
ACTA SCIENTIARUM MATHEMATICARUM (Szeged)
Dijana Ilišević,
Bojan Kuzma,
ChiKwong Li,
Edward Poon

317

Abstract. Let $\gamma $ be a unimodular complex number, and let $k$ be an integer. Then $\gamma A^k$ is an isometry for any isometry $A$ of a complex Banach space. It is shown that if $f$ is an analytic function on the unit circle sending an isometry to an isometry for any norm, then $f$ has the form $z \mapsto \gamma z^k$ for some unimodular $\gamma $ and integer $k$. The same conclusion on $f$ can be deduced if $f$ is merely continuous and preserves the isometries of some special classes of norms on a fixed finitedimensional complex Banach space. The result is extended to real Banach spaces $X$ with $\dim {X} \geq 4$, and it is shown that one cannot get the same conclusion on $f$ if $\dim {X}<4$. Further extensions of these results are also considered.
DOI: 10.14232/actasm0170566
AMS Subject Classification
(1991): 47B49, 15A60, 15A86, 46B04
Keyword(s):
Banach space,
isometry,
continuous function,
generalized permutation
Received September 10, 2017 and in final form February 5, 2018. (Registered under 56/2017.)
A. E. Guterman,
A. M. Maksaev

1938

Abstract. We prove that additive transformations on matrices over the binary Boolean semiring that preserve the scrambling index are automatically bijective. As a consequence we characterize such maps for matrices over an arbitrary antinegative semiring with identity and without zerodivisors.
DOI: 10.14232/actasm0170922
AMS Subject Classification
(1991): 15A04, 15A48
Keyword(s):
graphs,
semirings,
scrambling index
Received December 31, 2017. (Registered under 92/2017.)
M. Bendaoud,
A. Benyouness,
M. Sarih

3947

Abstract. Let $\mathcal H$ be a complex Hilbert space of dimension greater than $2$, and denote by $\mathcal L(\mathcal H)$ the algebra of all bounded linear operators on $\mathcal H$. For $\varepsilon >0$ and $T\in \mathcal L(\mathcal H)$, let $r_\varepsilon (T)$ denote the $\varepsilon $pseudo spectral radius of $T$. Let $\mathfrak {S}_1$ and $\mathfrak {S}_2$ be subsets of $\mathcal L(\mathcal H)$ which contain all rank one operators and the identity. A characterization is obtained for surjective maps $\phi \colon \mathfrak {S}_1\rightarrow \mathfrak {S}_2$ satisfying $r_{\varepsilon }(\phi (T)\phi (S)^*\phi (T))=r_{\varepsilon }(TS^*T)$ ($T, S\in \mathfrak {S}_1$). An analogous description is also obtained for the pseudo spectrum of operators.
DOI: 10.14232/actasm0178258
AMS Subject Classification
(1991): 47B49, 47A10, 47A25
Keyword(s):
pseudo spectral radius,
skew product of operators,
nonlinear preservers
Received November 22, 2017, and in revised form January 12, 2018. (Registered under 75/2017.)
Zine El Abidine Abdelali,
Abdellatif Bourhim

4964

Abstract. Let $\mn $ denote the algebra of all $n\times n$ complex matrices, and fix a nonzero vector $x_0$ in $\C ^n$. For any matrix $T\in \mn $, let $\sigma _T(x_0)$ denote the local spectrum of $T$ at $x_0$. Given three scalars $\mu ,~\nu $ and $\xi $ simultaneously nonzero, we study maps $\varphi $ on $\mn $ satisfying $ \sigma _{\mu STS + \nu T S +\xi ST}(x_0)= \sigma _{\mu \varphi (S)\varphi (T)\varphi (S) + \nu \varphi (T)\varphi (S)+\xi \varphi (S)\varphi (T)}(x_0) $ for all $S,~T\in \mn $. Our main result extends and unifies the main results of several papers on maps on $\mn $ preserving the local spectrum of different products.
DOI: 10.14232/actasm0175900
AMS Subject Classification
(1991): 47B49; 47A10, 47A11
Keyword(s):
nonlinear preservers,
local spectrum,
SVEP,
quadratic product,
Jordan product,
matrices
Received December 30, 2017 and in final form March 8, 2018. (Registered under 90/2017.)
Abstract. Borel probability measures living on metric spaces are fundamental mathematical objects. There are several meaningful distance functions that make the collection of the probability measures living on a certain space a metric space. We are interested in the description of the structure of the isometries of such metric spaces. We overview some of the recent results of the topic and we also provide some new ones concerning the Wasserstein distance. More specifically, we consider the space of all Borel probability measures on the unit sphere of a Euclidean space endowed with the Wasserstein metric $W_p$ for arbitrary $p \geq 1,$ and we show that the action of a Wasserstein isometry on the set of Dirac measures is induced by an isometry of the underlying unit sphere.
DOI: 10.14232/actasm018753y
AMS Subject Classification
(1991): 46E27, 54E40
Keyword(s):
Wasserstein isometries,
unit sphere
Received January 4, 2018 and in final form March 19, 2018. (Registered under 3/2018.)
Antonio M. Peralta

81123

Abstract. We survey the most recent results on the extension of isometries between special subsets of the unit spheres of $C^*$algebras, von Neumann algebras, trace class operators, preduals of von Neumann algebras, and $p$Schattenvon Neumann spaces, with special interest on Tingley's problem.
DOI: 10.14232/actasm0182550
AMS Subject Classification
(1991): 47B49; 46A22, 46B20, 46B04, 46A16, 46E40
Keyword(s):
Tingley's problem,
extension of isometries,
von Neumann algebra,
$p$Schattenvon Neumann,
trace class operators
Received January 5, 2018 and in final form March 7, 2018. (Registered under 5/2018.)
Abstract. The known descriptions of the groups of order automorphisms of operator intervals are very simple with only one exception: there are two known results describing the general form of order automorphisms of the effect algebra and they both look quite complicated. It is the aim of this paper to show that the group of order automorphisms of the effect algebra is isomorphic to the group of order automorphisms of any other proper operator interval. After proving this statement we will present a new description of order automorphisms of the effect algebra explaining better that the case of the effect algebra is not more complicated than the other operator intervals.
DOI: 10.14232/actasm0175629
AMS Subject Classification
(1991): 47B49
Keyword(s):
operator interval,
order isomorphism
Received October 22, 2017. (Registered under 62/2017.)
Manuel González,
Mostafa Mbekhta

137149

Abstract. We consider the linear maps $\varphi \colon \mathcal B(X)\to \mathcal B(Y)$ that preserve the semiFredholm operators in both directions or the essential spectrum of an operator, where $\mathcal B(X)$ is the algebra of all bounded linear operators on an infinitedimensional Banach space $X$. We describe some known results in the Hilbert space case, provide some basic results and examples in the general case, and state several open problems.
DOI: 10.14232/actasm017327x
AMS Subject Classification
(1991): 47B48, 47A10, 46H05
Keyword(s):
semiFredholm operators,
Calkin algebra,
linear preservers
Received November 30, 2017, and in revised form January 6, 2018. (Registered under 77/2017.)
Osamu Hatori,
Shiho Oi

151183

Abstract. We propose a unified approach to the study of isometries on algebras of vectorvalued Lipschitz maps and those of continuously differentiable maps by means of the notion of admissible quadruples. We describe isometries on function spaces of some admissible quadruples that take values in unital commutative $C^*$algebras. As a consequence we confirm the statement of \cite [Example 8]{jp} on Lipschitz algebras and show that isometries on such algebras indeed take the canonical form.
DOI: 10.14232/actasm0175586
AMS Subject Classification
(1991): 46E40, 46B04, 46J10, 46J15
Keyword(s):
isometries,
vectorvalued maps,
admissible quadruples,
vectorvalued Lipschitz algebras,
continuously differentiable maps
Received September 18, 2017, and in final form February 11, 2018. (Registered under 58/2017.)
Rumi Shindo Togashi

185200

Abstract. Let $X$ and $Y$ be locally compact Hausdorff spaces, where $X$ is firstcountable. Fix a positive integer $n \geq 3$ and a nonzero complex number $\lambda $. If a surjective map $T\colon C_{0}(X) \to C_{0}(Y)$ satisfies the condition $\sup _{y \in Y}\big \big (\prod _{ k= 1}^{n}T(f_k)\big )(y)+\lambda \big  = \sup _{x \in X}\big \big (\prod _{ k= 1}^{n}f_k \big )(x)+\lambda \big $
DOI: 10.14232/actasm0170765
AMS Subject Classification
(1991): 46J10; 46H40, 46J20, 47B49
Keyword(s):
function algebra,
normpreserving,
peripheral spectrum,
weighted composition operator
Received November 30, 2017 and in final form March 21, 2018. (Registered under 76/2017.)
Marcell Gaál,
Gergő Nagy

201208

Abstract. We prove that every bijective transformation on the set of Hilbert space effects which preserves a symmetric norm of a given KuboAndo mean is necessarily implemented by either a unitary or an antiunitary operator.
DOI: 10.14232/actasm018016z
AMS Subject Classification
(1991): 46L60, 47B49
Keyword(s):
nonlinear preservers,
Hilbert space effects,
operator interval,
KuboAndo mean,
order automorphism,
unitary operator
Received January 24, 2018 and in final form February 20, 2018. (Registered under 16/2018.)
Abstract. This paper describes bicontractive projections on spaces of vectorvalued continuous functions on a compact metric space.
DOI: 10.14232/actasm0173352
AMS Subject Classification
(1991): 47B38, 46B04, 46E40
Keyword(s):
contractive projections,
bicontractive projections,
vector measures,
spaces of vectorvalued functions
Received December 14, 2017 and in final form March 20, 2018. (Registered under 85/2017.)
Javad Mashreghi,
Thomas Ransford

227238

Abstract. The GleasonKahaneŻelazko theorem states that a linear functional on a Banach algebra that is nonzero on invertible elements is necessarily a scalar multiple of a character. Recently this theorem has been extended to certain Banach function spaces that are not algebras. In this article we present a brief survey of these extensions.
DOI: 10.14232/actasm0173238
AMS Subject Classification
(1991): 15A86; 30H10, 46H05, 46H40, 47B49
Keyword(s):
linear functional,
character,
automatic continuity,
Banach algebra,
function space,
Hardy space,
Dirichlet space,
reproducing kernel
Received November 17, 2017, and in revised form January 25, 2018. (Registered under 73/2017.)
Kazuhiro Kawamura,
Hironao Koshimizu,
Takeshi Miura

239261

Abstract. We present a unified framework to study isometries, with respect to various norms, on $C^{1}([0,1])$, the space of continuously first differentiable functions on the unit interval $[0,1]$. We also discuss continuous deformations of isometry groups induced by perturbations of norms on $C^{1}([0,1])$.
DOI: 10.14232/actasm0173310
AMS Subject Classification
(1991): 46E15, 57S10
Keyword(s):
isometry,
weighted composition operator
Received December 7, 2017 and in final form March 21, 2018. (Registered under 81/2017.)
Abstract. The context poset of Abelian $C^\ast $subalgebras of a given $C^\ast $algebra is an operator theoretic invariant of growing interest. We review recent results describing order isomorphisms between context posets in terms of Jordan type maps (linear or not) between important types of operator algebras. We discuss the important role of the generalized Gleason theorem on linearity of maps preserving linear combinations of commuting elements for studying symmetries of context posets. Related results on maps multiplicative with respect to commuting elements are investigated.
DOI: 10.14232/actasm0175828
AMS Subject Classification
(1991): 46L40
Keyword(s):
context posets,
piecewise Jordan maps,
MackeyGleason problem
Received December 9, 2017, and in revised form January 8, 2018. (Registered under 82/2017.)
JungHui Liu,
ChunYen Chou,
ChingJou Liao,
NgaiChing Wong

277307

Abstract. We survey some recent studies of linear zero product or orthogonality preservers between $C^*$/$W^*$algebras, their dual or predual spaces, and holomorphic disjointness preservers of $C^*$algebras. Such maps are expected to provide algebra or linear Jordan ($*$) homomorphisms between the underlying operator algebras. We also study orthogonality preservers between Hilbert $C^*$modules and Fourier algebras. A few open problems are stated.
DOI: 10.14232/actasm0182677
AMS Subject Classification
(1991): 46L40, 46L10, 46H40
Keyword(s):
zero product preservers,
orthogonality preservers,
Jordan homomorphisms,
Fourier algebras,
Hilbert $C^*$modules,
holomorphic maps of $C^*$algebras
Received February 5, 2018 and in final form April 12, 2018. (Registered under 17/2018.)
Abstract. We discuss a new approach to the problem of recovering signal from frame coefficients with erasures. It is known that, under the assumption that the erasure set of indices for a given frame satisfies the minimal redundancy condition, there exists a synthesizing dual frame which enables us to perfectly reconstruct the original signal without recovering the lost coefficients. In this paper we describe further properties of such dual frames compensating for erasures.
DOI: 10.14232/actasm0178372
AMS Subject Classification
(1991): 42C15; 47A05
Keyword(s):
frame,
dual frame,
erasure
Received December 28, 2017, and in revised form February 6, 2018. (Registered under 87/2017.)
Abstract. We provide an ordertheoretic characterization of algebraic orthogonality among positive elements of a general C$^{\ast }$algebra by proving a statement conjectured in [12]. Generalizing this idea, we describe absolutely ordered $p$normed spaces for $1 \le p \le \infty $ which present a model for ``noncommutative vector lattices''. This notion includes ordertheoretic orthogonality. We generalize algebraic orthogonality by introducing the notion of {\it absolute compatibility} among positive elements in absolute order unit spaces and relate it to the symmetrized product in the case of a C$^{\ast }$algebra. In the latter case, whenever one of the elements is a projection, the elements are absolutely compatible if and only if they commute. We develop an ordertheoretic prototype of the results. For this purpose, we introduce the notion of {\it order projections} and extend the results related to projections in a unital C$^{\ast }$algebra to order projections in an absolute order unit space. As an application, we describe the spectral decomposition theory for elements of an absolute order unit space.
DOI: 10.14232/actasm0175743
AMS Subject Classification
(1991): 46B40; 46L05, 46L30
Keyword(s):
absolute $\infty $orthogonality,
absolute order unit space,
absolute compatibility,
order projection
Received November 19, 2017, and in revised form February 1, 2018. (Registered under 74/2017.)

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