2017
2
4
6
188
Homomorphic conditional expectations as noncommutative retractions
2
2
Let $A$ be a $C^*$algebra and $mathcal{E}colon A to A$ a conditional expectation. The KadisonSchwarz inequality for completely positive maps, $$mathcal{E}(x)^* mathcal{E}(x) leq mathcal{E}(x^* x),$$implies that$$leftVertmathcal{E}(x)rightVert^2 leq leftVertmathcal{E}(x^* x)rightVert.$$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$$leftVertmathcal{E}(x)rightVert^2 = leftVertmathcal{E}(x^*x)rightVert,$$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.
1

396
408


Robert
Pluta
Department of Mathematics, University of California, Irvine
Department of Mathematics, University of
USA
plutar@tcd.ie


Bernard
Russo
Department of Mathematics, University of
California, Irvine
Department of Mathematics, University of
Californ
USA
brusso@uci.edu
conditional expectation
Kadison inequality
Retraction
triple homomorphism
JC*triple
Variants of Weyl's theorem for direct sums of closed linear operators
2
2
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 aWeyl'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.
1

409
418


Anuradha
Gupta
University of Delhi, Delhi.
University of Delhi, Delhi.
India
dishna2@yahoo.in


Karuna
Mamtani
India
karunamamtani@gmail.com
operators with compact resolvent
Direct sums
Weyl's Theorem
aWeyl's Theorem
Browder's Theorem
On orthogonal decomposition of a Sobolev space
2
2
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.
1

419
427


Dejenie
Lakew
Bryant & Stratton College
Bryant & Stratton College
USA
dejenieal@gmail.com
Sobolev space
orthogonal decomposition,
inner product
Distance
On symmetry of BirkhoffJames orthogonality of linear operators
2
2
A bounded linear operator $T$ on a normed linear space $mathbb{X}$ is said to be right symmetric (left symmetric) if $Aperp_{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), 189195] 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 BirkhoffJames of bounded linear operators on $ (mathbb{R}^n, cdot_{infty}) $ and characterize the right symmetric and left symmetric operators on $(mathbb{R}^n,cdot_{infty}).$
1

428
434


Puja
Ghosh
Jadavpur UNiversity
Jadavpur UNiversity
India
ghosh.puja1988@gmail.com


Debmalya
Sain
India
saindebmalya@gmail.com


Kallol
Paul
Jadavpur UNiversity
Jadavpur UNiversity
India
kalloldada@gmail.com
BirkhoffJames orthogonality
left symmetric operator
right symmetric operator
Traces for fractional Sobolev spaces with variable exponents
2
2
In this note we prove a trace theorem in fractional spaces with variable exponents. To be more precise, we show that if $pcolonoverline{Omega }times overline{Omega } rightarrow (1,infty )$ and $qcolonpartial Omegarightarrow (1,infty )$ are continuous functions such that[ frac{(n1)p(x,x)}{nsp(x,x)}>q(x) qquad mbox{ in } partial Omega cap {xinoverline{Omega}colon nsp(x,x) >0}, ]then the inequality $$ Vert fVert _{scriptstyle L^{q(cdot)}(partial Omega )} leq C left{ Vert fVert _{scriptstyle L^{bar{p}(cdot)}(Omega )}+ [f]_{s,p(cdot,cdot)} right} $$ holds. Here $bar{p}(x)=p(x,x)$ and $lbrack frbrack_{s,p(cdot,cdot)} $ denotes the fractional seminorm with variable exponent, that is given by begin{equation*} lbrack frbrack_{s,p(cdot,cdot)} := inf left{ lambda >0colon int_{Omega}int_{Omega }frac{f(x)f(y)^{p(x,y)}}{lambda ^{p(x,y)} xy^{n+sp(x,y)}}dxdy<1right} end{equation*}and $Vert fVert _{scriptstyle L^{q(cdot)}(partial Omega )}$ and $Vert fVert _{scriptstyle L^{bar{p}(cdot)}(Omega )}$ are the usual Lebesgue norms with variable exponent.
1

435
446


Leandro
Del Pezzo
U Buenos Aires
U Buenos Aires
Argentina
ldpezzo@dm.uba.ar


Julio
Rossi
Universidad de Buenos Aires
Facultad de Ciencias Exactas y Naturales
Depto Matematica
Ciudad Universitaria, pab 1,
Buenos Aires, Argentina
Universidad de Buenos Aires
Facultad de
Argentina
jrossi@dm.uba.ar
$p$Laplacian
fractional operators
variable exponents
Structures on the way from classical to quantum spaces and their tensor products
2
2
We study tensor products of two structures situated, in a sense, between normed spaces and (abstract) operator spaces. We call them Lambert and protoLambert 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 protoLambert tensor product is especially nice for spaces with the maximal protoLambert 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.
1

447
467


Alexander
Helemskii
USA
helemskii@rambler.ru
protoLambert space
Lbounded operator
protoLambert tensor product
Lambert space
Lambert tensor product
On skew [m,C]symmetric operators
2
2
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 $[m+2n2,C]$symmetric operator. Finally, we show that if $T$ is skew $[m,C]$symmetric and $S$ is $[n,D]$symmetric, then $Totimes S$ is skew $[m+n1, C otimes D]$symmetric.
1

468
474


Muneo
Cho
Kanagawa University
Kanagawa University
Serbia
chiyom01@kanagawau.ac.jp


Biljana
NacevskaNastovska
Department of Mathematics and Physics
Faculty of Electrical Engineering and Information Technology
Ss. Cyril and Methodius University in Skopje
Department of Mathematics and Physics
Faculty
Macedonia (Republic of)
bibanmath@gmail.com


Jun
Tomiyama
Japan
jtomiyama@fc.jwu.ac.jp
Hilbert space
linear operator
conjugation
misometric operator
msymetric
Pseudospectra of elements of reduced Banach algebras
2
2
Let $A$ be a Banach algebra with identity $1$ and $pin A$ be a nontrivial idempotent. Then $q=1p$ is also an idempotent. The subalgebras $pAp$ and $qAq$ are Banach algebras, called reduced Banach algebras, with identities $p$ and $q$ respectively. For $ain A$ and $varepsilon>0$, we examine the relationship between the $varepsilon$pseudospectrum $Lambda_{varepsilon}(A,a)$ of $ain A$, and $varepsilon$pseudospectra of $papin pAp$ and $qaqin qAq$. 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.
1

475
493


Arundhathi
Krishnan
India
arundhathi.krishnan@gmail.com


S. H.
Kulkarni
Indian Institute of Technology Madras
Indian Institute of Technology Madras
India
shk@iitm.ac.in
Banach algebra
Direct Sum
Reduced Banach algebra
idempotent
pseudospectrum
spectrum
2Local derivations on matrix algebras and algebras of measurable operators
2
2
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 2local derivation from the algebra $M_n(mathcal{A})$ into $M_n(mathcal{M}),,(ngeq 3)$ is a derivation. We apply this result to show that any 2local derivation on the algebra of locally measurable operators affiliated with a von Neumann algebra without direct abelian summands is a derivation.
1

494
505


Shavkat
Ayupov
Uzbekistan
sh_ayupov@mail.ru


Karimbergen
Kudaybergenov
Uzbekistan
karim2006@mail.ru


Amir
Alauadinov
Uzbekistan
amir_85@mail.ru
Matrix algebra
derivation
inner derivation
$2$local derivation
measurable operator
A formulation of the Jacobi coefficients $c^l_j(alpha, beta)$ via Bell polynomials
2
2
The Jacobi polynomials $(mathscr{P}^{(alpha, beta)}_k: kge0, alpha, beta>1)$ 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.
1

506
515


Stuart
Day
Department of Mathematics, University of Sussex
Department of Mathematics, University of
United Kingdom
s.day@sussex.ac.uk


Ali
Taheri
Department of Mathematics, University of Sussex
Department of Mathematics, University of
United Kingdom
a.taheri@sussex.ac.uk
Jacobi polynomials
LaplaceBeltrame operators
Heat kernel
Bell polynomials
Rank one symmetric spaces
BesovDunkl spaces connected with generalized Taylor formula on the real line
2
2
In the present paper, we define for the Dunkl tranlation operators on the real line, the BesovDunkl 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.
1

516
530


Chokri
Abdelkefi
Department of Mathematics, University of Tunis
Department of Mathematics, University of
Tunisia
chokri.abdelkefi@yahoo.fr


Faten
Rached
Tunisia
rached@math.jussieu.fr
Dunkl operator
Dunkl transform
Dunkl translation operators
Dunkl convolution
Generalized Taylor formula
BesovDunkl spaces
Stability of the cosinesine functional equation with involution
2
2
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 $psi : Gto Bbb R^+$ 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)nonumberend{align}for all $ x,y in S$, where $f, g : S to Bbb C$ are unknown functions to be determined. Secondly, we consider the HyersUlam stability of the equation,i.e., we study the functional inequalitybegin{align}g(x+sigma y)g(x)g(y)f(x)f(y)le psi(y)nonumberend{align}for all $ x,y in G$, where $f, g : Gto Bbb C$.
1

531
546


Jeongwook
Chang
Korea, Republic Of
jchang@dankook.ac.kr


ChangKwon
Choi
Korea, Republic Of
ck38@kunsan.ac.kr


Jongjin
Kim
Korea, Republic Of
jjkim@jbnu.ac.kr


Prasanna. K
Sahoo
USA
sahoo@louisville.edu
additive function
cosinesine functional equation
exponential function
Involution
Stability
$L^p$ Fourier transformation on nonunimodular locally compact groups
2
2
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 $f in L^p(G)$, $1le p le 2$, 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 HausdorffYoung 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 $M=lambda(G)^{primeprime}$ 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.
1

547
583


Marianne
Terp
Denmark
mtgmohr@gmail.com
$L^p$ Fourier transformation
locally compact group
Fourier algebra
positive definite function