operators
orthogonal projections
von Neumann algebras
2018
01
01
117
122
http://www.aot-math.org/article_45166_aed3194d347d41e51b89267a4029d5d6.pdf
Advances in Operator Theory
Adv. Operator Theory (AOT)
2018
3
1
Permanence of nuclear dimension for inclusions of unital $C^*$-algebras with the Rokhlin property
Hiroyuki
Osaka
Tamotsu
Teruya
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.
Rokhlin property
C*-index
nuclear dimension
2018
01
01
123
136
http://www.aot-math.org/article_45177_609d8347a1a4c02639504efeafda0dce.pdf
Advances in Operator Theory
Adv. Operator Theory (AOT)
2018
3
1
Almost Hadamard matrices with complex entries
Teodor
Banica
Ion
Nechita
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.
Hadamard matrix
Fourier matrix
Unitary group
2018
01
01
137
177
http://www.aot-math.org/article_45905_9d673bba71c41fc688aba52b6f8a1896.pdf
Advances in Operator Theory
Adv. Operator Theory (AOT)
2018
3
1
Non-commutative rational functions in strong convergent random variables
Sheng
Yin
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 $r$.
Strong convergence
non-commutative rational functions
random matrices
2018
01
01
178
192
http://www.aot-math.org/article_46452_614d056f5f7799b607d6277111157ff4.pdf
Advances in Operator Theory
Adv. Operator Theory (AOT)
2018
3
1
Fourier multiplier norms of spherical functions on the generalized Lorentz groups
Troels
Steenstrup
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.
Lie group
completely bounded Fourier multiplier norm
generalized Lorentz group
Representation
spherical function
2018
01
01
193
230
http://www.aot-math.org/article_47035_188d9dff266918afd8450b07fe22a042.pdf
Advances in Operator Theory
Adv. Operator Theory (AOT)
2018
3
1
On a class of Banach algebras associated to harmonic analysis on locally compact groups and semigroups
Anthony To-Ming
Lau
Hung
Pham
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.
Fourier algebra
locally compact group
Group Algebra
Fourier--Stieltjes algebra
$F$-algebra
2018
01
01
231
246
http://www.aot-math.org/article_47586_4d00ddd2b10646cbc0d558bf63b4c156.pdf
Advances in Operator Theory
Adv. Operator Theory (AOT)
2018
3
1
Uniformly bounded representations and completely bounded multipliers of ${rm SL}(2,mathbb{R})$
Francesca
Astengo
Michael
Cowling
Bianca
Di Blasio
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.
Completely bounded multipliers
Fourier algebra
${rm SL}(2,mathbb{R})$
2018
01
01
247
270
http://www.aot-math.org/article_49322_22d66b67cb2a0a25c1f990a92fdf1ff4.pdf
Advances in Operator Theory
Adv. Operator Theory (AOT)
2018
3
1
Completely positive contractive maps and partial isometries
Berndt
Brenken
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.
completely positive dynamical system
partial isometry
$C^$-correspondence
Cuntz--Pimsner $C^*$-algebra
Morita equivalence
2018
01
01
271
294
http://www.aot-math.org/article_49352_b35db7c10682e142099c3a89ec189db7.pdf
Advances in Operator Theory
Adv. Operator Theory (AOT)
2018
3
1
Uffe Haagerup - his life and mathematics
Mohammad Sal
Moslehian
Erling
Stormer
Steen
Thorbjoernsen
Carl
Winslow
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.
Uffe Haagerup
operator algebras
history of mathematics
2018
01
01
295
325
http://www.aot-math.org/article_50017_53b4c3d7f46e44ffb517cab097d0a9ae.pdf