Complex interpolation and non-commutative integration
Klaus
Werner
author
text
article
2018
eng
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.
Advances in Operator Theory
Tusi Mathematical Research Group (TMRG)
2538-225X
3
v.
1
no.
2018
1
16
http://www.aot-math.org/article_42356_71777b147346763bfff24fc7d39d965f.pdf
dx.doi.org/10.22034/aot.1611-1061
Semicontinuity and closed faces of C*-algebras
Lawrence
Brown
Purdue University
author
text
article
2018
eng
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.
Advances in Operator Theory
Tusi Mathematical Research Group (TMRG)
2538-225X
3
v.
1
no.
2018
17
41
http://www.aot-math.org/article_43918_bf8da69fd044f09da9c3e4f4db9277c1.pdf
dx.doi.org/10.22034/aot.1611-1048
The closure of ideals of $\ell^1(\Sigma)$ in its enveloping $\mathrm{C}^*$-algebra
Marcel
Jeu
author
Jun
Tomiyama
author
text
article
2018
eng
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 ${\mathrm C}^\ast(\Sigma)$.
Advances in Operator Theory
Tusi Mathematical Research Group (TMRG)
2538-225X
3
v.
1
no.
2018
42
52
http://www.aot-math.org/article_44047_f65a8f1062ea283744db5848a9363ba9.pdf
dx.doi.org/10.22034/aot.1702-1116
Positive map as difference of two completely positive or super-positive maps
Tsuyoshi
Ando
author
text
article
2018
eng
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.
Advances in Operator Theory
Tusi Mathematical Research Group (TMRG)
2538-225X
3
v.
1
no.
2018
53
60
http://www.aot-math.org/article_44116_531ea9ae7786a407c42b2866cb0dd368.pdf
dx.doi.org/10.22034/aot.1702-1129
Some natural subspaces and quotient spaces of $L^1$
Gilles
Godefroy
author
Nicolas
Lerner
Universite Pierre et Marie Curie
author
text
article
2018
eng
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.
Advances in Operator Theory
Tusi Mathematical Research Group (TMRG)
2538-225X
3
v.
1
no.
2018
61
74
http://www.aot-math.org/article_44924_ba9c3c5c2f3766d6635df15b74db8914.pdf
dx.doi.org/10.22034/aot.1702-1124
Partial isometries: a survey
Antonio
Peralta
Universidad de Granada
author
Francisco
Fernandez-Polo
Departamento de An&aacute;lisis Matem&aacute;tico, Facultad de Ciencias
author
text
article
2018
eng
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.
Advances in Operator Theory
Tusi Mathematical Research Group (TMRG)
2538-225X
3
v.
1
no.
2018
75
116
http://www.aot-math.org/article_45165_94afcb414af03a75e4a512f171c4db10.pdf
dx.doi.org/10.22034/aot.1703-1149
Operators with compatible ranges in an algebra generated by two orthogonal projections
Ilya
Spitkovsky
NYUAD
author
text
article
2018
eng
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.
Advances in Operator Theory
Tusi Mathematical Research Group (TMRG)
2538-225X
3
v.
1
no.
2018
117
122
http://www.aot-math.org/article_45166_aed3194d347d41e51b89267a4029d5d6.pdf
dx.doi.org/10.22034/aot.1702-1111
Permanence of nuclear dimension for inclusions of unital $C^*$-algebras with the Rokhlin property
Hiroyuki
Osaka
Ritsumeikan University
author
Tamotsu
Teruya
author
text
article
2018
eng
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.
Advances in Operator Theory
Tusi Mathematical Research Group (TMRG)
2538-225X
3
v.
1
no.
2018
123
136
http://www.aot-math.org/article_45177_609d8347a1a4c02639504efeafda0dce.pdf
dx.doi.org/10.22034/aot.1703-1145
Almost Hadamard matrices with complex entries
Teodor
Banica
Cergy-Pontoise University
author
Ion
Nechita
Dept. of Math. TU Munich
author
text
article
2018
eng
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.
Advances in Operator Theory
Tusi Mathematical Research Group (TMRG)
2538-225X
3
v.
1
no.
2018
137
177
http://www.aot-math.org/article_45905_9d673bba71c41fc688aba52b6f8a1896.pdf
dx.doi.org/10.22034/aot.1702-1114
Non-commutative rational functions in strong convergent random variables
Sheng
Yin
Faculty of Mathematics, Saarland University
author
text
article
2018
eng
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 $r$.
Advances in Operator Theory
Tusi Mathematical Research Group (TMRG)
2538-225X
3
v.
1
no.
2018
178
192
http://www.aot-math.org/article_46452_614d056f5f7799b607d6277111157ff4.pdf
dx.doi.org/10.22034/aot.1702-1126
Fourier multiplier norms of spherical functions on the generalized Lorentz groups
Troels
Steenstrup
author
text
article
2018
eng
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.
Advances in Operator Theory
Tusi Mathematical Research Group (TMRG)
2538-225X
3
v.
1
no.
2018
193
230
http://www.aot-math.org/article_47035_188d9dff266918afd8450b07fe22a042.pdf
dx.doi.org/10.22034/aot.1706-1172
On a class of Banach algebras associated to harmonic analysis on locally compact groups and semigroups
Anthony To-Ming
Lau
University of Alberta
author
Hung
Pham
author
text
article
2018
eng
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.
Advances in Operator Theory
Tusi Mathematical Research Group (TMRG)
2538-225X
3
v.
1
no.
2018
231
246
http://www.aot-math.org/article_47586_4d00ddd2b10646cbc0d558bf63b4c156.pdf
dx.doi.org/10.22034/aot.1702-1115
Uniformly bounded representations and completely bounded multipliers of ${\rm SL}(2,\mathbb{R})$
Francesca
Astengo
author
Michael
Cowling
author
Bianca
Di Blasio
author
text
article
2018
eng
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.
Advances in Operator Theory
Tusi Mathematical Research Group (TMRG)
2538-225X
3
v.
1
no.
2018
247
270
http://www.aot-math.org/article_49322_22d66b67cb2a0a25c1f990a92fdf1ff4.pdf
dx.doi.org/10.22034/aot.1707-1207
Completely positive contractive maps and partial isometries
Berndt
Brenken
author
text
article
2018
eng
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.
Advances in Operator Theory
Tusi Mathematical Research Group (TMRG)
2538-225X
3
v.
1
no.
2018
271
294
http://www.aot-math.org/article_49352_b35db7c10682e142099c3a89ec189db7.pdf
dx.doi.org/10.22034/aot.1703-1131
Uffe Haagerup - his life and mathematics
Mohammad Sal
Moslehian
Department of Pure Mathematics, Ferdowsi University of Mashhad, P. O. Box 1159, Mashhad 91775, Iran
author
Erling
Stormer
Department of Mathematics, The Faculty of Mathematics and Natural Sci-
ences, University of Oslo, Norway.
author
Steen
Thorbjoernsen
Department of Mathematics, Faculty of Science and Technology, University of Aarhus, Denmark
author
Carl
Winslow
Department of Science Education, Faculty of Science, University of Copen-
hagen, Denmark.
author
text
article
2018
eng
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.
Advances in Operator Theory
Tusi Mathematical Research Group (TMRG)
2538-225X
3
v.
1
no.
2018
295
325
http://www.aot-math.org/article_50017_53b4c3d7f46e44ffb517cab097d0a9ae.pdf
dx.doi.org/10.22034/aot.1708-1213