ACTA SCIENTIARUM MATHEMATICARUM (Szeged)

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 finite-dimensional 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/actasm-017-056-6

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.)

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 zero-divisors.

DOI: 10.14232/actasm-017-092-2

AMS Subject Classification (1991): 15A04, 15A48

Keyword(s): graphs, semirings, scrambling index

Received December 31, 2017. (Registered under 92/2017.)

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/actasm-017-825-8

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.)

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/actasm-017-590-0

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/actasm-018-753-y

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.)

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 $p$-Schatten--von Neumann spaces, with special interest on Tingley's problem.

DOI: 10.14232/actasm-018-255-0

AMS Subject Classification (1991): 47B49; 46A22, 46B20, 46B04, 46A16, 46E40

Keyword(s): Tingley's problem, extension of isometries, von Neumann algebra, $p$-Schatten--von 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/actasm-017-562-9

AMS Subject Classification (1991): 47B49

Keyword(s): operator interval, order isomorphism

Received October 22, 2017. (Registered under 62/2017.)

Abstract. We consider the linear maps $\varphi \colon \mathcal B(X)\to \mathcal B(Y)$ that preserve the semi-Fredholm 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 infinite-dimensional 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/actasm-017-327-x

AMS Subject Classification (1991): 47B48, 47A10, 46H05

Keyword(s): semi-Fredholm operators, Calkin algebra, linear preservers

Received November 30, 2017, and in revised form January 6, 2018. (Registered under 77/2017.)

Abstract. We propose a unified approach to the study of isometries on algebras of vector-valued 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/actasm-017-558-6

AMS Subject Classification (1991): 46E40, 46B04, 46J10, 46J15

Keyword(s): isometries, vector-valued maps, admissible quadruples, vector-valued Lipschitz algebras, continuously differentiable maps

Received September 18, 2017, and in final form February 11, 2018. (Registered under 58/2017.)

Abstract. Let $X$ and $Y$ be locally compact Hausdorff spaces, where $X$ is first-countable. Fix a positive integer $n \geq 3$ and a non-zero 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/actasm-017-076-5

AMS Subject Classification (1991): 46J10; 46H40, 46J20, 47B49

Keyword(s): function algebra, norm-preserving, peripheral spectrum, weighted composition operator

Received November 30, 2017 and in final form March 21, 2018. (Registered under 76/2017.)

Abstract. We prove that every bijective transformation on the set of Hilbert space effects which preserves a symmetric norm of a given Kubo--Ando mean is necessarily implemented by either a unitary or an antiunitary operator.

DOI: 10.14232/actasm-018-016-z

AMS Subject Classification (1991): 46L60, 47B49

Keyword(s): nonlinear preservers, Hilbert space effects, operator interval, Kubo--Ando mean, order automorphism, unitary operator

Received January 24, 2018 and in final form February 20, 2018. (Registered under 16/2018.)

Abstract. This paper describes bi-contractive projections on spaces of vector-valued continuous functions on a compact metric space.

DOI: 10.14232/actasm-017-335-2

AMS Subject Classification (1991): 47B38, 46B04, 46E40

Keyword(s): contractive projections, bi-contractive projections, vector measures, spaces of vector-valued functions

Received December 14, 2017 and in final form March 20, 2018. (Registered under 85/2017.)

Abstract. The Gleason--Kahane--Żelazko theorem states that a linear functional on a Banach algebra that is non-zero 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/actasm-017-323-8

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.)

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/actasm-017-331-0

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/actasm-017-582-8

AMS Subject Classification (1991): 46L40

Keyword(s): context posets, piecewise Jordan maps, Mackey-Gleason problem

Received December 9, 2017, and in revised form January 8, 2018. (Registered under 82/2017.)

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/actasm-018-267-7

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/actasm-017-837-2

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 order-theoretic 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 ``non-commutative vector lattices''. This notion includes order-theoretic 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 order-theoretic 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/actasm-017-574-3

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.)