Homomorphic conditional expectations as noncommutative retractions
Robert
Pluta
Department of Mathematics, University of California, Irvine
author
Bernard
Russo
Department of Mathematics, University of
California, Irvine
author
text
article
2017
eng
Let $A$ be a $C^*$-algebra and $\mathcal{E}\colon A \to A$ a conditional expectation. The Kadison-Schwarz inequality for completely positive maps, $$\mathcal{E}(x)^* \mathcal{E}(x) \leq \mathcal{E}(x^* x),$$implies that$$\left\Vert\mathcal{E}(x)\right\Vert^2 \leq \left\Vert\mathcal{E}(x^* x)\right\Vert.$$In this note we show that $\mathcal{E}$ is homomorphic (in the sense that $\mathcal{E}(xy) = \mathcal{E}(x)\mathcal{E}(y)$ for every $x, y$ in $A$) if and only if$$\left\Vert\mathcal{E}(x)\right\Vert^2 = \left\Vert\mathcal{E}(x^*x)\right\Vert,$$for every $x$ in $A$. We also prove that a homomorphic conditional expectation on a commutative $C^*$-algebra $C_0(X)$ is given by composition with a continuous retraction of $X$. One may therefore consider homomorphic conditional expectations as noncommutative retractions.
Advances in Operator Theory
Tusi Mathematical Research Group (TMRG)
2538-225X
2
v.
4
no.
2017
396
408
http://www.aot-math.org/article_46633_0951f9fe6e9ebddc2336176992f3fa2a.pdf
dx.doi.org/10.22034/aot.1705-1161
Variants of Weyl's theorem for direct sums of closed linear operators
Anuradha
Gupta
University of Delhi, Delhi.
author
Karuna
Mamtani
author
text
article
2017
eng
If $T$ is an operator with compact resolvent and $S$ is any densely defined closed linear operator, then the orthogonal direct sum of $T$ and $S$ satisfies various Weyl type theorems if some necessary conditions are imposed on the operator $S$. It is shown that if $S$ is isoloid and satisfies Weyl's theorem, then $T \oplus S$ satisfies Weyl's theorem. Analogous result is proved for a-Weyl's theorem. Further, it is shown that Browder's theorem is directly transmitted from $S$ to $T \oplus S$. The converse of these results have also been studied.
Advances in Operator Theory
Tusi Mathematical Research Group (TMRG)
2538-225X
2
v.
4
no.
2017
409
418
http://www.aot-math.org/article_46634_1d7bcdb1b169b981bcfa677780fe1a1b.pdf
dx.doi.org/10.22034/aot.1701-1087
On orthogonal decomposition of a Sobolev space
Dejenie
Lakew
Bryant & Stratton College
author
text
article
2017
eng
The theme of this short article is to investigate an orthogonal decomposition of the Sobolev space $W^{1,2}\left( \Omega \right) $ as $ W^{1,2}\left( \Omega \right) =A^{2,2}\left( \Omega \right) \oplus D^{2}\left( W_{0}^{3,2}\left( \Omega \right) \right)$ and look at some of the properties of the inner product therein and the distance defined from the inner product. We also determine the dimension of\ the orthogonal difference space $W^{1,2}\left( \Omega \right) \ominus \left(W_{0}^{1,2}\left( \Omega \right) \right) ^{\perp }$ and show the expansion of Sobolev spaces as their regularity increases.
Advances in Operator Theory
Tusi Mathematical Research Group (TMRG)
2538-225X
2
v.
4
no.
2017
419
427
http://www.aot-math.org/article_46656_2f36d0c24bbc89d4e269167986e59a54.pdf
dx.doi.org/10.22034/aot.1703-1135
On symmetry of Birkhoff-James orthogonality of linear operators
Puja
Ghosh
Jadavpur UNiversity
author
Debmalya
Sain
author
Kallol
Paul
Jadavpur UNiversity
author
text
article
2017
eng
A bounded linear operator $T$ on a normed linear space $\mathbb{X}$ is said to be right symmetric (left symmetric) if $A\perp_{B} T \Rightarrow T \perp_B A $ ($T \perp_{B} A \Rightarrow A \perp_B T $) for all $ A \in B(\mathbb{X}),$ the space of all bounded linear operators on $\mathbb{X}$. Turnsek [Linear Algebra Appl., 407 (2005), 189-195] proved that if $\mathbb{X}$ is a Hilbert space then $T$ is right symmetric if and only if $T$ is a scalar multiple of an isometry or coisometry. This result fails in general if the Hilbert space is replaced by a Banach space. The characterization of right and left symmetric operators on a Banach space is still open. In this paper we study the orthogonality in the sense of Birkhoff-James of bounded linear operators on $ (\mathbb{R}^n, |\cdot|_{\infty}) $ and characterize the right symmetric and left symmetric operators on $(\mathbb{R}^n,|\cdot|_{\infty}).$
Advances in Operator Theory
Tusi Mathematical Research Group (TMRG)
2538-225X
2
v.
4
no.
2017
428
434
http://www.aot-math.org/article_46810_689fe803bc0e591e2c5b81af0d53b265.pdf
dx.doi.org/10.22034/aot.1703-1137
Traces for fractional Sobolev spaces with variable exponents
Leandro
Del Pezzo
U Buenos Aires
author
Julio
Rossi
Universidad de Buenos Aires
Facultad de Ciencias Exactas y Naturales
Depto Matematica
Ciudad Universitaria, pab 1,
Buenos Aires, Argentina
author
text
article
2017
eng
In this note we prove a trace theorem in fractional spaces with variable exponents. To be more precise, we show that if $p\colon\overline{\Omega }\times \overline{\Omega } \rightarrow (1,\infty )$ and $q\colon\partial \Omega\rightarrow (1,\infty )$ are continuous functions such that\[ \frac{(n-1)p(x,x)}{n-sp(x,x)}>q(x) \qquad \mbox{ in } \partial \Omega \cap \{x\in\overline{\Omega}\colon n-sp(x,x) >0\}, \]then the inequality $$ \Vert f\Vert _{\scriptstyle L^{q(\cdot)}(\partial \Omega )} \leq C \left\{ \Vert f\Vert _{\scriptstyle L^{\bar{p}(\cdot)}(\Omega )}+ [f]_{s,p(\cdot,\cdot)} \right\} $$ holds. Here $\bar{p}(x)=p(x,x)$ and $\lbrack f\rbrack_{s,p(\cdot,\cdot)} $ denotes the fractional seminorm with variable exponent, that is given by \begin{equation*} \lbrack f\rbrack_{s,p(\cdot,\cdot)} := \inf \left\{ \lambda >0\colon \int_{\Omega}\int_{\Omega }\frac{|f(x)-f(y)|^{p(x,y)}}{\lambda ^{p(x,y)} |x-y|^{n+sp(x,y)}}dxdy<1\right\} \end{equation*}and $\Vert f\Vert _{\scriptstyle L^{q(\cdot)}(\partial \Omega )}$ and $\Vert f\Vert _{\scriptstyle L^{\bar{p}(\cdot)}(\Omega )}$ are the usual Lebesgue norms with variable exponent.
Advances in Operator Theory
Tusi Mathematical Research Group (TMRG)
2538-225X
2
v.
4
no.
2017
435
446
http://www.aot-math.org/article_47208_88cb54f5c152f3cc09373bb798ea5a26.pdf
dx.doi.org/10.22034/aot.1704-1152
Structures on the way from classical to quantum spaces and their tensor products
Alexander
Helemskii
author
text
article
2017
eng
We study tensor products of two structures situated, in a sense, between normed spaces and (abstract) operator spaces. We call them Lambert and proto-Lambert spaces and pay more attention to the latter ones. The considered two tensor products lead to essentially different norms in the respective spaces. Moreover, the proto-Lambert tensor product is especially nice for spaces with the maximal proto-Lambert norm and in particular, for $L_1$-spaces. At the same time the Lambert tensor product is nice for Hilbert spaces with the minimal Lambert norm.
Advances in Operator Theory
Tusi Mathematical Research Group (TMRG)
2538-225X
2
v.
4
no.
2017
447
467
http://www.aot-math.org/article_48029_0d40409e1cfb1b74fe5c9c32cf76cdbe.pdf
dx.doi.org/10.22034/aot.1706-1189
On skew [m,C]-symmetric operators
Muneo
Cho
Kanagawa University
author
Biljana
Nacevska-Nastovska
Department of Mathematics and Physics
Faculty of Electrical Engineering and Information Technology
Ss. Cyril and Methodius University in Skopje
author
Jun
Tomiyama
author
text
article
2017
eng
In this paper, first we characterize the spectra of skew $[m,C]$-symmetric operators and we also prove that if operators $T$ and $S$ are $C$-doubly commuting operators, $T$ is a skew $[m,C]$-symmetric operator and $Q$ is an $n$-nilpotent operator, then $T+Q$ is a skew $[m+2n-2,C]$-symmetric operator. Finally, we show that if $T$ is skew $[m,C]$-symmetric and $S$ is $[n,D]$-symmetric, then $T\otimes S$ is skew $[m+n-1, C \otimes D]$-symmetric.
Advances in Operator Theory
Tusi Mathematical Research Group (TMRG)
2538-225X
2
v.
4
no.
2017
468
474
http://www.aot-math.org/article_48114_7ad52fbbb1299adae9980e6759e8f244.pdf
dx.doi.org/10.22034/aot.1703-1147
Pseudospectra of elements of reduced Banach algebras
Arundhathi
Krishnan
author
S. H.
Kulkarni
Indian Institute of Technology Madras
author
text
article
2017
eng
Let $A$ be a Banach algebra with identity $1$ and $p\in A$ be a non-trivial idempotent. Then $q=1-p$ is also an idempotent. The subalgebras $pAp$ and $qAq$ are Banach algebras, called reduced Banach algebras, with identities $p$ and $q$ respectively. For $a\in A$ and $\varepsilon>0$, we examine the relationship between the $\varepsilon$-pseudospectrum $\Lambda_{\varepsilon}(A,a)$ of $a\in A$, and $\varepsilon$-pseudospectra of $pap\in pAp$ and $qaq\in qAq$. We also extend this study by considering a finite number of idempotents $p_{1},\cdots,p_{n}$, as well as an arbitrary family of idempotents satisfying certain conditions.
Advances in Operator Theory
Tusi Mathematical Research Group (TMRG)
2538-225X
2
v.
4
no.
2017
475
493
http://www.aot-math.org/article_48321_79bf8c4df3d69120cd091ee229b6bc45.pdf
dx.doi.org/10.22034/aot.1702-1112
2-Local derivations on matrix algebras and algebras of measurable operators
Shavkat
Ayupov
author
Karimbergen
Kudaybergenov
author
Amir
Alauadinov
author
text
article
2017
eng
Let $\mathcal{A}$ be a unital Banach algebra such that any Jordan derivation from $\mathcal{A}$ into any $\mathcal{A}$-bimodule $\mathcal{M}$ is a derivation. We prove that any 2-local derivation from the algebra $M_n(\mathcal{A})$ into $M_n(\mathcal{M})\,\,(n\geq 3)$ is a derivation. We apply this result to show that any 2-local derivation on the algebra of locally measurable operators affiliated with a von Neumann algebra without direct abelian summands is a derivation.
Advances in Operator Theory
Tusi Mathematical Research Group (TMRG)
2538-225X
2
v.
4
no.
2017
494
505
http://www.aot-math.org/article_43482_5c7c7ad78c756b7cccec7aecf018feb2.pdf
dx.doi.org/10.22034/aot.1612-1074
A formulation of the Jacobi coefficients $c^l_j(\alpha, \beta)$ via Bell polynomials
Stuart
Day
Department of Mathematics, University of Sussex
author
Ali
Taheri
Department of Mathematics, University of Sussex
author
text
article
2017
eng
The Jacobi polynomials $(\mathscr{P}^{(\alpha, \beta)}_k: k\ge0, \alpha, \beta>-1)$ are deeply intertwined with the Laplacian on compact rank one symmetric spaces. They represent the {\it spherical} or zonal functions and as such constitute the main ingredients in describing the spectral measures and spectral projections associated with the Laplacian on these spaces. In this note we strengthen this connection by showing that a set of spectral and geometric quantities associated with Jacobi operator fully describe the Maclaurin coefficients associated with the heat and other related Schwartzian kernels and present an explicit formulation of these quantities using the Bell polynomials.
Advances in Operator Theory
Tusi Mathematical Research Group (TMRG)
2538-225X
2
v.
4
no.
2017
506
515
http://www.aot-math.org/article_48949_48b57b4882775471cdf742aeac4354e6.pdf
dx.doi.org/10.22034/aot.1705-1163
Besov-Dunkl spaces connected with generalized Taylor formula on the real line
Chokri
Abdelkefi
Department of Mathematics, University of Tunis
author
Faten
Rached
author
text
article
2017
eng
In the present paper, we define for the Dunkl tranlation operators on the real line, the Besov-Dunkl space of functions for which the remainder in the generalized Taylor's formula has a given order. We provide characterization of these spaces by the Dunkl convolution.
Advances in Operator Theory
Tusi Mathematical Research Group (TMRG)
2538-225X
2
v.
4
no.
2017
516
530
http://www.aot-math.org/article_49228_6a57f1b28dc7c614b886ecd7150ded5c.pdf
dx.doi.org/10.22034/aot.1704-1154
Stability of the cosine-sine functional equation with involution
Jeongwook
Chang
author
Chang-Kwon
Choi
author
Jongjin
Kim
author
Prasanna. K
Sahoo
author
text
article
2017
eng
Let $S$ and $G$ be a commutative semigroup and a commutative group respectively, $\Bbb C$ and $\Bbb R^+$ the sets of complex numbers and nonnegative real numbers respectively, $\sigma : S \to S$ or $\sigma : G \to G$ an involution and $\psi : G\to \Bbb R^+$ be fixed. In this paper, we first investigate general solutions of the equation \begin{align}g(x+\sigma y)=g(x)g(y)+f(x)f(y)\nonumber\end{align}for all $ x,y \in S$, where $f, g : S \to \Bbb C$ are unknown functions to be determined. Secondly, we consider the Hyers-Ulam stability of the equation,i.e., we study the functional inequality\begin{align}|g(x+\sigma y)-g(x)g(y)-f(x)f(y)|\le \psi(y)\nonumber\end{align}for all $ x,y \in G$, where $f, g : G\to \Bbb C$.
Advances in Operator Theory
Tusi Mathematical Research Group (TMRG)
2538-225X
2
v.
4
no.
2017
531
546
http://www.aot-math.org/article_50055_de62150c980277715846bddca92ea10a.pdf
dx.doi.org/10.22034/aot.1706-1190
$L^p$ Fourier transformation on non-unimodular locally compact groups
Marianne
Terp
author
text
article
2017
eng
Let $G$ be a locally compact group with modular function $\Delta$ and left regular representation $\lambda$. We define the $L^p$ Fourier transform of a function $f \in L^p(G)$, $1\le p \le 2$, to be essentially the operator $\lambda(f)\Delta^{\frac{1}{q}}$ on $L^2(G)$ (where $\frac{1}{p}+\frac{1}{q}=1$) and show that a generalized Hausdorff--Young theorem holds. To do this, we first treat in detail the spatial $L^p$ spaces $L^p(\psi_0)$, $1 \le p \le \infty$, associated with the von Neumann algebra $M=\lambda(G)^{\prime\prime}$ on $L^2(G)$ and the canonical weight $\psi_0$ on its commutant. In particular, we discuss isometric isomorphisms of $L^2(\psi_0)$ onto $L^2(G)$ and of $L^1(\psi_0)$ onto the Fourier algebra $A(G)$. Also, we give a characterization of positive definite functions belonging to $A(G)$ among all continuous positive definite functions.
Advances in Operator Theory
Tusi Mathematical Research Group (TMRG)
2538-225X
2
v.
4
no.
2017
547
583
http://www.aot-math.org/article_50287_73922bd1a3c72852c000ad5144599937.pdf
dx.doi.org/10.22034/AOT.1709-1231