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Monday morning
8:00 Registration
8:40 Opening
Chair: Lajos Molnár
9:00 Invited Talk: Chi-Kwong LiSome results and problems in preservers
9:50 Invited Talk: Alexander E. GutermanFrom preservers to converters
10:40 Coffee break
Chair: Alexander E. Guterman
11:20 Mohamed BendaoudCondition spectra of special operators and preserver problems
11:50 Roksana SłowikSpectrum preservers on matrix algebras
12:10 Arundhathi KrishnanDetermining elements of a Banach algebra through pseudospectra
12:30 Lunch break
Monday afternoon
Chair: Bojan Kuzma
14:00 Zine El Abidine AbdelaliMaps preserving the local spectrum of some products of matrices
14:30 Antonio Morales CampoyOn mappings preserving the diamond order
15:00 Janko MarovtPartial orders in Rickart rings
15:30 Coffee break
Chair: Ngai-Ching Wong
16:10 Raymond Nung-Sing SzeUnitary similarity invariant function preservers of products of operators
16:40 Rachid El HartiAdjointable operators on a Hilbert C*-module, numerical range valued in aC*-algebra and positivity preserving linear maps
17:10 Ying-Fen LinOn positive Schur multipliers
17:40 Dániel VirosztekCharacterization of centrality by a local order-preserving property of certainfunctions on C∗-algebras
18:15 Sightseeing tour
Some results and problems in preservers
Chi-Kwong LiCollege of William and Mary, Williamsburg, USA
We describe some results and problems in preservers. Comments will be made aboutthe history, motivations, techniques, and connections of the study to other areas.
From preservers to converters
Alexander E. GutermanLomonosov Moscow State University, Moscow, Russia
The theory of transformations preserving different matrix properties and invariantsdates back to [1, Frobenius, 1897], [5, Schur, 1925], [2, Dieudonné, 1949] and is anintensively developing part of algebra nowadays. Another interesting problem is tocharacterize converters, i.e. maps sending one invariant into another. Correspondingquestion for permanent and determinant was posed in [3, Pólya, 1913]. The detailedand self-contained exposition on preservers can be found in a number of good surveysand monographs. See for example [4].
In the talk we present our recent results on this subject in three different directions.The first group of results is related to the preservers for matrices over semirings andtheir applications to preservers of the graph theory invariants. In particular, we discussgeneralizations of Frobenius and Dieudonné theorems for matrices over semirings andpreservers of Green relations. As for the graph theory invariants, we consider scram-bling index, cyclicity index, primitivity for colored graphs, etc. The second directionis related to preservers over fields. Here we include the results concerning the sets ofmatrices for which equality in Marcus-Oliveira conjecture holds and different monotonematrix maps. Finally we discuss the linear and non-linear converters, in particular thesolution of permanent Pólya problem.
This research was supported by the RFBR Grant 15-01-01132.
[1] G. Frobenius, Uber die Darstellung der endlichen Gruppen durch lineare Substi-tutionen, Sitzungsber., Preuss. Akad. Wiss (Berlin), Berlin, 1897, pp. 994-1015.
[2] J. Dieudonné, Sur une généralisation du groupe orthogonal à quatre variables,Arch. Math. 1 (1949), 282–287.
[3] G. Pólya, Aufgabe 424, Arch. Math. Phys. 20, 3 (1913), 271.
[4] S. Pierce and others, A survey of linear preserver problems, Linear and Mul-tilinear Algebra 33 (1992), 1–119.
[5] I. Schur, Einige Bemerkungen zur Determinantentheorie, Akad. Wiss. Berlin: S.-Ber. Preuß, 1925, 454–463.
Condition spectra of special operators andpreserver problems
Mohamed BendaoudMoulay Ismail University, Meknès, Morocco
In this talk, we present descriptions of the condition spectra of some special classesof operators. As application, characterization is obtained for nonlinear transformationson the algebra of all bounded linear operators on a Hilbert space that leave invariantthe condition spectrum or the condition spectral radius of certain products of operatorpairs. The corresponding preserver problems for the pseudo spectra of operators arealso discussed, and particularly we give an answer to a question raised by Molnár atthe 20th ILAS Conference 2016.
Spectrum preservers on matrix algebras
Roksana SłowikSilesian University of Technology, Gliwice, Poland
Let A be an algebra and let φ be a map defined on A. We call φ a spectrumpreserver if
sp(x) = sp(φ(x)) for all x ∈ A.Clearly, there is a lot of generalizations of the spectrum preservers. Moreover, variousconditions (e.g. linearity, injectivity) can be set on φ. During the talk we are goingto review the most classical results in the subject of spectrum preservers. We are alsogoing to present some our considerations. Namely, we will consider some continuousmaps φ defined on full matrix algebra over C with the property that the spectra oftwo matrices have nonempty intersections if and only if their images under φ also havenonempty intersections.
[1] L. Baribeau, T. Ransford, Non-linear spectrum-preserving maps, Bull. LondonMath. Soc. 32, (2000), 8–14.
[2] C. Costara, Maps on matrices that preserve the spectrum, Linear Algebra Appl.435, (2011), 2674–2680.
[3] M. Marcus, B.N. Moyls, Linear transformations on algebras of matrices, Canad.J. Math. 11, (1959), 61–66.
[4] J. Mrčun, Lipschitz spectrum preserving mappings on algebras of matrices, LinearAlgebra Appl. 215, (1995), 113–120.
[5] R. Słowik, Some variations on continuous spectrum preservers on matrix algebras,submitted.
Determining elements of a Banach algebrathrough pseudospectra
Arundhathi KrishnanIndian Institute of Technology Madras, Chennai, India
Let A be a Banach algebra with identity 1. For ε > 0, the ε-pseudospectrum ofan element a ∈ A is defined as Λε(A, a) = σ(A, a) ∪ λ ∈ C : ‖(λ − a)−1‖ ≥ 1
ε,
where σ(A, a) denotes the spectrum of a in A. Suppose for some ε > 0 and a, b ∈ A,Λε(A, ax) = Λε(A, bx) ∀x ∈ A. It is shown that a = b in each of the following cases:
1. a is invertible.
2. a is Hermitian idempotent.
3. a is the product of a Hermitian idempotent and an invertible element.
4. A is semisimple and a is the product of an idempotent and an invertible element.
5. A = B(X) for a Banach space X.
6. A is a C∗-algebra.
7. A is a commutative semisimple Banach algebra.
Next, let A and B be Banach algebras and let T : A → B be a bounded linearoperator. Suppose that for some ε > 0, T is an ε-pseudospectrum preserving map, i.e.
Λε(B, Ta) = Λε(A, a) ∀a ∈ A.
Then it is shown that T also preserves the numerical range V (a), i.e.,
V (Ta) = V (a) ∀a ∈ A.
Here V (a) = f(a) : f ∈ A′, f(1) = 1 = ‖f‖, with A′ the dual of A. Hence T is aninvertible operator onto the range of T , with T and T−1 state preserving.
This is joint work with Professor S. H. Kulkarni, Indian Institute of TechnologyMadras, Chennai.
[1] M. Brešar and Š. Špenko, Determining elements in Banach algebras throughspectral properties, J. Math. Anal. Appl. 393 (2012), 144–150.
[2] A. Krishnan and S. H. Kulkarni, Pseudospectrum of an element of a Banachalgebra, Oper Matrices 11 (2017), 263–287.
[3] V. J. Pellegrini, Numerical range preserving operators on a Banach algebra,Studia Math 54 (1975), 143–147.
Maps preserving the local spectrum of someproducts of matrices
Zine El Abidine AbdelaliMohammed V University, Rabat, Morocco
LetMn(C) denote the algebra of the algebra of all n×n complex matrices, and x0
a nonzero vector in Cn. For two fixed scalars µ and ν in C for which (µ, ν) 6= (0, 0),we characterize all maps ϕ onMn(C) satisfying
σµST ∗S+νT ∗S(x0) = σµϕ(S)ϕ(T )∗ϕ(S)+νϕ(T )∗ϕ(S)(x0), (S, T ∈Mn(C)).
This provides, in particular, a complete description of all maps onMn(C) preservingthe local spectrum of the skew double product "TS∗" and the skew triple product"TS∗T" of matrices. It also unifies and extends several known results on local spectrumpreservers.
On mappings preserving the diamond order
Antonio Morales CampoyUniversity of Almería, Spain
The diamond partial order on Mn(C) was defined in [1] by Baksalary and Hauke.In [3] Lebtahi, Patrício and Thome extended this partial order to the setting of regular*-rings. We show in [2] that this is a partial order in every C∗-algebra, and we studyother interesting properties of this relation. There are many results concerning maps,between either matrix algebras or operator algebras, preserving some partial order.Recently this study has been extended to semisimple Banach algebras and C∗-algebras.Along this line, we give in [2] some characterizations of linear maps between C∗-algebraspreserving the diamond partial order.
[1] J. K. Baksalary, J. Hauke, A further algebraic version of Cochran’s theoremand matrix partial orderings, Linear Algebra Appl. 127 (1990), 157–169.
[2] M. Burgos, A. C. Márquez-García, A. Morales-Campoy, Maps preservingthe diamond partial order, Appl. Math. Comput. 296 (2017), 137–147.
[3] L. Lebtahi, P. Patricio, N. Thome, The diamond partial order in rings, Lin.Mult. Alg. 62 (2014), 386–395.
Partial orders in Rickart rings
Janko MarovtUniversity of Maribor, Faculty of Economics and Business, Maribor, Slovenia
We consider the generalized concept of order relations on B(H), the algebra of allbounded linear operators on a Hilbert space H, which was proposed by Šemrl andwhich covers the star partial order, the left-star partial order, the right-star partialorder, and the minus partial order. The forms of the maps that preserve these ordersare presented. We extend the concept from B(H) to Rickart rings or to Rickart *-ringsand generalize some well-known results.
[1] P. Šemrl, Automorphisms of B(H) with respect to minus partial order, J. Math.Anal. Appl. 369 (2010), 205–213.
[2] G. Dolinar, A. E. Guterman, J. Marovt, Monotone transformations on B(H)with respect to the left-star and the right-star partial order, Math. Inequal. Appl.17 (No. 2) (2014), 573–589.
[3] J. Marovt, D. S. Rakić, D. S. Djordjević, Star, left-star, and right-starpartial orders in Rickart ∗-rings, Linear Multilinear Algebra 63 (No. 2) (2015),343–365.
[4] J. Marovt, On partial orders in Rickart rings, Linear Multilinear Algebra 63 (No.9) (2015), 1707–1723.
[5] G. Dolinar, B. Kuzma, J. Marovt, A note on partial orders of Hartwig, Mitsch,and Šemrl, Appl. Math. Comput. 270 (2015), 711–713.
[6] J. Marovt, On star, sharp, core, and minus partial orders in Rickart rings, BanachJ. Math. Anal. 10 (No. 3) (2016), 495–508.
Unitary similarity invariant functionpreservers of products of operators
Raymond Nung-Sing SzeThe Hong Kong Polytechnic University, Hong Kong
Let B(H) denote the Banach algebra of all bounded linear operators on a complexHilbert space H with dimH ≥ 3, and let A and B subsets of B(H) which contain allrank one operators. Suppose F (·) is an unitary invariant norm, the pseudo spectra, thepseudo-spectrum radius, the C-numerical range, or the C-numerical radius for somefinite rank operator C. In this talk, we determined for surjective maps Φ : A → Bsatisfying F (A∗B) = F (Φ(A)∗Φ(B)) for all A,B ∈ A.
Adjointable operators on a Hilbert C*-module,numerical range valued in a C*-algebra and
positivity preserving linear maps
Rachid El HartiUniversity Hassan First, FST of Settat, Morocco
Let A be a C∗-algebra and E is an A-Hilbert C∗-module. Denote by L(E), the setof all bounded adjointable operators on E . The set L(E) is defined a C∗-algebras withoperator norm over E . In this talk, we define the new version of numerical range of anoperator T in L(E) valued in a C*-algebra and we investigate its properties and thestatements of its numerical rediua. We will discuss the linear maps which preserve thisnumerical range and also some perspectives.
[1] U. E,C, Lance, Hilbert C∗-Modules, Londan Mathematical society lecture notesSeries 210 (Combridge University Press),
[2] V. M. Manuilov and E. V. Troitsky, Hilbert C∗-Modules and their morphisms,Journal of Mathematical Sciences, Vol 98, No. 2. 2000.
On positive Schur multipliers
Ying-Fen LinQueen’s University Belfast, Belfast, U.K.
Let T be an n by n matrix. A Schur multiplier ϕT : Mn → Mn is given byA 7→ T ? A, where ? is the Schur product. In this talk, we will show that Schurmultipliers, which can be viewed as support preserving maps, can be used to studypositive completion/extension problems. More precisely, the existence of a positiveextension can be realised through the positivity of Schur multipliers, even in a generalmeasurable setting.
This is a joint work with Rupert Levene and Ivan Todorov.
Characterization of centrality by a localorder-preserving property of certain functions
on C∗-algebras
Dániel VirosztekBudapest University of Technology and Economics and MTA-DE "Lendület"
Functional Analysis Research Group, Budapest and Debrecen, Hungary
Connections between the commutativity of a C∗-algebra A and the order-preservingproperty (that is, monotonicity) of some functions defined on some subsets of A havebeen investigated widely. The first result related to this topic is due to Ogasawara whoshowed in 1955 that a C∗-algebra A is commutative if and only if the square functionpreserves the order on the positive cone of A [3]. It was observed later by Pedersenthat the above statement remains true for any power function with exponent greaterthan one [4]. Wu proved a similar result for the exponential function in 2001 [5]. Ji andTomiyama showed in 2003 that for any function f which is monotone but not matrixmonotone of order 2, a C∗-algebra A is commutative if and only if f is monotone onthe positive cone of A [1].
Very recently, Molnár proved a local theorem, namely, that a self-adjoint elementa of a C∗-algebra A is central if and only if a ≤ b implies exp a ≤ exp b [2].
Motivated by the work of Molnár, we show the following. If I = (γ,∞) is a realinterval and f is a continuously differentiable function on I such that the derivativeof f is positive, strictly monotone increasing and logarithmically concave, then a self-adjoint element a of a C∗-algebra A with spectrum in I is central if and only if a ≤ bimplies f(a) ≤ f(b), that is, f is locally monotone at the point a. This result easilyimplies the results of Ogasawara, Pedersen, Wu, and Molnár.
[1] G. Ji and J. Tomiyama, On characterizations of commutativity of C∗-algebras,Proc. Amer. Math. Soc. 131 (2003), 3845–3849.
[2] L. Molnár, A characterization of central elements in C∗-algebras, Bull. Austral.Math. Soc. 95 (2017), 138–143.
[3] T. Ogasawara, A theorem on operator algebras, J. Sci. Hiroshima Univ. Ser. A.18 (1955), 307–309.
[4] G.K. Pedersen, C∗-Algebras and Their Automorphism Groups, London Mathe-matical Society Monographs, 14, Academic Press, Inc., London-New York, 1979.
[5] W. Wu, An order characterization of commutativity for C∗-algebras, Proc. Amer.Math. Soc. 129 (2001), 983–987.
Tuesday morning
Chair: Peter Šemrl
9:00 Invited Talk: Antonio M. PeraltaJordan techniques to solve Tingley’s problem in the case of compact operators
9:50 Invited Talk: Clément de Seguins PazzisRange-compatible homomorphisms on matrix spaces
10:40 Coffee break
Chair: Alexander E. Guterman
11:20 Francisco Javier Garcia-PachecoIsometric representations of group
11:50 Marko OrelPreservers on finite structures
12:10 Seda Oguz UnalFiniteness conditions for some semigroup and monoid constructions
12:30 Lunch break
Tuesday afternoon
Chair: Mostafa Mbekhta
14:00 Dijana IliševićOn square roots of isometries
14:30 Francisco J. Fernández-PoloTingley problem in B(H)
15:00 Gergő NagyIsometries on Hilbert space operators
15:30 Coffee break
Chair: Takeshi Miura
16:00 Bojan KuzmaFundamental theorem on orthomaps over certain projective spaces
16:30 Gregor DolinarMaximal doubly stochastic matrix centralizers
17:00 Moisés Villegas-VallecillosCompact composition operators on Lipschitz spaces without compactness as-sumptions
17:30 Antonio Jiménez-VargasWeighted Banach spaces of Lipschitz functions
Jordan techniques to solve Tingley’s problem inthe case of compact operators
Antonio M. PeraltaUniversidad de Granada, Granada, Spain
The nowadays known as Tingley’s problem asks whether every surjective isometryf : S(X)→ S(Y ) between the unit spheres of two normed spaces X and Y admits anextension to a surjective real linear isometry T : X → Y . The origins of this problemgo back to the paper [6], where D. Tingley made the first contribution in the study ofsurjective isometries between the unit spheres of two finite dimensional normed spaces.
A solution to Tingley’s problem has been pursued by many researchers duringthe last thirty years. Most of positive answers to this problem correspond to infinitedimensional Banach spaces which are very close to commutative C∗-algebras and theirdual spaces. Quite recently, R. Tanaka revitalized the interest on Tingley’s problemin the case of non-commutative C∗-algebras by providing a positive solution for finitedimensional C∗-algebras [4].
In this talk we shall present some new results extending Tanaka’s solution to thecase of surjective isometries between the unit spheres of two compact operators betweenarbitrary complex Hilbert spaces [4, 3]. The new results are based on the advantageof applying Jordan techniques in the study of Tingley’s problem, and more concretely,the results determining the facial structure of the closed unit ball of a C∗-algebra [1]and of a JB∗-triple [2] provide new tools to tackle this old problem.
[1] C.A. Akemann, G.K. Pedersen, Facial structure in operator algebra theory,Proc. Lond. Math. Soc. 64, 418-448 (1992).
[2] C.M. Edwards, F.J. Fernández-Polo, C.S. Hoskin, A.M. Peralta, On thefacial structure of the unit ball in a JB∗-triple, J. Reine Angew. Math. 641 (2010)123-144.
[3] F.J. Fernández-Polo, A.M. Peralta, Low rank compact operators and Tin-gley’s problem, preprint 2016. arXiv:1611.10218v1
[4] A.M. Peralta, R. Tanaka, A solution to Tingley’s problem for isometries be-tween the unit spheres of compact C∗-algebras and JB∗-triples, preprint 2016.arXiv:1608.06327v1.
[5] R. Tanaka, Spherical isometries of finite dimensional C∗-algebras, J. Math. Anal.Appl. 451 (2017), no. 1, 319-326.
[6] D. Tingley, Isometries of the unit sphere, Geom. Dedicata 22 (1987) 371-378.
Range-compatible homomorphisms on matrixspaces
Clément de Seguins PazzisUniversité de Versailles-Saint-Quentin-en-Yvelines, France
Let U and V be vector spaces, and let S be a linear subspace of the space L(U, V )of all linear maps from U to V . A map F : S → V is called range-compatible wheneverit maps every operator s in S to a vector of the range of s. Among such maps are theevaluation maps s 7→ s(x) for fixed x ∈ U , also called local maps.
The problem of classifying range-compatible (group) homomorphisms is a very re-cent topic whose motivation is rooted in its connection to:
• full-rank preserving maps on large spaces of rectangular matrices;
• the structure of large spaces of matrices with rank bounded above;
• the notion of algebraic reflexivity.
It was long known that every range-compatible linear map on the whole spaceL(U, V ) is local. The recent research on range-compatible homomorphisms essentiallypurports to extend this result to linear subspaces of L(U, V ) with small codimension.
In this talk, I will explain the motivation for studying range-compatible maps, Iwill review some recent results on the topic and I will outline some key techniques forproving them.
[1] C. de Seguins Pazzis, Range-compatible homomorphisms on matrix spaces, Lin-ear Algebra Appl. 484 (2015), 237–289.
[2] C. de Seguins Pazzis, Range-compatible homomorphisms over the field with twoelements, preprint, arXiv: http://arxiv.org/abs/1407.4077
[3] C. de Seguins Pazzis, Quasi-range-compatible affine maps on large operatorspaces, Linear Multilinear Algebra 64 (2016) 1056–1085.
[4] C. de Seguins Pazzis, Range-compatible homomorphisms on spaces of symmetricor alternating matrices, Linear Algebra Appl. 503 (2016) 135–163.
Isometric representations of group
Francisco Javier Garcia-PachecoUniversity of Cadiz, Puerto Real, Spain
Let π : G → GX be an isometric representation of a group G in a Banach spaceX over a normalizing non-discrete absolute valued division ring K. If π and π∗ aresupportive and X verifies the separation property, then XG is 1-complemented in Xalong XG. As an immediate consequence, in an isometric representation of a group ina smooth Banach space whose dual is also smooth, the subspace of G-invariant vectorsis 1-complemented.
This research was supported by the Grant MTM2014-58984-P.
[1] F. J. Garcia-Pacheco, Complementation of the subspace of G-invariant vectors,J. Alg. Appl. (2016) in press
[2] P. W. Nowak, Group Actions on Banach Spaces, Handbook of group actions. Vol.II, 121–149, Adv. Lect. Math. (ALM) 32, Int. Press, Somerville, MA, (2015).
Preservers on finite structures
Marko Orel(1) University of Primorska, Koper, Slovenia (2) IMFM, Ljubljana, Slovenia
Preservers of some binary relations are quite important in matrix theory, since theycan be applied to solve other preserver problems. Such maps can be interpreted also asgraph homomorphisms between appropriate graphs. Consequently, tools from varioussubareas in graph theory can be applied to solve certain preserver problems. In thetalk I will survey few such techniques together with some recent results from preserverproblems and the study of graph homomorphisms. I will also point out how both theseresearch areas overlap with certain open problems in finite geometry.
Finiteness conditions for some semigroup andmonoid constructions
Seda Oguz UnalCumhuriyet University, Sivas, Turkey
The study of finiteness conditions for semigroups (monoids) consists in giving someconditions which are satisfied by finite semigroups (monoids) and which are such asto assure the finiteness of them. Periodicity, residually finiteness, finite generation,finite presentability and locally finiteness are some examples of finiteness conditions.In this talk, I will give some important results on finite presentability preserved undersome semigroup and monoid constructions, namely Bruck-Reilly extension of a monoid,HNN extension, direct product of semigroups (see for example [2] and [3]). I will givenecessary and sufficient conditions for generalized Bruck-Reilly *-extension of a groupto be finitely generated and finitely presented and I will present some related resultson generalized Bruck-Reilly *-extension by using the presentation given in [1].
This is a joint work with Eylem Guzel Karpuz, Karamanoglu Mehmetbey Univer-sity, Karaman, Turkey.
[1] C. Kocapinar, E. G. Karpuz, F. Ates, A. S. Cevik, Grobner - Shirshov basesof the generalized Bruck - Reilly *-extension, Algebra Colloquium 12 (2002), 19–31.
[2] I. M. Araujo, N. Ruskuc, Finite presentability of Bruck - Reilly extensions ofgroups, International Journal of Algebra 242 (2001), 20–30.
[3] I. M. Araujo, Finite presentability of semigroup constructions, InternationalJournal of Algebra and Computation 12 (2002), 19–31.
On square roots of isometries
Dijana IliševićUniversity of Zagreb, Zagreb, Croatia
The square of any isometry is again an isometry. What about the converse of thisfact? More precisely, is it true that a given isometry is a square of some isometry(with respect to the same norm)? The aim of this talk is to answer this question inthe setting of various normed spaces. The answer depends on a given norm, it differsin real and in complex spaces, and it also differs in finite and in infinite dimensionalspaces.
This is a joint work with Bojan Kuzma from the University of Primorska, Koper,Slovenia.
Tingley problem in B(H)
Francisco J. Fernández-PoloUniversidad de Granada, Granada, Spain
Let X and Y be normed spaces, whose unit spheres are denoted by S(X) andS(Y ), respectively. The so-called Tingley’s problem, named after the contribution ofD. Tingley [1], asks if every surjective isometry f : S(X)→ S(Y ) admits an extensionto a surjective real linear isometry T : X → Y .
Given two complex Hilbert spaces H and K, let S(B(H)) and S(B(K)) denotethe unit spheres of the C∗-algebras B(H) and B(K) of all bounded linear oper-ators on H and K, respectively. We prove that for each surjective isometry f :S(B(K))→ S(B(H)) there exist a surjective complex linear or conjugate linear isom-etry T : B(K) → B(H) such that T|S(B(H)) = f . This provides a positive answer tothe isometric extension problem (Tingley’s problem) in the setting of B(H) spaces.
We also prove that every surjective isometry between the unit spheres of two atomicJBW∗-triples E and B admits a unique extension to a surjective real linear isometryfrom E into B. This result constitutes a new positive answer to Tingley’s problem inthe Jordan setting.
[1] D. Tingley, Isometries of the unit sphere, Geom. Dedicata 22 (1987), 371–378.
Isometries on Hilbert space operators
Gergő NagyUniversity of Debrecen, Debrecen, Hungary
In this talk, we collect some known results concerning the general forms of isometrieson classical structures of bounded linear operators acting on Hilbert spaces. Besidethese statements, we present a general question of Tingley about the extendibility ofisometries between unit spheres in normed spaces. This problem has been solved forseveral normed spaces, e.g. the space B(H) of linear operators on a finite dimensionalcomplex Hilbert space H. We review the theorem which provides the solution, itdescribes the structure of surjective isometries of the unit sphere in B(H). Finally, wepresent a new result of ours in which the general form of all isometries of the spaceformed by the positive operators in that sphere was determined.
Fundamental theorem on orthomaps overcertain projective spaces
Bojan Kuzma1Institute of Mathematics, Physics, and Mechanics and 2University of Primorska
A Cayley plane is a non-Desarguesian projective plane which consists of 3-by-3trace-one Hermitian matrices over octonions. It comes naturally equipped with or-thogonality relation A⊥B whenever A B := 1
2(AB + BA) = 0. Its importance lies
in the fact that, as an orthoset, Cayley plane can be embedded into Albert algebra ofall 3-by-2 Hermitian matrices over octonions with the standard Jordan product whichin turn constitutes one of the five families of formally real finite dimensional simpleJordan algebras.
In the talk we present our recent result where we classified maps which preserveorthogonality on Cayley plane. In addition, we also classify orthogonality preservingmaps on finite dimensional projective spaces over reals, complexes, or quaternions.Unlike similar results which extend Uhlhorns’s theorem (see [1]) we assume neitherinjectivity/surjectivity nor that orthogonality is preserved in both directions. Ourresults have direct application to maps which preserve Jordan orthogonality on realfinite-dimensional simple Jordan algebra.
This is a joint work with G. Dolinar and N. Stopar.
[1] U. Uhlhorn, Representation of symmetry transformations in quantum mechanics,Ark. Fysik 23 (1963), 307–340.
Maximal doubly stochastic matrix centralizers
Gregor DolinarUniversity of Ljubljana, Faculty of Electrical Engineering, Ljubljana, Slovenia
A doubly stochastic matrix is a square matrix of nonnegative real numbers witheach row and column summing to 1. For A ∈ Mn its centralizer, denoted by C(A), isthe set of all matrices commuting with A, that is C(A) = X ∈Mn : AX = XA, andfor a set S ⊆ Mn its centralizer, denoted by C(S), is the intersection of centralizersof all its elements, that is C(S) = X ∈ Mn : AX = XA, for every A ∈ S. Thecentralizer induces a preorder relation A B if C(A) ⊆ C(B). A non-scalar matrix Ais maximal if for every non-scalar matrix X with C(X) ⊇ C(A) it follows C(A) = C(X).
We describe doubly stochastic matrices with maximal centralizers and matriceswhich are maximal when their centralizers are restricted to doubly stochastic matrices.
This is a joint work with Henrique F. da Cruz (Departamento de Matemáticada Universidade da Beira Interior, Portugal), Rosário Fernandes (Departamento deMatemática, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, Por-tugal), and Bojan Kuzma (University of Primorska, Slovenia).
[1] H. F. da Cruz, G. Dolinar, R. Fernandes, and B. Kuzma, Maximal doublystochastic matrix centralizers, submitted to publication.
Compact composition operators on Lipschitzspaces without compactness assumptions
Moisés Villegas-VallecillosUniversity of Cadiz, Puerto Real, Spain
Kamowitz and Scheinberg [1] studied compact composition operators on Lip(X),the Banach space of all bounded Lipschitz functions f from a metric space (X, d) intothe field of real or complex numbers K, endowed with one of the following two norms:
‖f‖ = max ‖f‖∞, L(f) , ‖f‖s = ‖f‖∞ + L(f),
where ‖f‖∞ is the supremum norm of f and L(f) is the Lipschitz constant of f . Theyproved that if (X, d) is a compact metric space and φ : X → X is a Lipschitz mapping,then the composition operator Cφ : Lip(X) → Lip(X), defined by Cφ(f) = f φ, iscompact if and only if φ is supercontractive.
Our first aim is to give a more complete characterization of compact compositionoperators on Lip(X) without assuming compactness on X. Our approach lies in tack-ling the problem for pointed Lipschitz spaces Lip0 which generalize the Lipschitz spacesLip. Moreover, we obtain the analogous result for compact composition operators onlittle Lipschitz spaces lip(X) satisfying a kind of uniform point separation property.
Secondly, we determine the spectrum of compact composition operators on Lipschitzspaces Lip(X) and lip(X).
This is a joint work with Antonio Jiménez-Vargas from the University of Almeria(Spain).
[1] H. Kamowitz and S. Scheinberg, Some properties of endomorphisms of Lips-chitz algebras, Studia Math. 96 (1990), 61–67.
Weighted Banach spaces of Lipschitz functionsAntonio Jiménez-Vargas
University of Almería, Almería, Spain
We introduce the weighted Banach spaces of Lipschitz functions and study someproblems concerning their duality theory.
Let (X, d) be a pointed metric space with a basepoint denoted by e. A real-valuedfunction v on the set X := (x, y) ∈ X2 : x 6= y is said to be a weight on X if it is(strictly) positive and continuous.
The weighted Lipschitz space Lipv(X) is the Banach space of all Lipschitz scalar-valued functions f on X for which f(e) = 0 such that
sup
v(x, y)
|f(x)− f(y)|d(x, y)
: (x, y) ∈ X<∞,
endowed with the weighted Lipschitz norm:
Lipv(f) = sup
v(x, y)
|f(x)− f(y)|d(x, y)
: (x, y) ∈ X.
The weighted little Lipschitz space lipv(X) is the closed linear subspace of Lipv(X)consisting of all those functions f with the property that for every ε > 0, there existsa δ > 0 such that v(x, y)|f(x)− f(y)|/d(x, y) < ε whenever 0 < d(x, y) < δ.
Thus Lipv(X) may be regarded as all Lipschitz scalar-valued functions f on Xvanishing at the basepoint such that |f(x)− f(y)|/d(x, y) satisfies a growth conditionof order O(1/v(x, y)) while lipv(X) are those functions for which |f(x)− f(y)|/d(x, y)has a growth rate of order o(1/v(x, y)).
The study of Lipv spaces is new and interesting. We analyse the proof of the Ng–Dixmier theorem to describe an isometric predual Fv(X) of the space Lipv(X). ViewingLipv(X) as the dual of Fv(X), we study the bounded weak* topology τbw∗ on Lipv(X)and prove that (Lipv(X), τbw∗) = (Fv(X)∗, τc) and Fv(X) = ((Lipv(X), τbw∗)′, τc),where τc denotes the topology of uniform convergence on compact sets.
We also give a process of linearization of the elements of Lipv(X) which is a lin-earizing construction stronger than a predual space, and characterize the space Fv(X)by means of an universal property.
Assuming X is compact, we address the question as to when Lipv(X) is canonicallyisometrically isomorphic to lipv(X)∗∗, and show that this is the case whenever lipv(X)is an M-ideal in Lipv(X) and the called associated weights vL and vl coincide.
The study of the biduals of weighted Banach spaces of analytic functions in [1, 2]motivates our approach to the subject.
This research was partially supported by the Spanish Ministry of Economy andCompetitiveness project no. MTM2014-58984-P and the European Regional Develop-ment Fund (ERDF), and Junta of Andalucía grant FQM-194.
[1] K. D. Bierstedt and W. H. Summers, Biduals of weighted Banach spaces ofanalytic functions, J. Austral. Math. Soc. Ser. A 54 (1993), 70–79.
[2] C. Boyd and P. Rueda, The biduality problem and M-ideals in weighted spacesof holomorphic functions, J. Convex Anal. 18 (2011), 1065–1074.
Wednesday
Chair: Antonio M. Peralta
9:00 Invited Talk: Peter ŠemrlOrder isomorphisms of operator intervals
9:50 Invited Talk: Mostafa MbekhtaNonlinear maps commuting with the λ-Aluthge transform under certain prod-ucts
10:40 Coffee break
Chair: Fernanda M. Botelho
11:20 Invited Talk: Osamu HatoriHomomorphisms and isometries on Banach algebras of vector-valued maps
12:10 Rumi Shindo TogashiStructures of maps with the properties of finite products
12:40 Lei LiKaplansky theorem for completely regular spaces
13:00 Lunch break and free program
17:00 Ship tour
18:30 Conference dinner
Order isomorphisms of operator intervals
Peter ŠemrlUniversity of Ljubljana, Ljubljana, Slovenia
We will present a general theory of order isomorphisms of operator intervals whichunifies and extends several known results, among others the famous Ludwig’s descrip-tion of ortho-order automorphisms of effect algebras and Molnár’s characterization ofbijective order preserving maps on bounded observables. All the proofs are elementaryand mostly self-contained. The optimality of the results will be demonstrated by coun-terexamples. Only here we need to use a non-trivial tool, that is, the Löwner’s theoryof operator monotone functions. For our results we need to understand only the struc-ture of strictly operator monotone functions. One of the consequences of our resultsand proofs is that the full understanding of strictly operator monotone functions canbe achieved using rather simple linear algebra arguments.
Nonlinear maps commuting with the λ-Aluthgetransform under certain products
Mostafa MbekhtaUniversity Lille1, France
In this talk we give a complete form of the bijective (not necessarily linear) mapsΦ : B(H) → B(K), where H,K are Hilbert spaces of dimension greater than 2, thatsatisfy
∆λ(Φ(A) ? Φ(B)) = Φ(∆λ(A ? B)) for all A,B ∈ B(H),
where ∆λ(T ) is the λ-Aluthge transform of T and the operation A ? B means one ofthe following products:
1. The standard product A ? B = AB
2. The jordan product A ? B = A B = 12(AB +BA).
3. The start Jordan product A ? B = A B∗ = 12(AB∗ +B∗A).
4. The (n,m)-Jordan triple product A ? B = AnBAm with n,m ∈ N such thatn+m ≥ 1.
This is a joint work with Chabbabi Fadil, from the University Lille 1.
[1] F. Botelho; L. Molnár; G. Nagy, Linear bijections on von Neumann factorscommuting with λ-Aluthge transform, Bull. Lond. Math. Soc. 48 (2016), 74–84.
[2] F. Chabbabi, Product commuting maps with the λ-Aluthge transform, J. Math.Anal. Appl. 449 (2017), 589–600.
[3] F. Chabbabi and M.Mbekhta, Jordan product maps commuting with the λ-Aluthge transform, J. Math. Anal. Appl. 450 (2017), 293–313.
[4] F. Chabbabi and M.Mbekhta, General product nonlinear maps commutingwith the λ-Aluthge transform, Mediterr. J. Math. 14 (2017), 14–42.
[5] F. Chabbabi and M.Mbekhta, Nonlinear maps commuting with the λ-AluthgeTransform under (n,m)−Jordan-Triple product, 2017, (preprint).
Homomorphisms and isometries on Banachalgebras of vector-valued maps
Osamu HatoriNiigata University, Niigata, Japan
We study maps on certain admissible quadruples including Lipschitz algebras ofvector-valued maps, where the range of maps are unital commutative Banach algebrasincluding unital commutative C∗-algebras. We consider:
• maps which preserve algebraic structures,
• maps which preserve metric structures.
This is a joint work with Shiho Oi and Hiroyuki Takagi.
Structures of maps with the properties of finiteproducts
Rumi Shindo TogashiNational Institute of Technology, Nagaoka College, Nagaoka, Japan
Let X and Y be locally compact Hausdorff spaces and C0(X) and C0(Y ) be Banachalgebras of all continuous complex valued functions which vanish at infinity. Let X befirst countable.
Let n ≥ 3 be a fixed natural number. We show that if T : C0(X) → C0(Y ) is asurjection such that
supy∈Y
∣∣∣∣∣(
n∏k=1
T (fk)
)(y) + 1
∣∣∣∣∣ = supx∈X
∣∣∣∣∣(
n∏k=1
fk
)(x) + 1
∣∣∣∣∣for all f1, · · · , fn ∈ C0(X), then there exist a homeomorphism φ : Y → X, a continuousfunction α : Y → e 2
nπi, · · · , e
2(n−1)n
πi, 1, and a clopen set K ⊂ Y such that T (f)(y) =α(y)f(φ(y)) if y ∈ K or f(φ(y)) if y ∈ Y \K for all f ∈ C0(X). This is a joint workwith Takeshi Miura from Niigata University. In the case of uniform algebras relatedresults were obtained in [1, 3, 4, 5].
We describe also the structure of weakly peripherally-multiplicative surjections andthe following is proved: If T : C0(X) → C0(Y ) is a surjection with the property that,for a fixed natural number n ≥ 2,
σπ
(n∏k=1
T (fk)
)∩ σπ
(n∏k=1
fk
)6= ∅
for all f1, · · · , fn ∈ C0(X), where σπ(f) is the peripheral spectrum of f , then there exista homeomorphism φ : Y → X and a continuous function α : Y → e 2
nπi, · · · , e
2(n−1)n
πi, 1such that T (f)(y) = α(y)f(φ(y)) for all f ∈ C0(X) and y ∈ Y . One of the consequencesof this result is that, for the first countable spacesX and mappings T : C0(X)→ C0(Y ),the answer to Question 1 in [2] is affirmative.
[1] O. Hatori, T. Miura and H. Takagi, Multiplicatively spectrum-preserving and norm-preserving maps between invertible groups of commutative Banach algebras, (2006),preprint.
[2] O. Hatori, S. Lambert, A. Luttman, T. Miura, T. Tonev and R. Yates, Spectralpreservers in commutative Banach algebras, Contemp. Math., 547 (2011), 103–123.
[3] A. Luttman and S. Lambert, Norm conditions for uniform algebra isomorphisms,Cent. Eur. J. Math., 6(2) (2008), 272–280.
[4] R. Shindo, Norm conditions for real-algebra isomorphisms between uniform alge-bras, Cent. Eur. J. Math., 8(1) (2010), 135–147.
[5] R. Shindo, Maps between uniform algebras preserving norms of rational functions,Mediterr. J. Math., 8 (2011), 81–95.
Kaplansky theorem for completely regularspaces
Lei LiNankai University, Tianjin, China
Let X, Y be realcompact spaces. Let φ be a linear bijection from C(X) onto C(Y ).We show that if φ preserves nonvanishing functions, then φ is a weighted compositionoperator. This result is applied also to other nice function spaces, e.g., uniformly orLipschitz continuous functions on metric spaces. This is joint work with ProfessorNgai-Ching Wong.
Thursday morning
Chair: Chi-Kwong Li
9:00 Invited Talk: Fernanda BotelhoContractive and bi-contractive projections on vector valued spaces of contin-uous functions
9:50 Invited Talk: Javad MashreghiOuter preserving endomorphisms
10:40 Coffee break
Chair: Osamu Hatori
11:20 Takeshi MiuraNorms on C1([0, 1]) and their surjective isometries
11:50 Ya-Shu WangIsometries between vector-value function spaces
12:10 Tamás TitkosNon-linear preservers for probability measures
12:30 Lunch break
Thursday afternoon
Chair: György Pál Gehér
14:00 Miroslav KurešAutomorphisms of Weil algebras and differential geometry
14:30 Jiří TomášProduct preserving bundle functors on Mfm ×Mf and their identificationwith the bundles of (p,A)-jets
15:00 Marko StošićCombinatorics of HOMFLY invariants for knots
15:30 Coffee break
Chair: Damir Bakić
16:10 Edward PoonContinuous multiplicative isometries of Toeplitz algebras
16:40 Jani VirtanenTransition asymptotics of Toeplitz determinants and their applications in ran-dom matrix theory
17:10 Marija DodigMinimal completion problems for matrix pencils
17:40 Kenier CastilloOn an open-ended problem proposed by P. Turán
Contractive and bi-contractive projections onvector valued spaces of continuous functions
Fernanda BotelhoThe University of Memphis, Memphis TN, USA
We shall present results on the structure of contractive and bi-contractive projec-tions on the space of continuous functions defined on a compact metric space withvalues in a uniformly convex Banach space. The approach followed in this presenta-tion will use techniques developed by Friedman-Russo for the scalar case and Singer’sidentification of the dual space C(Ω, E)∗ with the space of regular vector measures ofbounded variation defined on the Borel subsets of Ω.
Outer preserving endomorphisms
Javad MashreghiLaval University, Quebec City, Canada
Let T : Hp → Hp be a linear mapping (no continuity assumption). What can wesay about T if we assume that “it preserves outer functions”? Another related questionis to consider linear functionals T : Hp → C (again, no continuity assumption) andask about those functionals whose kernels do not include any outer function. Westudy such questions via an abstract result which can be interpreted as the generalizedGleason–Kahane–Żelazko theorem for modules. In particular, we see that continuity ofendomorphisms and functionals is a part of the conclusion. We discuss, further stepsin treating other function spaces, e.g., Bergman, Dirichlet, Besov, the little Bloch, andVMOA.
This is a joint work with T. Ransford.
[1] J. Mashreghi, T. Ransford A Gleason–Kahane–Żelazko theorem for modulesand applications to holomorphic function spaces. J. Lond. Math. Soc., 47 (6):1014–1020, 2015.
Norms on C1([0, 1]) and their surjective isometriesTakeshi Miura
Niigata University, Niigata, Japan
Let C1([0, 1]) be the complex linear space of all continuously differentiable complexvalued functions on the closed unit interval [0, 1]. The space C1([0, 1]) is a Banachspace with respect to several norms, say ‖f‖C = supt∈[0,1](|f(t)| + |f ′(t)|), ‖f‖Σ =‖f‖∞+‖f ′‖∞ and ‖f‖σ = |f(0)|+‖f ′‖∞ for f ∈ C1([0, 1]). Here ‖g‖∞ = supt∈[0,1] |g(t)|for a continuous function g on [0, 1].
A mapping S : M → N between normed linear spaces (M, ‖ · ‖M) and (N, ‖ · ‖N) isan isometry if and only if
‖S(f)− S(g)‖N = ‖f − g‖M (∀f, g ∈M).
Here, we do not assume linearity of the map.Cambern [1], Rao and Roy [4] and Koshimizu [3] gave the characterizations of
surjective, complex linear isometries on C1([0, 1]) with respect to ‖ · ‖C , ‖ · ‖Σ and‖ · ‖σ, respectively. The proofs are all based on the extreme point method, yet detailsvary depending on norms they study. To unify those results, we introduce a norm onC1([0, 1]) as follows:Definition 1. Let D be a compact connected subset of [0, 1] × [0, 1]. For each f ∈C1([0, 1]) we define ‖f‖〈D〉 = sup(s,t)∈D(|f(s)| + |f ′(t)|). We see that ‖f‖〈D〉 is a normon C1([0, 1]) if and only if p1(D)∪ p2(D) = [0, 1], where pj denotes the projection fromD to the j-th coordinate of [0, 1]× [0, 1].Theorem 2 ([2]). Let D be a compact connected subset of [0, 1] × [0, 1] such thatp1(D) ∪ p2(D) = [0, 1]. Let p1(D) = [a, b] and p2(D) = [c, d] with a ≤ b and c ≤ d. IfS : C1([0, 1])→ C1([0, 1]) is a surjective isometry, then there exist continuous functionsα, β : [0, 1] → C, constants ε0, ε1 ∈ ±1, a C1-diffeomorphism ϕ : [a, b] → [a, b] anda homeomorphism ψ : [c, d] → [c, d] such that |α| = 1 on [a, b], α is constant on [c, d],|β| = 1 on [c, d] and
S(f)(t) = S(0)(t) + α(t)[f(ϕ(t))]ε0 (∀f ∈ C1([0, 1]), ∀t ∈ [a, b]),
(S(f))′(t) = (S(0))′(t) + β(t)[f(ψ(t))]ε0ε1 (∀f ∈ C1([0, 1]),∀t ∈ [c, d]).
Here, we define [f(s)]ε = f(s) if ε = 1, and [f(s)]ε = f(s) if ε = −1.This is a joint work with Kazuhiro Kawamura from University of Tsukuba and
Hironao Koshimizu from National Institute of Technology, Yonago College.This research was supported by JSPS KAKENHI Grant Number 15K04921.
[1] M. Cambern, Isometries of certain Banach algebras, Studia Math. 25 (1964-1965)217–225.
[2] K. Kawamura, H. Koshimizu and T. Miura, Norms on C1([0, 1]) and their isome-tries, preprint.
[3] H. Koshimizu, Linear isometries on spaces of continuously differentiable and Lips-chitz continuous functions, Nihonkai Math. J. 22 (2011), 73–90.
[4] N.V. Rao and A.K. Roy, Linear isometries of some function spaces, Pacific J. Math.38 (1971), 177–192.
Isometries between vector-value functionspaces
Ya-Shu WangNational Chung Hsing University, Taichung, Taiwan
Let X be a Hausdorff topological space and E be a Banach space. Denote byCb(X,E) the Banach space of all bounded continuous functions from X into E withsupremum norm. We study linear isometries between various types of function spaces.In this talk, I will present a representation for a general linear isometry on a certainsubspace of Cb(X,E), and then apply the representation to the spaces of Lipschitzfunctions or differentiable functions.
Non-linear preservers for probability measures
Tamás TitkosAlféd Rényi Institute of Mathematics, Budapest, Hungary
There is a long history and vast literature of distance preserving maps (i.e. isome-tries) on different kind of linear and non-linear spaces. The starting point of this work isLajos Molnár’s paper Lévy isometries of the space of probability distribution functions.In that paper Molnár presented the complete description of surjective Lévy-isometriesof the non-linear space of all cumulative distribution functions of one-dimensional ran-dom variables. Since Prokhorov provided a concrete metrisation of the topology ofweak convergence (today known as the Lévy-Prokhorov distance), a natural questionarises:
Is there an analogous description of all onto mappings that preserve the Lévy-Prokhorov distance of probability measures on Polish spaces?
Since stochastic processes depending upon a continuous parameter are basicallyprobability measures on certain subspaces of the space of all functions of a real variable,a particularly important case of this theory is when the underlying metric space hasa linear structure. In this talk we give the complete description of the structure ofsurjective Lévy-Prokhorov isometries on the space of all Borel probability measures onan arbitrary separable real Banach-space.
This is a joint work with György Pál Gehér.This research was supported by the “Lendület” Program (LP2012-46/2012) of the
Hungarian Academy of Sciences.
Automorphisms of Weil algebras anddifferential geometry
Miroslav KurešBrno University of Technology, Brno, Czechia
Motivated by algebraic geometry, André Weil suggested the treatment of infinitesi-mal objects as homomorphisms from algebras of smooth functions into some real finite-dimensional commutative algebra with unit in 1950’s. In fact, he follows a certain ideaof Sophus Lie: so-called A-near points represent the ‘parametrized infinitesimal sub-manifolds’. More precisely, letM be a smooth manifold and let C∞(M,R) be its ring ofsmooth functions into R: A-near points of M are defined as R-algebra homomorphismC∞(M,R)→ A, where A is a certain local R-algebra A (precisely defined below) nowcalled the Weil algebra. This can be regarded as the first notable occurrence of local R-algebras in differential geometry. New concepts, such as Weil algebras, Weil functors,Weil bundles were introduced and they are widely studied, even to this day, because oftheir considerable generality. In a modern categorical approach to differential geometry,if we interpret geometric objects as bundle functors, then natural transformations rep-resent a number of geometric constructions. In this context, finding a bijection betweennatural transformations of two Weil functors TA, TB (generalizing well-known functorsof higher order velocities and, of course, the tangent functor as the first of them) andcorresponding morphisms of Weil algebras A and B, has fundamental importance.
The talk will focus on the description of groups of automorfisms of Weil algebras,their properties and subalgebras of fixed points. The author will summarize someresults (see e.g. the survey paper [1]) and also add some new ones.
[1] M. Kureš, Fixed point subalgebras of Weil algebras: from geometric to algebraicquestions. In: Complex and Differential Geometry. Springer Proceedings in Mathe-matics, Springer, 2011, pp. 183–192.
Product preserving bundle functors onMfm ×Mf and their identification with the
bundles of (p,A)-jets
Jiří TomášBrno University of Technology, Czech republic
Let A be a Weil algebra of width k,M be anm-dimensional manifold,Mfm the cat-egory of m-dimensional manifolds with local diffeomorphisms andMf the category ofsmooth manifolds with smooth maps. We generalize the concept of a (p,A)-covelocityon M introduced in [1] for the cases of m ≥ k to the lower-dimensional cases of m. Weintroduce (p,A)-covelocities by means of A-automorphism classes of the restrictions of(p,A)-covelocities from TA(M × Rk−m) to the domains determined by TA-respectingfoliations on TAM . We construct a general example of such foliation. From (p,A)-covelocities we immediately obtain the concept of a (p,A)-jet and we finally give theidentification of (p,A)=jets with product preserving bundle functors onMfm ×Mf ,applying essentialy the result of Kolář and Mikulski, [2].
[1] J. Tomáš, Bundles of (p,A)-covelocities and (p,A)-jets. Miskolc Math. Notes,2013(14), pp. 547–555.
[2] I. Kolář and W. M. Mikulski On the fiber product preserving bundle functors.Diff. Geom. and its Appl., 1999(11), pp. 105–115.
Combinatorics of HOMFLY invariants for knots
Marko StošićCAMGSD, Departamento de Matématica, Instituto Superior Técnico, Av. Rovisco
Pais, 1049-001 Lisbon, Portugal ([email protected]),and
Mathematical Institute SANU, Knez Mihajlova 36, 11000 Beograd, Serbia.
The main method to study topological properties of knots is by means of theirinvariants. There are numerous knot invariants, originating from different areas ofmathematics and physics – including algebra, representation theory, operator theory,quantum groups, string theory, etc. In this talk we shall focus on (colored) HOMFLYinvariants of knots and their relationship to the string theory motivated BPS invariants,giving rise to some surprising combinatorial results.
Continuous multiplicative isometries of Toeplitzalgebras
Edward PoonEmbry-Riddle University, Prescott, United States
We characterize continuous nondegenerate multiplicative maps on the algebra Tn ofn×n upper-triangular Toeplitz matrices. As a corollary, the continuous multiplicativeisometries of Tn (for any norm) are necessarily similarities or similarities composedwith complex conjugation; for unitarily invariant norms, such isometries are preciselythe unitary similarities, or unitary similarities composed with complex conjugation.
Transition asymptotics of Toeplitzdeterminants and their applications in random
matrix theory
Jani VirtanenUniversity of Reading, Reading, United Kingdom
The study of asymptotic behaviour of Toeplitz determinants goes back to Szegő,who described the behaviour for certain positive symbols in 1915. Since then the prob-lem has been widely studied in mathematics and physics communities. I discuss somerecent progress related to symbols that possess so-called Fisher-Hartwig singularitiesand double-scaling limits in two parameters, one in large n (the size of the matrices)and the other one related to the the number of Fisher-Hartwig singularities. I alsomention some applications of these asymptotic results that arise in random matrixtheory.
Minimal completion problems for matrix pencils
Marija DodigCEAFEL, Departamento de Matématica, Universidade de Lisboa, Edificio C6,
Campo Grande, 1749-016 Lisbon, Portugal ([email protected]),and
Mathematical Institute SANU, Knez Mihajlova 36, 11000 Beograd, Serbia.
We present some recent results in determining Kronecker invariants of a matrixpencil with a prescribed subpencil. These are based on a concept of minimal comple-tions - i.e. adding the minimal possible number of rows and columns in order to obtainthe prescribed set of Kronecker invariants. The techniques we use are combinatorial,based on LR sequences, Young diagrams and combinatorics of partitions, involvingrecently introduced concept of generalized majorization between three different parti-tions of integers. Among other results, we give a solution to the General Matrix PencilCompletion Problem in the minimal case.
On an open-ended problem proposed by P. Turán
Kenier CastilloUniversity of Coimbra, Coimbra, Portugal
In a problem-paper contained part of his lectures held in Summer 1975 at theUniversité de Montreal, P. Turán proposed the following open-ended question: “It isknown that the zeros of the nth orthogonal polynomial (with respect to a Lebesgue-integral function on an interval) separate the zeros of the (n + 1)th polynomial. Whatcorresponds to this fact on the unit circle?”. The answer to this question, the so-calledparaorthogonal polynomials on the unit circle (POPUC), appeared (in a somewhathidden form) in a serie of papers by Delsarte and Genin, when they were working insignal processing. After that, several authors stated additional properties of zeros ofPOPUC, the most recent one due to B. Simon. The purpose of this talk is to extendin a simple and unified way the known results on interlacing of zeros of POPUC. Thistalk is about a joint work with J. Petronilho (see [1]).
[1] K. Castillo and J. Petronilho, Refined interlacing properties for zeros ofparaorthogonal polynomials on the unit circle, Centre for Mathematics, Universityof Coimbra Technical Report 16-45 (2016).
Friday morning
Chair: Javad Mashreghi
9:00 Invited Talk: Jan HamhalterStructure of abelian parts of C∗-algebras and its preservers
9:50 Invited Talk: Ngai-Ching WongDisjointness preservers of operator algebras and related objects
10:40 Coffee break
Chair: Jan Hamhalter
11:20 Damir BakićPreserving the reconstruction property of frames from frame coefficients witherasures
11:50 Ljiljana ArambašićOuter frames for Hilbert C∗-modules
12:20 Ahlem Ben Ali EssalehExtreme-strong-local ∗-automorphism of B(H) satisfying the 3-local property
12:40 Martin BohataPreservers and Jordan isomorphisms
13:00 Lunch break
Friday afternoon
Chair: Dijana Ilišević
14:30 Anil Kumar KarnOrthogonality in C∗-algebras and a model for non-commutative vector lattices
15:00 László L. StachóOn the structure of C0-semigroups of holomorphic Carathéodory isometries
15:30 Marcell GaálMaps on positive operators preserving Rényi type relative entropies and max-imal f -divergences
15:50 György Pál GehérThe optimal version of Wigner’s theorem on the Grassmann space of all n-dimensional subspaces
16:10 Closing
Structure of abelian parts of C∗-algebras andits preservers
Jan HamhalterCzech Technical University in Prague, Prague, Czech Republic
The aim of this talk is to present recent results on morphisms of the poset ofabelian C∗-subalgebras and related problems concerning the structure of various mapsbehaving well with respect to commuting elements. Let A be a unital C∗-algebra.Denote by C(A) the poset of all unital abelian C∗-subalgebras of A ordered by settheoretic inclusion. This structure embodies Bohrification program in foundation ofquantum theory and represents a C∗-invariant of growing importance. We show thatisomorphisms of these structures result as set theoretic images of Jordan type maps.More specifically, we show that for any order isomorphism ϕ : C(A)→ C(B) there is a(usually unique) map Φ : A→ B that is a linear Jordan ∗-isomorphism when restrictedto any abelian C∗-subalgebra of A, such that
ϕ(C) = Φ(x) |x ∈ C
for all C ∈ C(A). Moreover, we show that Φ can be chosen to be a linear Jordan∗-isomorphism if A and B are typical von Neumann algebras. In other words, C(A)is a complete Jordan invariant for von Neumann algebras. We shall show that C(A)is even a complete C∗-invariant for important classes of C∗-algebras. We apply theseresults to Choquet theory. Especially, we describe preserves of the Choquet order oforthogonal measures on the state space that have a fixed barycenter in terms of Jordanmorphisms of the underlying von Neumann algebras. It implies interesting preserverstheorem on convex decompositions of states.
We discuss related Mackey-Gleason theorem for AW ∗-algebras and state non-bijec-tive generalization of Dye’s theorem on orthomorphisms of von Neumann projectionlattices to AW ∗-algebras. This enables us to extend the results on isomorphisms ofC(A)’s to AW ∗-algebras.
The structure C(A) is connected with recently studied piecewise structure of op-erator algebras. In this approach, motivated by foundation of physics, piecewise mor-phisms are maps well behaved with respect to commuting elements. We shall describea few variants of piecewise Jordan (triple) maps on the structures of self-adjoint oper-ators, positive invertible elements, and unitary groups.
[1] J. Hamhalter, Quantum Measure Theory, Kluwer Academic Publishers (2003),Boston, London.
[2] J.Hamhalter, Isomorphisms of ordered structures of abelian C*-subalgebras ofC*-algebras, J. Math. Analysis Appl. 383 (2011), no. 2, 391–399.
[3] J.Hamhalter, Dye’s Theorem and Gleason’s Theorem for AW*-algebras, J.Math. Anal. Appl. 422 (2015), 1103-1115.
[4] B.Lindenhovius, C(A), PhD Thesis, Radboud University, Netherlands, 2016.
Disjointness preservers of operator algebrasand related objects
Ngai-Ching WongNational Sun Yat-sen University, Kaohsiung, 80424, Taiwan.
Recall that a W*-algebraM is a C*-algebra with a predual. SoM carries many dif-ferent structures, including the geometric (i.e., norm) structure, the *-algebraic struc-ture, and the normal structure (i.e., weak* topology). As the norm of an element ain M is equal to the square root of the spectral radius of a∗a, the geometric structureof M can be recovered from its *-algebraic structure. It is further showed by Gardnerthat two W*-algebras are *-algebraic isomorphic if they are algebraic isomorphic, andall algebraic isomorphisms between W*-algebras are norm and σ-weakly bi-continuous.Indeed, every algebra isomorphism θ : M → N between W*-algebras carries the formθ(a) = π(hah−1) for some invertible positive element h in M and some *-isomorphismπ from M onto N . Therefore, W*-algebras are completely determined by their linearand product structures.
In this talk, we show that only the linear and zero product, or more generally, thelinear disjointness structure also suffice. If time allows, we will also discuss how thethe linear disjointness structure determines Hilbert C*-modules, Fourier algebras, andgeneral C*-algebras.
Preserving the reconstruction property offrames from frame coefficients with erasures
Damir BakićUniversity of Zagreb, Zagreb, Croatia
A sequence (xn)∞n=1 in a real or complex Hilbert space H is a frame for H if thereexist positive constants A and B, that are called frame bounds, such that
A‖x‖2 ≤∞∑n=1
|〈x, xn〉|2 ≤ B‖x‖2, ∀x ∈ H.
In general, frames are linearly dependent (hence, redundant) generating systems. Theyare often used in process of encoding and decoding signals. It is the redundancyproperty of frames that makes them useful in applications where some of the coefficientscould be corrupted or erased during the data transmission.
We shall discuss a new approach to the problem of recovering signal from framecoefficients with erasures. Provided that the erasure set satisfies the minimal redun-dancy condition, we construct a suitable synthesizing dual frame which enables us toperfectly reconstruct the original signal without recovering the lost coefficients. Suchdual frames which compensate for erasures will be described from various viewpoints.
This is a joint work with Ljiljana Arambašić from the University of Zagreb.This research was supported by the Croatian Science Foundation under the project
IP-2016-06-1046.
Outer frames for Hilbert C∗-modules
Ljiljana ArambašićUniversity of Zagreb, Zagreb, Croatia
A Hilbert C∗-module over a C∗-algebra A is a right A-module X equipped with anA-valued inner product 〈·, ·〉 : X×X → A such that X is a Banach space with respectto the norm ‖x‖ = ‖〈x, x〉‖ 1
2 . A (possibly finite) sequence (xn)n in X is called a framefor X if there exist positive constants A and B such that
A〈x, x〉 ≤∞∑n=1
〈x, xn〉〈xn, x〉 ≤ B〈x, x〉, ∀x ∈ X, (1)
where the sum in the middle converges in norm. When we take for the underlyingC∗-algebra of coefficients the field of complex numbers, i.e., when X is a Hilbert space,(1) becomes
A‖x‖2 ≤∞∑n=1
|〈xn, x〉|2 ≤ B‖x‖2, ∀x ∈ X,
which means that (xn)n is a standard Hilbert space frame.In this talk we present some new results in the theory of frames for countably
generated Hilbert C∗-modules over arbitrary C∗-algebras. In investigating the non-unital case we introduce the concept of an outer frame for X - in comparison withframes for X the difference is that the elements of an outer frame for X are merelymembers of a multiplier moduleM(X) and need not belong to X. Outer frames behavevery much like frames; for example, the reconstruction property for elements of X ispreserved, and the only adjointable operators on M(X) that preserve the class offrames and outer frames for X are the surjective ones. Building on a unified approachto frames and outer frames we obtain new results on dual frames, frame perturbations,tight approximations of frames and finite extensions of Bessel sequences.
This is a joint work with Damir Bakić from the University of Zagreb.This research was supported by the Croatian Science Foundation under the project
IP-2016-06-1046.
Extreme-strong-local ∗-automorphism of B(H)
satisfying the 3-local property
Ahlem Ben Ali EssalehUniversity of Monastir, Monastir, Tunisia
We show that, for a complex Hilbert space H with dimension ≥ 3, every extreme-strong-local ∗-automorphism (i.e. bilocal ∗-automorphism) T : B(H) → B(H) sat-isfying the 3-local property is a ∗-monomorphism, that is, every linear mapping T :B(H)→ B(H) satisfying that for every a in B(H) and every ξ, η in H, there exists a∗-automorphism πa,ξ,η : B(H)→ B(H), depending on a, ξ, and η, such that
T (a)(ξ) = πa,ξ,η(a)(ξ), and T (a)(η) = πa,ξ,η(a)(η),
is a ∗-monomorphism. This gives a complete positive answer to the question posed byL. Molnár in [Arch. Math. 102, 83-89 (2014)].
This is a joint work with Mohsen Niazi, University of Birjand, Iran & Prof. AntonioM. Peralta, University of Granada, Spain.
[1] A. Ben Ali Essaleh, M. Niazi, A.M. Peralta, Bilocal *-automorphisms ofB(H) satisfying the 3-local property, Arch. Math 104 (2015), 157–164.
[2] A. Ben Ali Essaleh, A.M. Peralta, M.I. Ramírez, Weak-local derivationsand homomorphisms on C∗-algebras, Linear and Multilinear Algebra 64 (2016),169–186.
[3] M. Brešar, P. Šemrl, On local automorphisms and mappings that preserveidempotents, Studia Math 113 (1995), 101–108.
[4] S.R. Garcia, J.E. Tener, Unitary equivalence of a matrix to its transpose, J.Operator Theory 68 (2012), 179–203.
[5] Kh.D. Ikramov, A.K. Abdikalykov, On unitary transposable matrices of orderthree, Translation of Mat. Zametki 91 (2012), 563–570.
[6] R.V. Kadison, Isometries of operator algebras, Ann. of Math 54 (1951), 325–338.
[7] L. Molnár, Selected Preserver Problems on Algebraic Structures of Linear Opera-tors and on Function Spaces, Lecture Notes in Mathematics 1895, Springer-Verlag,Berlin Heidelberg, 2007.
[8] L. Molnár, Bilocal ∗-automorphisms of B(H), Arch. Math 102 (2014), 83–89.
[9] A.M. Peralta, A note on 2-local representations of C∗-algebras, Operators andMatrices 9 (2015), 343–358.
Preservers and Jordan isomorphisms
Martin BohataCzech Technical University in Prague, Prague, Czech Republic
The Jordan isomorphisms are essential ingredients of many maps preserving variousmathematical structures on operator algebras. Classical example is the description ofthe surjective linear isometries between C*-algebras due to Kadison [4]. The main partof this talk is devoted to nonlinear star order preserving maps between AW*-algebrasand it is based on the results published in the paper [1]. We show that continuous starorder isomorphisms between normal parts of AW*-algebras (without Type I2 directsummand) preserving multiples of the unit are given by a composition of continuousfunction calculus with a Jordan *-isomorphism. The proof is based on a deep Dye’stype theorem for AW*-algebras [3]. The assumption of preserving multiples of the unitis essential in the proof of our result. In fact, it is known that not every continuousstar order isomorphism on B(H) ⊕ B(H), where H is an infinite dimensional Hilbertspace, satisfies this assumption. However, our next result describes the structure ofevery continuous star order isomorphism on normal part of a von Neumann algebrathat is a direct sum of von Neumann factors of Type In, where n 6= 2. In this case,star order isomorphisms act more or less componentwise and so the whole problem isreduced to continuous star order isomorphisms on corresponding factors. This resultcan be regarded as an extension of the first result about nonlinear continuous star orderisomorphisms obtained by Dolinar and Molnár for B(H) [2].
This is a joint work with Jan Hamhalter from the Czech Technical University inPrague.
[1] M. Bohata and J. Hamhalter, Star order on operator and function algebrasand its nonliner preservers, Lin. Multilin. Alg. 64 (2016), 2519–2532.
[2] G. Dolinar and L. Molnár, Maps on quantum observables preserving the Gud-der order, Rep. Math. Phys. 60 (2007), 159–166.
[3] J. Hamhalter, Dye’s theorem and Gleason’s theorem for AW*-algebras, J. Math.Anal. Appl. 422 (2015), 1103–1115.
[4] R. V. Kadison, Isometries of operator algebras, Annals of Math. 54 (1951), 325–338.
Orthogonality in C∗-algebras and a model fornon-commutative vector lattices
Anil Kumar KarnNational Institute of Science Education and Research, Bhubaneswar, India
Let A be a C∗-algebra. A pair a, b ∈ A+ is said to be orthogonal algebraically,(we write a ⊥a b,) if ab = 0. It follows from the functional calculus that for eachself adjoint element a ∈ Asa, there exists a unique pair a+, a− ∈ A+ with a+ ⊥a a−such that a = a+ − a−. In this presentation, we obtain a geometric equivalent ofalgebraic orthogonality. This form is free from multiplication and therefore, is suitableto be introduced in ordered normed spaces. Extending this idea, we introduce a pairof notions in ordered spaces which generalize the notions “join” and “meet”. Followingwhich we propose a model of non-commutative vector lattices. The presentation willprimarily be based on the following papers and books.
[1] R. V. Kadison and J. R. Ringrose, Fundamentals of the theory ofoperator algebras,I, Academic Press, New York, 1983.
[2] A. K. Karn, A p-theory of ordered normed spaces, Positivity 14 (2010), 441–458.
[3] A. K. Karn, Orthogonality in `p-spaces and its bearing on ordered normed spaces,Positivity 18 (2014), 223–234.
[4] A. K. Karn, Orthogonality in C∗-algebras, Positivity 20(3) (2016), 607-620.
[5] G. K. Pedersen, C∗-algebras and their automorphism groups, Academic Press, Lon-don, 1979.
[6] M. Takesaki, Theory of Operator Algebras I, Springer, 1979.
On the structure of C0-semigroups ofholomorphic Carathéodory isometries
László L. StachóUniversity of Szeged, Szeged, Hungary
We extend Vesentini’s description for the infinitesimal generators of strongly con-tinuous one-parameter semigroups of holomorphic Carathéodory isometries of the unitball of a complex Hilbert space to the setting of reflexive Cartan factors. Our treat-ment is based on intensive use of joint fixed points along with Kaup type ideas withpartial vector fields of second degree. In particular we establish closed formulas for theHilbert space case in terms of spectral resolutions of skew self-adjoint dilations relatedto the Reich-Shoikhet non-linear infinitesimal generator. We also provide partial re-sults toward a Hille-Yosida type theory for holomorphic self-maps of bounded domainsin Banach spaces.
Maps on positive operators preserving Rényitype relative entropies and maximal
f -divergences
Marcell GaálUniversity of Szeged, Szeged, Hungary
In this talk we deal with two quantum relative entropy preserver problems on thecones of positive operators. The first one is related to a quantum Rényi relative entropylike quantity which plays an important role in classical-quantum channel decoding.The second one is connected to the so-called maximal f -divergences introduced by D.Petz and M. B. Ruskai who considered this quantity as a generalization of the usualBelavkin-Staszewski relative entropy.
Joint work with Gergő Nagy.
[1] M. Gaál, G. Nagy, Maps on positive operators preserving Rényi type relativeentropies and maximal f -divergences
The optimal version of Wigner’s theorem on theGrassmann space of all n-dimensional subspaces
György Pál GehérUniversity of Reading, UK
Let H be a complex Hilbert space and P1(H) stand for the set of all rank-one pro-jections (or equivalently, the projective space over H, i.e. the space of all 1-dimensionalsubspaces/lines of H). Wigner’s theorem – in its original form – states the following:
Theorem 1 (Wigner). If ϕ : P1(H) → P1(H) is a bijective map that preserves thetransition probability, i.e.
Trϕ(P )ϕ(Q) = TrPQ (P,Q ∈ P1(H)),
then we can always find a unitary or an antiunitary operator U : H → H such that
ϕ(P ) = UPU∗ (P ∈ P1(H)).
Moreover, U is unique up to a scalar factor.
Nowadays, this theorem is often refereed to as the very first non-linear preserverproblem in the history of the field. Its importance lies in the foundations of quantummechanics.
Recently, Peter Šemrl initiated the project of finding the so-called optimal versionsof known theorems in the field of preserver problems. By optimality we mean that wewould like to prove these theorems under assumptions that are as weak as possible.Peter Šemrl obtained the optimal versions of several classical theorems (such as Hua’sfundamental theorem of the geometry of matrices, Ludwig’s theorem from quantummechanics, etc.), and by now it is apparent that these problems are usually extremelyhard and they require the development of a substantially novel technique.
Let Pn(H) denote the set of all rank-n projections (or equivalently, the space ofall n-dimensional subspaces of H), and ](P,Q) be the system of principal anglesbetween ranP and ranQ. (Note that for rank-one projections TrPQ = cos2 ](P,Q)holds). In 2001, Lajos Molnár provided a natural generalisation of Wigner’s theoremon Grassmann spaces. Namely, he proved that every (not necessarily bijective) mapϕ : Pn(H)→ Pn(H) which preserve the system of principal angles are induced by linearand conjugatelinear isometries, except for the case of dimH = 2n and n > 1, when themap of orthocomplementation also comes into the picture.
In this talk we will present a natural joint extension of Wigner’s theorem andMolnár’s result in its optimal form. Namely, we characterise all transformationsϕ : Pn(H)→ Pn(H) which preserve the transition probability TrPQ. Note that since
TrPQ =∑
ϑ∈](P,Q)
cos2 ϑ,
therefore this result indeed generalises Molnár’s above described theorem.
[1] Gy.P. Gehér, Wigner’s theorem on Grassmann spaces, J. Funct. Anal., acceptedfor publication.
