Tusi Mathematical Research Group (TMRG)
Advances in Operator Theory
2538-225X
3
1
2018
01
01
Complex interpolation and non-commutative integration
1
16
EN
Klaus
Werner
klaus.werner@sap.com
10.22034/aot.1611-1061
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
http://www.aot-math.org/article_42356.html
http://www.aot-math.org/article_42356_71777b147346763bfff24fc7d39d965f.pdf
Tusi Mathematical Research Group (TMRG)
Advances in Operator Theory
2538-225X
3
1
2018
01
01
Semicontinuity and closed faces of C*-algebras
17
41
EN
Lawrence
G.
Brown
Purdue University
lgb@math.purdue.edu
10.22034/aot.1611-1048
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
http://www.aot-math.org/article_43918.html
http://www.aot-math.org/article_43918_bf8da69fd044f09da9c3e4f4db9277c1.pdf
Tusi Mathematical Research Group (TMRG)
Advances in Operator Theory
2538-225X
3
1
2018
01
01
The closure of ideals of $ell^1(Sigma)$ in its enveloping $mathrm{C}^*$-algebra
42
52
EN
Marcel
de
Jeu
mdejeu@math.leidenuniv.nl
Jun
Tomiyama
juntomi@med.email.ne.jp
10.22034/aot.1702-1116
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
http://www.aot-math.org/article_44047.html
http://www.aot-math.org/article_44047_f65a8f1062ea283744db5848a9363ba9.pdf
Tusi Mathematical Research Group (TMRG)
Advances in Operator Theory
2538-225X
3
1
2018
01
01
Positive map as difference of two completely positive or super-positive maps
53
60
EN
Tsuyoshi
Ando
ando@es.hokudai.ac.jp
10.22034/aot.1702-1129
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
http://www.aot-math.org/article_44116.html
http://www.aot-math.org/article_44116_531ea9ae7786a407c42b2866cb0dd368.pdf
Tusi Mathematical Research Group (TMRG)
Advances in Operator Theory
2538-225X
3
1
2018
01
01
Some natural subspaces and quotient spaces of $L^1$
61
74
EN
Gilles
Godefroy
godefroy@math.jussieu.fr
Nicolas
Lerner
Universite Pierre et Marie Curie
nicolas.lerner@imj-prg.fr
10.22034/aot.1702-1124
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 $l^1$
http://www.aot-math.org/article_44924.html
http://www.aot-math.org/article_44924_ba9c3c5c2f3766d6635df15b74db8914.pdf
Tusi Mathematical Research Group (TMRG)
Advances in Operator Theory
2538-225X
3
1
2018
01
01
Partial isometries: a survey
75
116
EN
Antonio
Peralta
Universidad de Granada
aperalta@ugr.es
Francisco
J
Fernandez-Polo
Departamento de An&aacute;lisis Matem&aacute;tico, Facultad de Ciencias
pacopolo@ugr.es
10.22034/aot.1703-1149
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
http://www.aot-math.org/article_45165.html
http://www.aot-math.org/article_45165_94afcb414af03a75e4a512f171c4db10.pdf
Tusi Mathematical Research Group (TMRG)
Advances in Operator Theory
2538-225X
3
1
2018
01
01
Operators with compatible ranges in an algebra generated by two orthogonal projections
117
122
EN
Ilya
M
Spitkovsky
NYUAD
imspitkovsky@gmail.com
10.22034/aot.1702-1111
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
http://www.aot-math.org/article_45166.html
http://www.aot-math.org/article_45166_aed3194d347d41e51b89267a4029d5d6.pdf
Tusi Mathematical Research Group (TMRG)
Advances in Operator Theory
2538-225X
3
1
2018
01
01
Permanence of nuclear dimension for inclusions of unital $C^*$-algebras with the Rokhlin property
123
136
EN
Hiroyuki
Osaka
Ritsumeikan University
osaka@se.ritsumei.ac.jp
Tamotsu
Teruya
teruya@gunma-u.ac.jp
10.22034/aot.1703-1145
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
http://www.aot-math.org/article_45177.html
http://www.aot-math.org/article_45177_609d8347a1a4c02639504efeafda0dce.pdf
Tusi Mathematical Research Group (TMRG)
Advances in Operator Theory
2538-225X
3
1
2018
01
01
Almost Hadamard matrices with complex entries
137
177
EN
Teodor
Banica
Cergy-Pontoise University
teo.banica@gmail.com
Ion
Nechita
Dept. of Math. TU Munich
ion.nechita@gmail.com
10.22034/aot.1702-1114
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
http://www.aot-math.org/article_45905.html
http://www.aot-math.org/article_45905_9d673bba71c41fc688aba52b6f8a1896.pdf
Tusi Mathematical Research Group (TMRG)
Advances in Operator Theory
2538-225X
3
1
2018
01
01
Non-commutative rational functions in strong convergent random variables
178
192
EN
Sheng
Yin
Faculty of Mathematics, Saarland University
yin@math.uni-sb.de
10.22034/aot.1702-1126
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
http://www.aot-math.org/article_46452.html
http://www.aot-math.org/article_46452_614d056f5f7799b607d6277111157ff4.pdf
Tusi Mathematical Research Group (TMRG)
Advances in Operator Theory
2538-225X
3
1
2018
01
01
Fourier multiplier norms of spherical functions on the generalized Lorentz groups
193
230
EN
Troels
Steenstrup
aot@troelssj.dk
10.22034/aot.1706-1172
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
http://www.aot-math.org/article_47035.html
http://www.aot-math.org/article_47035_188d9dff266918afd8450b07fe22a042.pdf
Tusi Mathematical Research Group (TMRG)
Advances in Operator Theory
2538-225X
3
1
2018
01
01
On a class of Banach algebras associated to harmonic analysis on locally compact groups and semigroups
231
246
EN
Anthony To-Ming
Lau
University of Alberta
anthonyt@ualberta.ca
Hung
Le
Pham
hung.pham@vuw.ac.nz
10.22034/aot.1702-1115
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
http://www.aot-math.org/article_47586.html
http://www.aot-math.org/article_47586_4d00ddd2b10646cbc0d558bf63b4c156.pdf
Tusi Mathematical Research Group (TMRG)
Advances in Operator Theory
2538-225X
3
1
2018
01
01
Uniformly bounded representations and completely bounded multipliers of ${rm SL}(2,mathbb{R})$
247
270
EN
Francesca
Astengo
astengo@dima.unige.it
Michael
G.
Cowling
m.cowling@unsw.edu.au
Bianca
Di Blasio
bianca.diblasio@unimib.it
10.22034/aot.1707-1207
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,${rm SL}(2,mathbb{R})$
http://www.aot-math.org/article_49322.html
http://www.aot-math.org/article_49322_22d66b67cb2a0a25c1f990a92fdf1ff4.pdf
Tusi Mathematical Research Group (TMRG)
Advances in Operator Theory
2538-225X
3
1
2018
01
01
Completely positive contractive maps and partial isometries
271
294
EN
Berndt
Brenken
bbrenken@ucalgary.ca
10.22034/aot.1703-1131
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
http://www.aot-math.org/article_49352.html
http://www.aot-math.org/article_49352_b35db7c10682e142099c3a89ec189db7.pdf
Tusi Mathematical Research Group (TMRG)
Advances in Operator Theory
2538-225X
3
1
2018
01
01
Uffe Haagerup - his life and mathematics
295
325
EN
Mohammad Sal
Moslehian
Department of Pure Mathematics, Ferdowsi University of Mashhad, P. O. Box 1159, Mashhad 91775, Iran
moslehian@um.ac.ir
Erling
Stormer
Department of Mathematics, The Faculty of Mathematics and Natural Sci-
ences, University of Oslo, Norway.
erlings@math.uio.no
Steen
Thorbjoernsen
Department of Mathematics, Faculty of Science and Technology, University of Aarhus, Denmark
steenth@math.au.dk
Carl
Winslow
Department of Science Education, Faculty of Science, University of Copen-
hagen, Denmark.
winslow@ind.ku.dk
10.22034/aot.1708-1213
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
http://www.aot-math.org/article_50017.html
http://www.aot-math.org/article_50017_53b4c3d7f46e44ffb517cab097d0a9ae.pdf