eng
Tusi Mathematical Research Group (TMRG)
Advances in Operator Theory
2538-225X
2538-225X
2018-01-01
3
1
1
16
10.22034/aot.1611-1061
42356
Complex interpolation and non-commutative integration
Klaus Werner
klaus.werner@sap.com
1
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.
http://www.aot-math.org/article_42356_71777b147346763bfff24fc7d39d965f.pdf
Hilbert space
Interpolation
Banach space
eng
Tusi Mathematical Research Group (TMRG)
Advances in Operator Theory
2538-225X
2538-225X
2018-01-01
3
1
17
41
10.22034/aot.1611-1048
43918
Semicontinuity and closed faces of C*-algebras
Lawrence Brown
lgb@math.purdue.edu
1
Purdue University
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.
http://www.aot-math.org/article_43918_bf8da69fd044f09da9c3e4f4db9277c1.pdf
operator algebras
Semicontinuity
Closed projection
Operator convex
eng
Tusi Mathematical Research Group (TMRG)
Advances in Operator Theory
2538-225X
2538-225X
2018-01-01
3
1
42
52
10.22034/aot.1702-1116
44047
The closure of ideals of $ell^1(Sigma)$ in its enveloping $mathrm{C}^*$-algebra
Marcel Jeu
mdejeu@math.leidenuniv.nl
1
Jun Tomiyama
juntomi@med.email.ne.jp
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 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 ${mathrm C}^ast(Sigma)$.
http://www.aot-math.org/article_44047_f65a8f1062ea283744db5848a9363ba9.pdf
Primary 46K99
Secondary 46H10, 47L65, 54H20
eng
Tusi Mathematical Research Group (TMRG)
Advances in Operator Theory
2538-225X
2538-225X
2018-01-01
3
1
53
60
10.22034/aot.1702-1129
44116
Positive map as difference of two completely positive or super-positive maps
Tsuyoshi Ando
ando@es.hokudai.ac.jp
1
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.
http://www.aot-math.org/article_44116_531ea9ae7786a407c42b2866cb0dd368.pdf
Positive map
completely positive map
super-positive map
Norm
tensor product
eng
Tusi Mathematical Research Group (TMRG)
Advances in Operator Theory
2538-225X
2538-225X
2018-01-01
3
1
61
74
10.22034/aot.1702-1124
44924
Some natural subspaces and quotient spaces of $L^1$
Gilles Godefroy
godefroy@math.jussieu.fr
1
Nicolas Lerner
nicolas.lerner@imj-prg.fr
2
Universite Pierre et Marie Curie
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.
http://www.aot-math.org/article_44924_ba9c3c5c2f3766d6635df15b74db8914.pdf
nicely placed subspaces of $L^1$
Lipschitz-free spaces over $mathbb{R}^n$
subspaces of $l^1$
eng
Tusi Mathematical Research Group (TMRG)
Advances in Operator Theory
2538-225X
2538-225X
2018-01-01
3
1
75
116
10.22034/aot.1703-1149
45165
Partial isometries: a survey
Antonio Peralta
aperalta@ugr.es
1
Francisco Fernandez-Polo
pacopolo@ugr.es
2
Universidad de Granada
Departamento de An&aacute;lisis Matem&aacute;tico, Facultad de Ciencias
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.
http://www.aot-math.org/article_45165_94afcb414af03a75e4a512f171c4db10.pdf
partial isometry
von Neumann regularity
Moore-Penrose invertibility
tripotent
reduced minimum modulus
conorm
quadratic-conorm, extreme points
eng
Tusi Mathematical Research Group (TMRG)
Advances in Operator Theory
2538-225X
2538-225X
2018-01-01
3
1
117
122
10.22034/aot.1702-1111
45166
Operators with compatible ranges in an algebra generated by two orthogonal projections
Ilya Spitkovsky
imspitkovsky@gmail.com
1
NYUAD
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.
http://www.aot-math.org/article_45166_aed3194d347d41e51b89267a4029d5d6.pdf
hermitian operators
orthogonal projections
von Neumann algebras
eng
Tusi Mathematical Research Group (TMRG)
Advances in Operator Theory
2538-225X
2538-225X
2018-01-01
3
1
123
136
10.22034/aot.1703-1145
45177
Permanence of nuclear dimension for inclusions of unital $C^*$-algebras with the Rokhlin property
Hiroyuki Osaka
osaka@se.ritsumei.ac.jp
1
Tamotsu Teruya
teruya@gunma-u.ac.jp
2
Ritsumeikan University
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.
http://www.aot-math.org/article_45177_609d8347a1a4c02639504efeafda0dce.pdf
Rokhlin property
C*-index
nuclear dimension
eng
Tusi Mathematical Research Group (TMRG)
Advances in Operator Theory
2538-225X
2538-225X
2018-01-01
3
1
137
177
10.22034/aot.1702-1114
45905
Almost Hadamard matrices with complex entries
Teodor Banica
teo.banica@gmail.com
1
Ion Nechita
ion.nechita@gmail.com
2
Cergy-Pontoise University
Dept. of Math. TU Munich
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.
http://www.aot-math.org/article_45905_9d673bba71c41fc688aba52b6f8a1896.pdf
Hadamard matrix
Fourier matrix
Unitary group
eng
Tusi Mathematical Research Group (TMRG)
Advances in Operator Theory
2538-225X
2538-225X
2018-01-01
3
1
178
192
10.22034/aot.1702-1126
46452
Non-commutative rational functions in strong convergent random variables
Sheng Yin
yin@math.uni-sb.de
1
Faculty of Mathematics, Saarland University
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 $r$.
http://www.aot-math.org/article_46452_614d056f5f7799b607d6277111157ff4.pdf
Strong convergence
non-commutative rational functions
random matrices
eng
Tusi Mathematical Research Group (TMRG)
Advances in Operator Theory
2538-225X
2538-225X
2018-01-01
3
1
193
230
10.22034/aot.1706-1172
47035
Fourier multiplier norms of spherical functions on the generalized Lorentz groups
Troels Steenstrup
aot@troelssj.dk
1
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.
http://www.aot-math.org/article_47035_188d9dff266918afd8450b07fe22a042.pdf
Lie group
completely bounded Fourier multiplier norm
generalized Lorentz group
representation
spherical function
eng
Tusi Mathematical Research Group (TMRG)
Advances in Operator Theory
2538-225X
2538-225X
2018-01-01
3
1
231
246
10.22034/aot.1702-1115
47586
On a class of Banach algebras associated to harmonic analysis on locally compact groups and semigroups
Anthony To-Ming Lau
anthonyt@ualberta.ca
1
Hung Pham
hung.pham@vuw.ac.nz
2
University of Alberta
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.
http://www.aot-math.org/article_47586_4d00ddd2b10646cbc0d558bf63b4c156.pdf
Fourier algebra
locally compact group
Group Algebra
Fourier--Stieltjes algebra
$F$-algebra
eng
Tusi Mathematical Research Group (TMRG)
Advances in Operator Theory
2538-225X
2538-225X
2018-01-01
3
1
247
270
10.22034/aot.1707-1207
49322
Uniformly bounded representations and completely bounded multipliers of ${rm SL}(2,mathbb{R})$
Francesca Astengo
astengo@dima.unige.it
1
Michael Cowling
m.cowling@unsw.edu.au
2
Bianca Di Blasio
bianca.diblasio@unimib.it
3
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.
http://www.aot-math.org/article_49322_22d66b67cb2a0a25c1f990a92fdf1ff4.pdf
Completely bounded multipliers
Fourier algebra
${rm SL}(2,mathbb{R})$
eng
Tusi Mathematical Research Group (TMRG)
Advances in Operator Theory
2538-225X
2538-225X
2018-01-01
3
1
271
294
10.22034/aot.1703-1131
49352
Completely positive contractive maps and partial isometries
Berndt Brenken
bbrenken@ucalgary.ca
1
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.
http://www.aot-math.org/article_49352_b35db7c10682e142099c3a89ec189db7.pdf
completely positive dynamical system
partial isometry
$C^$-correspondence
Cuntz--Pimsner $C^*$-algebra
Morita equivalence
eng
Tusi Mathematical Research Group (TMRG)
Advances in Operator Theory
2538-225X
2538-225X
2018-01-01
3
1
295
325
10.22034/aot.1708-1213
50017
Uffe Haagerup - his life and mathematics
Mohammad Sal Moslehian
moslehian@um.ac.ir
1
Erling Stormer
erlings@math.uio.no
2
Steen Thorbjoernsen
steenth@math.au.dk
3
Carl Winslow
winslow@ind.ku.dk
4
Department of Pure Mathematics, Ferdowsi University of Mashhad, P. O. Box 1159, Mashhad 91775, Iran
Department of Mathematics, The Faculty of Mathematics and Natural Sci- ences, University of Oslo, Norway.
Department of Mathematics, Faculty of Science and Technology, University of Aarhus, Denmark
Department of Science Education, Faculty of Science, University of Copen- hagen, Denmark.
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.
http://www.aot-math.org/article_50017_53b4c3d7f46e44ffb517cab097d0a9ae.pdf
Uffe Haagerup
operator algebras
history of mathematics