2018
3
1
7
325
Complex interpolation and noncommutative integration
2
2
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 noncommutative $L^p$spaces of Leinert [Int. J. Math. textbf{2} (1991), no. 2, 177182] and Terp [J. Operator Theory textbf{8} (1982), 327360] 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), 151169.] and Haagerup [Colloq. Internat. CNRS, 274, CNRS, Paris, 1979].\Haagerup and Pisier [Canad. J. Math. textbf{41} (1989), no. 5, 882906.], 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. 14371440, 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.
1

1
16


Klaus
Werner
Germany
klaus.werner@sap.com
Hilbert space
interpolation
Banach space
Semicontinuity and closed faces of C*algebras
2
2
C. Akemann and G.K. Pedersen [Duke Math. J. 40 (1973), 785795.] defined three concepts of semicontinuity for selfadjoint 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 quasistate 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 interpolationextension 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.
1

17
41


Lawrence
Brown
Purdue University
Purdue University
USA
lgb@math.purdue.edu
operator algebras
Semicontinuity
Closed projection
Operator convex
The closure of ideals of $ell^1(Sigma)$ in its enveloping $mathrm{C}^*$algebra
2
2
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 twosided 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 twosided ideal of $ell^1(Sigma)$ in ${mathrm C}^ast(Sigma)$ is again a proper twosided ideal of ${mathrm C}^ast(Sigma)$.
1

42
52


Marcel
Jeu
Netherlands
mdejeu@math.leidenuniv.nl


Jun
Tomiyama
USA
juntomi@med.email.ne.jp
Primary 46K99
Secondary 46H10, 47L65, 54H20
Positive map as difference of two completely positive or superpositive maps
2
2
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 superpositivity in place of complete positivity.
1

53
60


Tsuyoshi
Ando
Japan
ando@es.hokudai.ac.jp
Positive map
completely positive map
superpositive map
Norm
tensor product
Some natural subspaces and quotient spaces of $L^1$
2
2
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 starshaped subset of $mathbb {R}^n$ and $X$ is a dilationstable closed subspace of $L^1(Omega)$ consisting of continuous functions, then the unit ball of $X$ is compact for the compactopen topology on $Omega$. It follows in particular that such spaces $X$, when they have Grothendieck's approximation property, have unconditional finitedimensional decompositions and are isomorphic to weak*closed subspaces of $l^1$. Numerous examples are provided where such results apply.
1

61
74


Gilles
Godefroy
France
godefroy@math.jussieu.fr


Nicolas
Lerner
Universite Pierre et Marie Curie
Universite Pierre et Marie Curie
France
nicolas.lerner@imjprg.fr
nicely placed subspaces of $L^1$
Lipschitzfree spaces over $mathbb{R}^n$
subspaces of $l^1$
Partial isometries: a survey
2
2
We survey the main results characterizing partial isometries in C$^*$algebras and tripotents in JB$^*$triples obtained in terms of regularity, conorm, quadraticconorm, and the geometric structure of the underlying Banach spaces.
1

75
116


Antonio
Peralta
Universidad de Granada
Universidad de Granada
Spain
aperalta@ugr.es


Francisco
FernandezPolo
Departamento de An&aacute;lisis Matem&aacute;tico, Facultad de Ciencias
Departamento de An&aacute;lisis Matem&
Spain
pacopolo@ugr.es
partial isometry
von Neumann regularity
MoorePenrose invertibility
tripotent
reduced minimum modulus
conorm
quadraticconorm, extreme points
Operators with compatible ranges in an algebra generated by two orthogonal projections
2
2
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.
1

117
122


Ilya
Spitkovsky
NYUAD
NYUAD
United Arab Emirates
imspitkovsky@gmail.com
hermitian operators
orthogonal projections
von Neumann algebras
Permanence of nuclear dimension for inclusions of unital $C^*$algebras with the Rokhlin property
2
2
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.
1

123
136


Hiroyuki
Osaka
Ritsumeikan University
Ritsumeikan University
Japan
osaka@se.ritsumei.ac.jp


Tamotsu
Teruya
Japan
teruya@gunmau.ac.jp
Rokhlin property
C*index
nuclear dimension
Almost Hadamard matrices with complex entries
2
2
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.
1

137
177


Teodor
Banica
CergyPontoise University
CergyPontoise University
France
teo.banica@gmail.com


Ion
Nechita
Dept. of Math. TU Munich
Dept. of Math. TU Munich
Germany
ion.nechita@gmail.com
Hadamard matrix
Fourier matrix
Unitary group
Noncommutative rational functions in strong convergent random variables
2
2
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, noncommutative polynomials can be extended to noncommutative 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 semicircular elements which lies in the domain of $r$.
1

178
192


Sheng
Yin
Faculty of Mathematics, Saarland University
Faculty of Mathematics, Saarland University
Germany
yin@math.unisb.de
Strong convergence
noncommutative rational functions
random matrices
Fourier multiplier norms of spherical functions on the generalized Lorentz groups
2
2
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.
1

193
230


Troels
Steenstrup
Denmark
aot@troelssj.dk
Lie group
completely bounded Fourier multiplier norm
generalized Lorentz group
Representation
spherical function
On a class of Banach algebras associated to harmonic analysis on locally compact groups and semigroups
2
2
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) FourierStieltjes algebras, and use this to give new characterisations the reduced FourierStieltjes algebras of discrete groups.
1

231
246


Anthony ToMing
Lau
University of Alberta
University of Alberta
Canada
anthonyt@ualberta.ca


Hung
Pham
New Zealand
hung.pham@vuw.ac.nz
Fourier algebra
locally compact group
Group Algebra
FourierStieltjes algebra
$F$algebra
Uniformly bounded representations and completely bounded multipliers of ${rm SL}(2,mathbb{R})$
2
2
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.
1

247
270


Francesca
Astengo
Italy
astengo@dima.unige.it


Michael
Cowling
Australia
m.cowling@unsw.edu.au


Bianca
Di Blasio
Italy
bianca.diblasio@unimib.it
Completely bounded multipliers
Fourier algebra
${rm SL}(2,mathbb{R})$
Completely positive contractive maps and partial isometries
2
2
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.
1

271
294


Berndt
Brenken
Canada
bbrenken@ucalgary.ca
completely positive dynamical system
partial isometry
$C^$correspondence
CuntzPimsner $C^*$algebra
Morita equivalence
Uffe Haagerup  his life and mathematics
2
2
In remembrance of Professor Uffe Valentin Haagerup (19492015), as a brilliant mathematician, we review some aspects of his life, and his outstanding mathematical accomplishments.
1

295
325


Mohammad Sal
Moslehian
Department of Pure Mathematics, Ferdowsi University of Mashhad, P. O. Box 1159, Mashhad 91775, Iran
Department of Pure Mathematics, Ferdowsi
Iran
moslehian@um.ac.ir


Erling
Stormer
Department of Mathematics, The Faculty of Mathematics and Natural Sci
ences, University of Oslo, Norway.
Department of Mathematics, The Faculty of
USA
erlings@math.uio.no


Steen
Thorbjoernsen
Department of Mathematics, Faculty of Science and Technology, University of Aarhus, Denmark
Department of Mathematics, Faculty of Science
Denmark
steenth@math.au.dk


Carl
Winslow
Department of Science Education, Faculty of Science, University of Copen
hagen, Denmark.
Department of Science Education, Faculty
Denmark
winslow@ind.ku.dk
Uffe Haagerup
operator algebras
history of mathematics