For the special case where $\cA$ is a

. The proof of existence of such a correspondence relies on a subtle characterization of smoothing operators of unitary representations previously studied by Neeb, Salmasian, and Zellner.

For a broad class of Lie supergroups, which includes nilpotent as well as classical simple ones, we prove that the associated C∗

-algebra is CCR. In particular, we obtain the uniqueness of direct integral decomposition for unitary representations of these Lie supergroup}, author = {Neeb, Karl Hermann and Salmasian, Hadi}, doi = {10.2140/pjm.2016.282.213}, faupublication = {yes}, journal = {Pacific Journal of Mathematics}, pages = {213-232}, peerreviewed = {Yes}, title = {{Crossed} product algebras and direct integral decomposition for {Lie} supergroups}, url = {https://arxiv.org/abs/1506.01558}, volume = {282}, year = {2016} } @article{faucris.108990244, abstract = {We consider group actions α:G→Aut(A) of topological groups

expG(t(x1+x2))=limn→∞(expG(tnx1)expG(tnx2))n

holds uniformly on compact subsets of R. All locally exponential Lie groups have this property, but also groups of automorphisms of principal bundles over compact smooth manifolds. A key result of the present article is that, if *G* has the Trotter property, π:G→GL(V) is a continuous representation of *G* on a locally convex space, and v∈V is a vector such that ¯dπ(x)v:=ddt|t=0π(expG(tx))v exists for every x∈g, then the map g→V,x↦¯dπ(x)v is linear. Using this result we conclude that, for a representation of a locally exponential Fréchet–Lie group *G* on a metrizable locally convex space, the space of Ck-vectors coincides with the common domain of the *k*-fold products of the operators ¯dπ(x). For unitary representations on Hilbert spaces, the assumption of local exponentiality can be weakened to the Trotter property. As an application, we show that for smooth (resp., analytic) unitary representations of Fréchet–Lie supergroups (G,g) where *G* has the Trotter property, the common domain of the operators of g=g¯0⊕g¯1 can always be extended to the space of smooth (resp., analytic) vectors for *G*.},
author = {Neeb, Karl Hermann and Salmasian, Hadi},
doi = {10.1007/s00209-012-1142-5},
faupublication = {yes},
journal = {Mathematische Zeitschrift},
keywords = {Infinite dimensional Lie group; Representation; Differentiable vector; Smooth vector; Analytic vector; Derived representation; Lie supergroup; Trotter property},
pages = {419-451},
peerreviewed = {Yes},
title = {{Differentiable} vectors and unitary representations of {Fréchet}-{Lie} supergroups},
volume = {275},
year = {2013}
}
@article{faucris.267814545,
abstract = {In this note, we study in a finite dimensional Lie algebra g the set of all those elements x for which the closed convex hull of the adjoint orbit contains no affine lines; this contains in particular elements whose adjoint orbits generates a pointed convex cone C-x. Assuming that g is admissible, i.e., contains a generating invariant convex subset not containing affine lines, we obtain a natural characterization of such elements, also for non-reductive Lie algebras. Motivated by the concept of standard (Borchers) pairs in QFT, we also study pairs (x, h) of Lie algebra elements satisfying [h, x] = x for which Cx pointed. Given x, we show that such elements h can be constructed in such a way that ad h defines a 5-grading, and characterize the cases where we even get a 3-grading.},
author = {Neeb, Karl Hermann and Oeh, Daniel},
doi = {10.1007/s41980-021-00671-y},
faupublication = {yes},
journal = {Bulletin of the Iranian Mathematical Society},
note = {CRIS-Team WoS Importer:2022-01-07},
peerreviewed = {Yes},
title = {{Elements} in {Pointed} {Invariant} {Cones} in {Lie} {Algebras} and {Corresponding} {Affine} {Pairs}},
year = {2021}
}
@article{faucris.123952224,
author = {Neeb, Karl Hermann and Hofmann, Karl H.},
doi = {10.1007/BF01270691},
faupublication = {no},
journal = {Archiv der Mathematik},
pages = {134--137},
peerreviewed = {Yes},
title = {{Epimorphisms} of {C}*-algebras are surjective},
volume = {65},
year = {1995}
}
@article{faucris.123030864,
author = {Neeb, Karl Hermann and Penkov, Ivan},
doi = {10.4153/CMB-2011-018-2},
faupublication = {yes},
journal = {Canadian Mathematical Bulletin-Bulletin Canadien De Mathematiques},
pages = {519},
peerreviewed = {Yes},
title = {{Erratum}: {Cartan} subalgebras of g∞},
volume = {54},
year = {2011}
}
@article{faucris.210157824,
author = {Neeb, Karl Hermann and Grundling, Henrik},
doi = {10.1142/S0129055X17920027},
faupublication = {yes},
journal = {Reviews in Mathematical Physics},
peerreviewed = {Yes},
title = {{Erratum}: {Full} regularity for a {C}-algebra of the canonical commutation relations},
url = {https://arxiv.org/abs/math/0605413},
volume = {30},
year = {2018}
}
@article{faucris.108988484,
abstract = {Motivated by the theory of unitary representations of finite dimensional Lie supergroups, we describe those Lie superalgebras which have a faithful finite dimensional unitary representation. We call these Lie superalgebras unitary. This is achieved by describing the classification of real finite dimensional compact simple Lie superalgebras, and analyzing, in a rather elementary and direct way, the decomposition of reductive Lie superalgebras (g is a semisimple -module) over fields of characteristic zero into ideals.},
author = {Azam, Saeid and Neeb, Karl Hermann},
doi = {10.1016/j.jpaa.2015.02.024},
faupublication = {yes},
journal = {Journal of Pure and Applied Algebra},
pages = {4422 - 4440},
peerreviewed = {Yes},
title = {{Finite} dimensional compact and unitary {Lie} superalgebras},
url = {http://www.sciencedirect.com/science/article/pii/S0022404915000511},
volume = {219},
year = {2015}
}
@article{faucris.106772644,
abstract = {We investigate the finite-dimensional Lie groups whose points are separated by the continuous homomorphisms into groups of invertible elements of locally convex algebras with continuous inversion that satisfy an appropriate completeness condition. We find that these are precisely the linear Lie groups, that is, the Lie groups which can be faithfully represented as matrix groups. Our method relies on proving that certain finite-dimensional Lie subalgebras of algebras with continuous inversion commute modulo the Jacobson radical.},
author = {Beltita, Daniel and Neeb, Karl Hermann},
faupublication = {no},
journal = {Studia Mathematica},
pages = {249-262},
peerreviewed = {Yes},
title = {{Finite}-dimensional {Lie} subalgebras of algebras with continuous inversion},
volume = {185},
year = {2008}
}
@article{faucris.258188498,
abstract = {Let V be a standard subspace in the complex Hilbert space H and G be a finite dimensional Lie group of unitary and antiunitary operators on H containing the modular group (Delta(it)(V))(t is an element of R) of V and the corresponding modular conjugation J(V). We study the semigroup},
author = {Neeb, Karl Hermann},
doi = {10.1090/ert/566},
faupublication = {yes},
journal = {Representation Theory},
note = {CRIS-Team WoS Importer:2021-05-14},
pages = {300-343},
peerreviewed = {Yes},
title = {{FINITE} {DIMENSIONAL} {SEMIGROUPS} {OF} {UNITARY} {ENDOMORPHISMS} {OF} {STANDARD} {SUBSPACES}},
volume = {25},
year = {2021}
}
@article{faucris.117699824,
abstract = {Flux homomorphisms for closed vector-valued differential forms on infinite dimensional manifolds are defined. We extend the relation between the kernel of the flux for a closed 2-form ω and Kostant’s exact sequence associated to a principal bundle with curvature ω to the context of infinite-dimensional fiber and base space. We then use these results to construct central extensions of infinite dimensional Lie groups.},
author = {Neeb, Karl Hermann and Vizman, Cornelia},
doi = {10.1007/s00605-002-0001-6},
faupublication = {no},
journal = {Monatshefte für Mathematik},
keywords = {Flux homomorphism; symplectic manifold; holonomy group; symplectomorphism; central extension},
pages = {309–333},
peerreviewed = {Yes},
title = {{Flux} {Homomorphisms} and {Principal} {Bundles} over {Infinite} {Dimensional} {Manifolds}},
volume = {139},
year = {2003}
}
@article{faucris.308586710,
abstract = {We discuss the interplay between causal structures of symmetric spaces and geometric aspects of Algebraic Quantum Field Theory (AQFT). The central focus is the set of Euler elements in a Lie algebra, i.e., elements whose adjoint action defines a 3-grading. In the first half of this article we survey the classification of reductive causal symmetric spaces from the perspective of Euler elements. This point of view is motivated by recent applications in AQFT. In the second half we obtain several results that prepare the exploration of the deeper connection between the structure of causal symmetric spaces and AQFT. In particular, we explore the technique of strongly orthogonal roots and corresponding systems of sl

In this paper, we give for each affinisation g of a simple Hilbert-Lie algebra k an explicit isomorphism from g to one of the standard affinisations of k. The existence of such an isomorphism could also be derived from the classification of locally affine root systems, but for representation theoretic purposes it is crucial to obtain it explicitely as a deformation between two twists which is compatible with the root decompositions. We illustrate this by applying our isomorphism theorem to the study of positive energy highest weight representations of g.

In subsequent work, the present paper will be used to obtain a complete classification of the positive energy highest weight representations of affinisations of},
author = {Marquis, Timothée and Neeb, Karl Hermann},
doi = {10.4153/CJM-2016-003-x},
faupublication = {yes},
journal = {Canadian Journal of Mathematics-Journal Canadien De Mathematiques},
peerreviewed = {Yes},
title = {{Isomorphisms} of twisted {Hilbert} loop algebras},
url = {https://arxiv.org/abs/1508.07938},
year = {2017}
}
@incollection{faucris.235591547,
address = {Boston},
author = {Neeb, Karl Hermann},
booktitle = {Tsinghua Lectures in Mathematics},
editor = {Yat-Sun Poon, Shing-Tung Yau, Lizhen Ji},
faupublication = {yes},
isbn = {978-1571463722},
pages = {361-391},
peerreviewed = {unknown},
publisher = {Higher Education Press in China
International Press of Boston},
series = {Advanced Lectures in Mathematics ALM},
title = {{Kähler} {Geometry}, {Momentum} {Maps} and {Convex} {Sets}},
url = {https://arxiv.org/abs/1510.03289},
volume = {45},
year = {2018}
}
@article{faucris.107789704,
author = {Neeb, Karl Hermann},
faupublication = {no},
journal = {Forum Mathematicum},
pages = {349-384},
peerreviewed = {Yes},
title = {{Kähler} structures and convexity properties of coadjoint orbits},
volume = {7},
year = {1995}
}
@article{faucris.210395657,
author = {Neeb, Karl Hermann and Olafsson, Gestur},
doi = {10.2140/pjm.2019.299.117},
faupublication = {yes},
journal = {Pacific Journal of Mathematics},
pages = {117–169},
peerreviewed = {Yes},
title = {{KMS} conditions, standard real subspaces and reflection positivity on the circle group},
url = {https://arxiv.org/abs/1611.00080},
volume = {299},
year = {2019}
}
@article{faucris.121130504,
author = {Neeb, Karl Hermann},
faupublication = {no},
journal = {Archivum Mathematicum},
pages = {465--489},
peerreviewed = {unknown},
title = {{Lie} group extensions associated to projective modules of continuous inverse algebras},
volume = {44},
year = {2008}
}
@incollection{faucris.116129464,
author = {Neeb, Karl Hermann},
booktitle = {Developments and trends in infinite-dimensional Lie theory},
doi = {10.1007/978-0-8176-4741-4{\_}9},
editor = {K.-H. Neeb; A. Pianzola},
faupublication = {yes},
pages = {281--338},
peerreviewed = {unknown},
publisher = {Birkhäuser Boston, Inc., Boston, MA},
series = {Progr. Math.},
title = {{Lie} groups of bundle automorphisms and their extensions},
url = {https://arxiv.org/abs/0709.1063},
volume = {288},
year = {2011}
}
@article{faucris.124037364,
author = {Neeb, Karl Hermann and Wagemann, Friedrich},
faupublication = {no},
journal = {Geometriae Dedicata},
pages = {17 - 60},
peerreviewed = {Yes},
title = {{Lie} {Group} structures on groups of smooth and holomorphic maps},
volume = {134},
year = {2008}
}
@book{faucris.120907644,
author = {Hilgert, Joachim and Neeb, Karl Hermann},
faupublication = {no},
peerreviewed = {unknown},
publisher = {Vieweg Verlag},
series = {Memoirs of the American Mathematical Society},
title = {{Lie}-{Gruppen} und {Lie}-{Algebren}},
year = {1991}
}
@book{faucris.124000184,
author = {Hilgert, Joachim and Neeb, Karl Hermann},
faupublication = {no},
peerreviewed = {unknown},
publisher = {Springer-Verlag},
series = {Lecture Notes in Math.},
title = {{Lie} {Semigroups} and their {Applications}},
volume = {1552},
year = {1993}
}
@incollection{faucris.116129024,
author = {Neeb, Karl Hermann and Salmasian, Hadi},
booktitle = {Supersymmetry in mathematics and physics},
doi = {10.1007/978-3-642-21744-9{\_}10},
editor = {R. Fioresi, S. Ferrara, V.S. Varadarajan},
faupublication = {yes},
pages = {195--239},
peerreviewed = {unknown},
publisher = {Springer, Heidelberg},
series = {Lecture Notes in Math.},
title = {{Lie} supergroups, unitary representations, and invariant cones},
url = {https://arxiv.org/abs/1012.2809},
volume = {2027},
year = {2011}
}
@article{faucris.109009604,
author = {Grundling, Hendrik and Neeb, Karl Hermann},
faupublication = {no},
journal = {Letters in Mathematical Physics},
pages = {169 - 185},
peerreviewed = {Yes},
title = {{Localization} via {Automorphisms} of the {CARs}: {Local} gauge invariance},
volume = {93},
year = {2010}
}
@article{faucris.121322564,
author = {Neeb, Karl Hermann},
faupublication = {no},
journal = {Travaux mathématiques},
pages = {25-120},
peerreviewed = {Yes},
title = {{Locally} convex root graded {Lie} algebras},
volume = {14},
year = {2003}
}
@article{faucris.121593164,
author = {Neeb, Karl Hermann},
faupublication = {no},
journal = {Manuscripta Mathematica},
pages = {343-358},
peerreviewed = {Yes},
title = {{Locally} finite {Lie} algebras with unitary highes weight representations},
volume = {104},
year = {2001}
}
@article{faucris.116273344,
abstract = {If *P*→*X* is a topological principal *K*-bundle and a central extension of *K* by *Z*, then there is a natural obstruction class in sheaf cohomology whose vanishing is equivalent to the existence of a -bundle over *X* with . In this paper, we establish a link between homotopy theoretic data and the obstruction class *δ*_{1}(*P*) which in many cases can be used to calculate this class in explicit terms. Writing for the connecting maps in the long exact homotopy sequence, two of our main results can be formulated as follows. If *Z* is a quotient of a contractible group by the discrete group Γ, then the homomorphism *π*_{3}(*X*)→Γ induced by coincides with and if *Z* is discrete, then induces the homomorphism . We also obtain some information on obstruction classes defining trivial homomorphisms on homotopy groups},
author = {Neeb, Karl Hermann and Wagemann, Friedrich and Wockel, Christoph},
doi = {10.1112/plms/pds047},
faupublication = {yes},
journal = {Proceedings of the London Mathematical Society},
pages = {589 - 620},
peerreviewed = {Yes},
title = {{Making} lifting obstructions explicit},
volume = {106},
year = {2013}
}
@article{faucris.124006784,
author = {Neeb, Karl Hermann and Hilgert, Johannes},
doi = {10.1007/BF02573517},
faupublication = {no},
journal = {Semigroup Forum},
pages = {205--222},
peerreviewed = {Yes},
title = {{Maximality} of compression semigroups},
volume = {50},
year = {1995}
}
@inproceedings{faucris.121201564,
author = {Neeb, Karl Hermann},
booktitle = {Andrejewski-Tage},
faupublication = {yes},
peerreviewed = {unknown},
title = {{Moments} problems and analytic vectors in unitary representations of infinite-dimensional {Lie} groups},
venue = {Konstanz},
year = {2016}
}
@incollection{faucris.106707744,
abstract = {For a smooth projective unitary representation of a locally convex Lie group G, the projective space of smooth vectors is a locally convex Kaehler manifold. We show that the action of G on this space is weakly Hamiltonian, and lifts to a Hamiltonian action of the central U(1)-extension of G obtained from the projective representation. We identify the non-equivariance cocycles obtained from the weakly Hamiltonian action with those obtained from the projective representation, and give some integrality conditions on the image of the momentum ma},
author = {Janssens, Bas and Neeb, Karl Hermann},
booktitle = {Geometric Methods in Physics. XXXIV Workshop 2015},
doi = {10.1007/978-3-319-31756-4{\_}12},
editor = {Kielanowski P., Bieliavsky P., Odesskii A., Odzijewicz A., Schlichenmaier M., Voronov T.},
faupublication = {yes},
isbn = {978-3-319-31755-7},
keywords = {momentum map; projective representation},
pages = {115-127},
peerreviewed = {unknown},
publisher = {Springer-Birkhäuser},
series = {Trends in Mathematics},
title = {{Momentum} maps for smooth projective unitary representations},
url = {https://arxiv.org/abs/1510.08257},
volume = {II},
year = {2016}
}
@article{faucris.110159544,
author = {Neeb, Karl Hermann},
faupublication = {no},
journal = {Mathematische Annalen},
pages = {261 - 273},
peerreviewed = {Yes},
title = {{Monotone} functions on symmetric spaces},
volume = {291},
year = {1991}
}
@article{faucris.245479871,
abstract = {We prove several results asserting that the action of a Banach-Lie group on Hilbert spaces of holomorphic sections of a holomorphic Hilbert space bundle over a complex Banach manifold is multiplicity-free. These results require the existence of compatible anti-holomorphic bundle maps and certain multiplicity-freeness assumptions for stabilizer groups. For the group action on the base, the notion of an -weakly visible action (generalizing T. Koboyashi's visible actions) provides an effective way to express the assumptions in an economical fashion. In particular, we derive a version for group actions on homogeneous bundles for larger groups. We illustrate these general results by several examples related to operator groups and von Neumann algebras.},
author = {Miglioli, Martin and Neeb, Karl Hermann},
doi = {10.1093/imrn/rny160},
faupublication = {yes},
journal = {International Mathematics Research Notices},
note = {CRIS-Team WoS Importer:2020-11-20},
pages = {4852-4889},
peerreviewed = {Yes},
title = {{Multiplicity}-freeness of {Unitary} {Representations} in {Sections} of {Holomorphic} {Hilbert} {Bundles}},
volume = {2020},
year = {2020}
}
@article{faucris.257692201,
abstract = {Let G be a Lie group with Lie algebra g, h∈g an element for which the derivation ad h defines a 3-grading of g and τG an involutive automorphism of G inducing on g the involution e^{πiadh}. We consider antiunitary representations (U,H) of the Lie group Gτ=G⋊{idG,τG} for which the positive cone CU={x∈g:−i∂U(x)≥0} and h span g. To a real subspace E⊆H^{−∞} of distribution vectors invariant under exp(Rh) and an open subset O⊆G, we associate the real subspace HE(O)⊆H, generated by the subspaces U^{−∞}(φ)E, where φ∈Cc^{∞}(O,R) is a real-valued test function on O. Then HE(O) generates the complex Hilbert space HE(G):=HE(G)+iHE(G)‾ for every non-empty open subset O⊆G (Reeh–Schlieder property). For the real standard subspace V⊆H, for which JV=U(τG) is the modular conjugation and ΔV^{−it/2π}=U(expth) is the modular group, we obtain sufficient conditions to be of the form HE(S) for an open subsemigroup S⊆G. If g is semisimple with simple hermitian ideals of tube type, we verify these criteria and obtain nets of cyclic subspaces HE(O), O⊆G, satisfying the Bisognano–Wichmann property in a suitable sense. Our construction also yields such nets on simple Jordan space-times and compactly causal symmetric spaces of Cayley type. By second quantization, these nets lead to free quantum fields in the sense of Haag–Kastler on causal homogeneous spaces whose groups are generated by modular groups and conjugations.},
author = {Neeb, Karl Hermann and Ólafsson, Gestur},
doi = {10.1016/j.aim.2021.107715},
faupublication = {yes},
journal = {Advances in Mathematics},
keywords = {Covariant net; Lie group; Modular theory; Quantum fields; Standard subspace},
note = {CRIS-Team Scopus Importer:2021-05-07},
peerreviewed = {Yes},
title = {{Nets} of standard subspaces on {Lie} groups},
volume = {384},
year = {2021}
}
@article{faucris.123313124,
abstract = {In this article we study non-abelian extensions of a Lie group $G$ modeled on a locally convex space by a Lie group $N$. The equivalence classes of such extension are grouped into those corresponding to a class of so-called smooth outer actions $S$ of $G$ on $N$. If $S$ is given, we show that the corresponding set $\mathrm{Ext}{(G,N)}_{S}$ of extension classes is a principal homogeneous space of the locally smooth cohomology group ${H}_{ss}^{2}{(G,Z\left(N\right))}_{S}$. To each $S$ a locally smooth obstruction class $\chi \left(S\right)$ in a suitably defined cohomology group ${H}_{ss}^{3}{(G,Z\left(N\right))}_{S}$ is defined. It vanishes if and only if there is a corresponding extension of $G$ by $N$. A central point is that we reduce many problems concerning extensions by non-abelian groups to questions on extensions by abelian groups, which have been dealt with in previous work. An important tool is a Lie theoretic concept of a smooth crossed module $\alpha :H\to G$, which we view as a central extension of a normal subgroup of $G$.},
author = {Neeb, Karl Hermann},
faupublication = {no},
journal = {Annales de l'Institut Fourier},
pages = {209-271},
peerreviewed = {Yes},
title = {{Non}-abelian extensions of infinite-dimensional {Lie} groups},
volume = {57},
year = {2007}
}
@article{faucris.123129424,
abstract = {In this paper we extend and adapt several results on extensions of Lie algebras to topological Lie algebras over topological fields of characteristic zero. In particular we describe the set of equivalence classes of extensions of the Lie algebra by the Lie algebra a disjoint union of affine spaces with translation group H^2(\g,\z(\n)){\_}{[S]}, where [S] denotes the equivalence class of the continuous outer action S \: \g \to \der \n. We also discuss topological crossed modules and explain how they are related to extensions of Lie algebras by showing that any continuous outer action gives rise to a crossed module whose obstruction class in H^3(\g,\z(\n)){\_}S is the characteristic class of the corresponding crossed module. The correspondence between crossed modules and extensions further leads to a description of \n-extensions of \g in terms of certain \z(\n)-extensions of a Lie algebra which is an extension of \g by \n/\z(\n). We discuss several types of examples, describe applications to Lie algebras of vector fields on principal bundles, and in two appendices we describe the set of automorphisms and derivations of topological Lie algebra extensions.},
author = {Neeb, Karl Hermann},
doi = {10.1080/00927870500441973},
faupublication = {no},
journal = {Communications in Algebra},
keywords = {Curvature, Gerbe; Lie algebra extensions; Non-abelian cohomology; Principal bundle},
pages = {991-1041},
peerreviewed = {Yes},
title = {{Non}-abelian extensions of topological {Lie} algebras},
volume = {34},
year = {2006}
}
@article{faucris.121008624,
abstract = {A real seminormed involutive algebra is a real associative algebra \({\mathcal{A}}\) endowed with an involutive antiautomorphism * and a submultiplicative seminorm *p* with *p*(*a**) = *p*(*a*) for \({a\in \mathcal{A}}\). Then \({\mathtt{ball}(\mathcal{A}, p) := \{ a \in \mathcal{A} \: p(a) < 1\}}\) is an involutive subsemigroup. For the case where \({\mathcal{A}}\) is unital, our main result asserts that a function \({\varphi \: \mathtt{ball}(\mathcal{A},p) \to B(V)}\), *V* a Hilbert space, is completely positive (defined suitably) if and only if it is positive definite and analytic for any locally convex topology for which \({\mathtt{ball}(\mathcal{A},p)}\) is open. If \({\eta{\_}\mathcal{A} \: \mathcal{A} \to C^{*}(\mathcal{A},p)}\) is the enveloping C*-algebra of \({(\mathcal{A},p)}\) and \({e^{C^{*}(\mathcal{A}, p)}}\) is the *c*_{0}-direct sum of the symmetric tensor powers \({S^n(C^{*}(\mathcal{A},p))}\), then the above two properties are equivalent to the existence of a factorization \({\varphi = \Phi \circ \Gamma}\), where \({\Phi \: e^{C^{*}(\mathcal{A}, p)} \to B(V)}\) is linear completely positive and \({\Gamma(a) = \sum{\_}{n = 0}^\infty \eta{\_}\mathcal{A}(a)^{\otimes n}}\). We also obtain a suitable generalization to non-unital algebras. An important consequence of this result is a description of the unitary representations of the unitary group \({{\rm U}(\mathcal{A})}\) with bounded analytic extensions to \({\mathtt{ball}(\mathcal{A},p)}\) in terms of representations of the C*-algebra \({e^{C^{*}(\mathcal{A}, p)}}\).},
author = {Beltita, Daniel and Neeb, Karl Hermann},
doi = {10.1007/s00020-015-2244-3},
faupublication = {yes},
journal = {Integral Equations and Operator Theory},
keywords = {C*-algebra; *-semigroup; involutive algebra; completely positive map; unitary group},
pages = {517 - 562},
peerreviewed = {Yes},
title = {{Nonlinear} completely positive maps and dilation theory for real involutive algebras},
volume = {83},
year = {2015}
}
@article{faucris.124024164,
abstract = {Let K→X be a smooth Lie algebra bundle over a σ-compact manifold X whose typical fiber is the compact Lie algebra k. We give a complete description of the irreducible bounded (i.e., norm continuous) unitary representations of the Fréchet–Lie algebra Γ(K) of all smooth sections of K, and of the LF-Lie algebra Γc(K) of compactly supported smooth sections. For Γ(K), irreducible bounded unitary representations are finite tensor products of so-called evaluation representations, hence in particular finite dimensional. For Γc(K), bounded unitary irreducible (factor) representations are possibly infinite tensor products of evaluation representations, which reduces the classification problem to results of Glimm and Powers on irreducible (factor) representations of UHF C∗-algebras. The key part in our proof is the result that every irreducible bounded unitary representation of a Lie algebra of the form k⊗RAR, where AR is a unital real complete continuous inverse algebra, is a finite product of evaluation representations. On the group level, our results cover in particular the bounded unitary representations of the identity component Gau(P)0 of the group of smooth gauge transformations of a principal fiber bundle P→X with compact base and structure group, and the groups SUn(A)0 with A a complete involutive commutative continuous inverse algebra.},
author = {Janssens, Bas and Neeb, Karl Hermann},
doi = {10.1093/imrn/rnu231},
faupublication = {yes},
journal = {International Mathematics Research Notices},
pages = {9081 - 9137},
peerreviewed = {Yes},
title = {{Norm} continuous unitary representations of {Lie} algebras of smooth sections},
year = {2015}
}
@article{faucris.121751344,
author = {Neeb, Karl Hermann},
faupublication = {no},
journal = {Monatshefte für Mathematik},
pages = {303 - 321},
peerreviewed = {Yes},
title = {{Objects} dual to subsemigroups of groups},
volume = {122},
year = {1991}
}
@article{faucris.121629904,
abstract = {Let $G$ be a connected and simply connected Banach–Lie group. On the complex enveloping algebra of its Lie algebra $\U0001d524$ we define the concept of an analytic functional and show that every positive analytic functional $\lambda $ is integrable in the sense that it is of the form $\lambda \left(D\right)=\langle \mathtt{d}\pi \left(D\right)v,v\rangle $ for an analytic vector $v$ of a unitary representation of $G$. On the way to this result we derive criteria for the integrability of $*$-representations of infinite dimensional Lie algebras of unbounded operators to unitary group representations.For the matrix coefficient ${\pi}^{v,v}\left(g\right)=\langle \pi \left(g\right)v,v\rangle $ of a vector $v$ in a unitary representation of an analytic Fréchet–Lie group $G$ we show that $v$ is an analytic vector if and only if ${\pi}^{v,v}$ is analytic in an identity neighborhood. Combining this insight with the results on positive analytic functionals, we derive that every local positive definite analytic function on a simply connected Fréchet–BCH–Lie group $G$ extends to a global analytic function.},
author = {Neeb, Karl Hermann},
faupublication = {yes},
journal = {Annales de l'Institut Fourier},
pages = {1441 - 1476},
peerreviewed = {Yes},
title = {{On} analytic vectors for unitary representations of infinite dimensional {Lie} groups},
url = {http://eudml.org/doc/219711},
volume = {61},
year = {2011}
}
@article{faucris.121522764,
author = {Neeb, Karl Hermann and Salmasian, Hadi and Zellner, Christoph},
doi = {10.1215/21562261-3089019},
faupublication = {yes},
journal = {Kyoto Journal of Mathematics},
keywords = {Infinite-dimensional Lie groups; unitary representation; integrated representations; smooth vectors; Arveson spectral theory},
pages = {501 - 515},
peerreviewed = {No},
title = {{On} an invariance property of the space of smooth vectors},
volume = {55},
year = {2015}
}
@article{faucris.115052344,
author = {Neeb, Karl Hermann},
faupublication = {no},
journal = {Journal of Lie Theory},
pages = {293--300},
peerreviewed = {Yes},
title = {{On} a theorem of {S}. {Banach}},
volume = {7},
year = {1997}
}
@article{faucris.121282524,
author = {Neeb, Karl Hermann and Penkov, Ivan},
faupublication = {no},
journal = {Canadian Mathematical Bulletin-Bulletin Canadien De Mathematiques},
pages = {597-616},
peerreviewed = {Yes},
title = {{On} {Cartan} {Subalgebras} of gl∞},
volume = {46},
year = {2003}
}
@article{faucris.123716384,
author = {Neeb, Karl Hermann},
faupublication = {no},
journal = {Manuscripta Mathematica},
pages = {51-65},
peerreviewed = {Yes},
title = {{On} closedness and simple connectedness of adjoint and coadjoint orbits},
volume = {82},
year = {1994}
}
@article{faucris.121221144,
abstract = {The notion of a linear Coxeter system introduced by Vinberg generalizes the geometric representation of a Coxeter group. Our main theorem asserts that if *G * is called semibounded if the corresponding operators idπ(x) from the derived representations are uniformly bounded from above on some non-empty open subset of the Lie algebra g. Not every Lie group has non-trivial semibounded unitary representations, so that it becomes an important issue to decide when this is the case. In the present paper we describe a complete solution of this problem for the class of generalized oscillator groups, which are semidirect products of Heisenberg groups with a one-parameter group *γ*. For these groups it turns out that the existence of non-trivial semibounded representations is equivalent to the existence of the so-called semi-equicontinuous non-trivial coadjoint orbits, a purely geometric condition on the coadjoint action. This in turn can be expressed by a positivity condition on the Hamiltonian function corresponding to the infinitesimal generator *D* of *γ*. A central point of our investigations is that we make no assumption on the structure of the spectrum of *D*. In particular, *D* can be any skew-adjoint operator on a Hilbert space.},
author = {Neeb, Karl Hermann and Zellner, Christoph},
doi = {10.1016/j.difgeo.2012.10.010},
faupublication = {yes},
journal = {Differential Geometry and its Applications},
keywords = {Infinite dimensional Lie group; Semibounded representation; Oscillator group; Coadjoint orbit; Complex structure},
pages = {268-283},
peerreviewed = {Yes},
title = {{Oscillator} algebras with semi-equicontinuous coadjoint orbits},
volume = {31},
year = {2013}
}
@incollection{faucris.124007664,
author = {Hilgert, Joachim and Neeb, Karl Hermann},
booktitle = {Generalized symmetries in physics (Clausthal, 1993)},
editor = {H.-D. Doebner et al.},
faupublication = {no},
pages = {395--410},
peerreviewed = {unknown},
publisher = {World Sci. Publ., River Edge, NJ},
title = {{Poisson} {Lie} groups and non-linear convexity theorems},
year = {1994}
}
@article{faucris.108356204,
abstract = {This is a sequel to our paper on nonlinear completely positive maps and dilation theory for real involutive algebras, where we have reduced all classification problems to the passage from a *C **∗*-algebra *A *to its symmetric powers *S**n*(*A*), resp., to holomorphic representations of the multiplicative *∗*-semigroup (*A**, **·*). Here we study the correspondence between representations of *A *and of *S**n*(*A*) in detail. As *S**n*(*A*) is the fixed point algebra for the natural action of the symmetric group *S**n *on *A**⊗**n*, this is done by relating representations of *S**n*(*A*) to those of the crossed product *A**⊗**n *{\_} *S**n *in which it is a hereditary subalgebra. For *C**∗ *-algebras of type I, we obtain a rather complete description of the equivalence classes of the irreducible representations of *S**n*(*A*) and we relate this to the Schur–Weyl theory for *C**∗*-algebras. Finally we show that if *A **⊆* *B*(*H*) is a factor of type II or III, then its corresponding multiplicative representation on *H**⊗**n *is a factor representation of the same type, unlike the classical case *A *= *B*(*H*},
author = {Beltita, Daniel and Neeb, Karl Hermann},
doi = {10.1007/s00020-016-2335-9},
faupublication = {yes},
journal = {Integral Equations and Operator Theory},
keywords = {C∗-algebra; ∗-semigroup; completely positive map},
pages = {545-578},
peerreviewed = {Yes},
title = {{Polynomial} representations of {C}*-algebras and their applications},
volume = {86},
year = {2016}
}
@article{faucris.116190404,
author = {Hilgert, Joachim and Neeb, Karl Hermann},
doi = {10.1090/S0002-9947-99-02184-4},
faupublication = {no},
journal = {Transactions of the American Mathematical Society},
pages = {1345--1380},
peerreviewed = {Yes},
title = {{Positive} definite spherical functions on {Olshanski} domains},
volume = {352},
year = {2000}
}
@article{faucris.108970664,
author = {Neeb, Karl Hermann and Salmasian, Hadi},
doi = {10.1007/s00031-013-9228-7},
faupublication = {yes},
journal = {Transformation Groups},
pages = {803-844},
peerreviewed = {Yes},
title = {{Positive} definite superfunctions and unitary representations of {Lie} supergroups},
volume = {18},
year = {2013}
}
@article{faucris.122398364,
author = {Neeb, Karl Hermann},
faupublication = {yes},
journal = {Glasgow Mathematical Journal},
pages = {295 - 316},
peerreviewed = {Yes},
title = {{Positive} energy representations and continuity of projective representations for general topological groups},
volume = {56},
year = {2014}
}
@article{faucris.107266764,
author = {Marquis, Timothée and Neeb, Karl Hermann},
doi = {10.1093/imrn/rnv367},
faupublication = {yes},
journal = {International Mathematics Research Notices},
pages = {6689-6712},
peerreviewed = {Yes},
title = {{Positive} energy representations for locally finite split {Lie} algebras},
url = {https://arxiv.org/abs/1507.06077},
volume = {21},
year = {2016}
}
@article{faucris.210382332,
author = {Neeb, Karl Hermann and Marquis, Timothée},
doi = {10.2969/jmsj/06941485},
faupublication = {yes},
journal = {Journal of the Mathematical Society of Japan},
pages = {1485-1518},
peerreviewed = {Yes},
title = {{Positive} energy representations of double extensions of {Hilbert} loop algebras},
url = {https://arxiv.org/abs/1511.03980},
volume = {69},
year = {2017}
}
@inproceedings{faucris.106777924,
author = {Neeb, Karl Hermann},
booktitle = {Fields Workshop on Representation Theory and Analysis over Local Fields},
faupublication = {yes},
peerreviewed = {unknown},
title = {{Positive} energy representations of gauge groups},
venue = {University of Ottawa},
year = {2015}
}
@inproceedings{faucris.119932824,
author = {Neeb, Karl Hermann},
faupublication = {yes},
peerreviewed = {unknown},
title = {{Positive} energy representations of gauge groups},
venue = {Ecole Polytechnique Fed. Lausanne, Centre Interfac. Bernoulli},
year = {2015}
}
@inproceedings{faucris.123699224,
author = {Neeb, Karl Hermann},
booktitle = {Darmstadt-Erlangen-Freiburg Seminar on Conformal-Field-Theory},
faupublication = {yes},
peerreviewed = {unknown},
title = {{Positive} energy representations of gauge groups},
venue = {Freiburg},
year = {2016}
}
@article{faucris.117702244,
abstract = {We define *symmetric spaces* in arbitrary dimension and over arbitrary non-discrete topological fields *continuous quasi-inverse Jordan pairs* and *-triple systems*. This class of spaces, called *smooth generalized projective geometries*, generalizes the well-known (finite or infinite-dimensional) bounded symmetric domains as well as their ‘compact-like’ duals. An interpretation of such geometries as models of Quantum Mechanics is proposed, and particular attention is paid to geometries that might be considered as ‘standard models’ – they are associated to *associative continuous inverse algebras* and to *Jordan algebras of hermitian elements* in such an algebra.},
author = {Bertram, Wolfgang and Neeb, Karl Hermann},
doi = {10.1007/s10711-004-4197-6},
faupublication = {no},
journal = {Geometriae Dedicata},
keywords = {Jordan algebra; Jordan pair; Jordan triple; symmetric space; conformal completion; projective completion; Lie group},
pages = {75-115},
peerreviewed = {Yes},
title = {{Projective} {Completions} of {Jordan} {Pairs}, {Part} {II}: {Manifold} {Structures} and {Symmetric} {Spaces}},
volume = {112},
year = {2005}
}
@article{faucris.110582164,
abstract = {A geometric realization of the projective completion of the Jordan pair corresponding to a three-graded Lie algebra is given which permits to develop a geometric structure theory of the projective completion. This will be used in Part II of this work to define a manifold structure on the projective completion (in arbitrary dimension and over quite general base fields and -rings).},
author = {Bertram, Wolfgang and Neeb, Karl Hermann},
faupublication = {no},
journal = {Journal of Algebra},
pages = {474-519},
peerreviewed = {Yes},
title = {{Projective} completions of {Jordan} pairs. {Part} {I}: {The} generalized projective geometry of a {Lie} algebra},
volume = {277},
year = {2004}
}
@article{faucris.220870002,
abstract = {For an infinite-dimensional Lie group G modeled on a locally convex Lie algebra g, we prove that every smooth projective unitary representation of G corresponds to a smooth linear unitary representation of a Lie group extension G(#) of G. (The main point is the smooth structure on G(#)) For infinite-dimensional Lie groups G which are 1-connected, regular, and modeled on a barreled Lie algebra g, we characterize the unitary g-representations which integrate to G. Combining these results, we give a precise formulation of the correspondence between smooth projective unitary representations of G, smooth linear unitary representations of G(#), and the appropriate unitary representations of its Lie algebra g(#).},
author = {Janssens, Bas and Neeb, Karl Hermann},
doi = {10.1215/21562261-2018-0016},
faupublication = {yes},
journal = {Kyoto Journal of Mathematics},
note = {CRIS-Team WoS Importer:2019-06-18},
pages = {293-341},
peerreviewed = {Yes},
title = {{Projective} unitary representations of infinite-dimensional {Lie} groups},
volume = {59},
year = {2019}
}
@article{faucris.123916364,
author = {Hofmann, Karl H. and Neeb, Karl Hermann},
doi = {10.1017/S030500410800128X},
faupublication = {no},
journal = {Mathematical Proceedings of the Cambridge Philosophical Society},
pages = {351--378},
peerreviewed = {Yes},
title = {{Pro}-{Lie} groups which are infinite-dimensional {Lie} groups},
volume = {146},
year = {2009}
}
@inproceedings{faucris.119885744,
author = {Neeb, Karl Hermann},
booktitle = {ESI Programm "Infinite Dimensional Riemannian Geometry"},
faupublication = {yes},
peerreviewed = {unknown},
title = {{Realization} of reducible positive energy representations of the {Virasoro} group in spaces of holomorphic sections},
venue = {Wien},
year = {2015}
}
@article{faucris.210154272,
author = {Neeb, Karl Hermann and Jorgensen, Palle E.T. and Olafsson, Gestur},
doi = {10.3390/sym10060191},
faupublication = {yes},
journal = {Symmetry},
peerreviewed = {unknown},
title = {{Reflection} negative kernels and fractional {Brownian} motion},
url = {https://arxiv.org/abs/1805.02593},
volume = {10},
year = {2018}
}
@article{faucris.108873644,
abstract = {The concept of reflection positivity has its origins in the work of Osterwalder–Schrader on constructive quantum field theory. It is a fundamental tool to construct a relativistic quantum field theory as a unitary representation of the Poincaré group from a non-relativistic field theory as a representation of the euclidean motion group. This is the second article in a series on the mathematical foundations of reflection positivity. We develop the theory of reflection positive one-parameter groups and the dual theory of dilations of contractive hermitian semigroups. In particular, we connect reflection positivity with the outgoing realization of unitary one-parameter groups by Lax and Phillips. We further show that our results provide effective tools to construct reflection positive representations of general symmetric Lie groups, including the

A reflection positive Hilbert space is a triple (E,E_{+},θ), where E is a Hilbert space, *θ * a unitary involution and E_{+} a closed subspace on which the hermitian form 〈v,w〉_{θ}:=〈θv,w〉 is positive semidefinite. For a triple (G,τ,S), where *G* is a Lie group, *τ* an involutive automorphism of *G* and *S * a subsemigroup invariant under the involution s↦s^{♯}=τ(s)^{−1}, a unitary representation *π* of *G * on (E,E_{+},θ) is called reflection positive if θπ(g)θ=π(τ(g)) and π(S)E_{+}⊆E_{+}. This is the first in a series of papers in which we develop a new and systematic approach to reflection positive representations based on reflection positive distributions and reflection positive distribution vectors. This approach is most natural to obtain classification results, in particular in the abelian case. Among the tools we develop is a generalization of the Bochner–Schwartz Theorem to positive definite distributions on open convex cones. We further illustrate our techniques with a non-abelian example by constructing reflection positive distribution vectors for complementary series representations of the conformal group of the sphere S^{n}.

These are precisely the ones for which there exists an anti-unitary involution J commuting with Uc. This provides an interesting link with the modular data arising in Tomita{Takesaki theory.

Introducing the concept of a positive de nite function with values in the space of sesquilinear

forms, we further establish a link between KMS states and re ection positivity on the circle.},
author = {Neeb, Karl Hermann and Olafsson, Gestur},
doi = {10.1088/1742-6596/597/1/012004},
faupublication = {yes},
journal = {Journal of Physics : Conference Series},
peerreviewed = {unknown},
title = {{Reflection} positivity for the circle group},
volume = {597},
year = {2015}
}
@article{faucris.122241504,
abstract = {We study functions f : (a,b) ---> R on open intervals in R with respect to various kinds of positive and negative definiteness conditions. We say that f is positive definite if the kernel f((x + y)/2) is positive definite. We call f negative definite if, for every h > 0, the function e^{-hf} is positive definite. Our first main result is a L\'evy--Khintchine formula (an integral representation) for negative definite functions on arbitrary intervals. For (a,b) = (0,\infty) it generalizes classical results by Bernstein and Horn.

On a symmetric interval (-a,a), we call f reflection positive if it is positive definite and, in addition, the kernel f((x - y)/2) is positive definite. We likewise define reflection negative functions and obtain a L\'evy--Khintchine formula for reflection negative functions on all of R. Finally, we obtain a characterization of germs of reflection negative functions on 0-neighborhoods in},
author = {Jorgensen, Palle E.T. and Neeb, Karl Hermann and Olafsson, Gestur},
doi = {10.1007/s00233-017-9847-8},
faupublication = {yes},
journal = {Semigroup Forum},
pages = {31-48},
peerreviewed = {Yes},
title = {{Reflection} positivity on real intervals},
volume = {96},
year = {2018}
}
@article{faucris.231299153,
abstract = {In this article we specialize a construction of a reflection positive Hilbert space due to Dimock and Jaffe–Ritter to the sphere S^{n}. We determine the resulting Osterwalder–Schrader Hilbert space, a construction that can be viewed as the step from euclidean to relativistic quantum field theory. We show that this process gives rise to an irreducible unitary spherical representation of the orthochronous Lorentz group G^{c}= O 1 , n(R) ^{↑} and that the representations thus obtained are the irreducible unitary spherical representations of this group. A key tool is a certain complex domain Ξ , known as the crown of the hyperboloid, containing a half-sphere S+n and the hyperboloid H^{n} as totally real submanifolds. This domain provides a bridge between those two manifolds when we study unitary representations of G^{c} in spaces of holomorphic functions on Ξ. We connect this analysis with the boundary components which are the de Sitter space and a bundle over the space of future pointing lightlike vectors.},
author = {Neeb, Karl Hermann and Ólafsson, Gestur},
doi = {10.1007/s13324-019-00353-3},
faupublication = {yes},
journal = {Analysis and Mathematical Physics},
keywords = {Dissecting involutions; Positive definite kernels; Reflection positivity; Spherical representations; Symmetric spaces},
note = {CRIS-Team Scopus Importer:2020-01-10},
peerreviewed = {Yes},
title = {{Reflection} positivity on spheres},
volume = {10},
year = {2020}
}
@article{faucris.122579424,
abstract = {In this paper we study representations of the automorphism groups of classical infinite-dimensional tube domains. In particular we construct the *L*^{2}-realization of all unitary highest weight representations, including the vector-valued case. We also find a projective representation of the full identity component of the affine automorphism group of the Hilbert–Schmidt version of the tube domain with trivial cocycle on the subgroup corresponding to the trace class version, but non-trivial on the large group. Finally we show that the operator-valued measures corresponding to the vector valued highest weight representations have densities of a rather weak type with respect to Wishart distributions which makes it possible to determine their “supports.”},
author = {Neeb, Karl Hermann and Ørsted, Bent},
doi = {10.1006/jfan.2001.3884},
faupublication = {no},
journal = {Journal of Functional Analysis},
pages = {133-178},
peerreviewed = {Yes},
title = {{Representations} in {L2}-{Spaces} on {Infinite}-{Dimensional} {Symmetric} {Cones}},
volume = {190},
year = {2002}
}
@article{faucris.116231324,
author = {Neeb, Karl Hermann},
faupublication = {no},
journal = {Semigroup Forum},
pages = {197 - 218},
peerreviewed = {Yes},
title = {{Representations} of involutive semigroups},
volume = {48},
year = {1994}
}
@article{faucris.121588324,
author = {Neeb, Karl Hermann},
faupublication = {no},
journal = {Transformation Groups},
pages = {325-350},
peerreviewed = {Yes},
title = {{Representation} theory and convexity},
volume = {5},
year = {2000}
}
@book{faucris.107626464,
abstract = {From April 2009 until March 2016, the German Science Foundation supported

generously the Priority Program SPP 1388 in Representation Theory. The core

principles of the projects realized in the framework of the priority program have been

categorification and geometrization, this is also reflected by the contributions to this

volume.

Apart from the articles by former postdocs supported by the priority program, the

volume contains a number of invited research and survey articles, many of them are

extended versions of talks given at the last joint meeting of the priority program in

Bad Honnef in March 2015. This volume is covering current research topics from the

representation theory of finite groups, of algebraic groups, of Lie superalgebras, of

finite dimensional algebras and of infinite dimensional Lie groups.

Graduate students and researchers in mathematics interested in representation

theory will find this volume inspiring. It contains many stimulating contributions to

the development of this broad and extremely diverse subject.},
address = {Zürich},
editor = {Littelmann, Peter and Krause, Henning and Malle, Gunter and Neeb, Karl Hermann and Schweigert, Christoph},
faupublication = {yes},
isbn = {978-3-03719-171-2},
keywords = {algebraic groups; bounded and semibounded representations; categorification; character formulae; cluster algebras; Deligne–Lusztig theory; flat degenerations; geometrization; higher representation theory; highest weight categories; infinite dimensional Lie groups; local-global conjectures; special varieties; topological field theory},
peerreviewed = {automatic},
publisher = {European Mathematical Society},
series = {EMS Congress Reports},
title = {{Representation} {Theory} - {Current} {Trends} and {Perspectives}},
year = {2017}
}
@article{faucris.116326144,
abstract = {To each irreducible infinite dimensional representation of a *C**-algebra , we associate a collection of irreducible norm-continuous unitary representations of its unitary group , whose equivalence classes are parameterized by highest weights in the same way as the irreducible bounded unitary representations of the group are. These are precisely the representations arising in the decomposition of the tensor products under . We show that these representations can be realized by sections of holomorphic line bundles over homogeneous Kähler manifolds on which acts transitively and that the corresponding norm-closed momentum sets distinguish inequivalent representations of this type.},
author = {Beltita, Daniel and Neeb, Karl Hermann},
doi = {10.1002/mana.201100114},
faupublication = {yes},
journal = {Mathematische Nachrichten},
pages = {1170 - 1198},
peerreviewed = {Yes},
title = {{Schur}-{Weyl} {Theory} for {C}*-algebras},
volume = {285},
year = {2012}
}
@inproceedings{faucris.112223144,
author = {Neeb, Karl Hermann},
booktitle = {IVth School and Workshop on Lie Theory},
faupublication = {yes},
peerreviewed = {unknown},
title = {{Semiboundedness} in the representation theory of infinite-dimensional {Lie} groups},
venue = {Campinas},
year = {2015}
}
@article{faucris.119159524,
author = {Neeb, Karl Hermann},
doi = {10.1142/S1793744210000132},
faupublication = {yes},
journal = {Confluentes Mathematici},
pages = {37--134},
peerreviewed = {No},
title = {{Semibounded} representations and invariant cones in infinite dimensional {Lie} algebras},
volume = {2},
year = {2010}
}
@article{faucris.116127044,
author = {Neeb, Karl Hermann},
faupublication = {yes},
journal = {Travaux mathématiques},
pages = {29-109},
peerreviewed = {unknown},
title = {{Semibounded} representations of {Hermitian} {Lie} groups},
volume = {21},
year = {2012}
}
@inproceedings{faucris.107716004,
author = {Neeb, Karl Hermann},
faupublication = {yes},
peerreviewed = {unknown},
title = {{Semibounded} unitary representations and smoothing operators},
venue = {Institute of Mathematics of the Romanian Academy, Bukarest},
year = {2016}
}
@article{faucris.121158224,
abstract = {A unitary representation $\pi $ of a, possibly infinite dimensional, Lie group $G$ is called semibounded if the corresponding operators $i\mathtt{d}\pi \left(x\right)$ from the derived representation are uniformly bounded from above on some non-empty open subset of the Lie algebra $\U0001d524$ of $G$. We classify all irreducible semibounded representations of the groups ${\widehat{\mathcal{L}}}_{\phi}\left(K\right)$ which are double extensions of the twisted loop group ${\mathcal{L}}_{\phi}\left(K\right)$, where $K$ is a simple Hilbert–Lie group (in the sense that the scalar product on its Lie algebra is invariant) and $\phi $ is a finite order automorphism of $K$ which leads to one of the $7$ irreducible locally affine root systems with their canonical $\mathbb{Z}$-grading. To achieve this goal, we extend the method of holomorphic induction to certain classes of Fréchet–Lie groups and prove an infinitesimal characterization of analytic operator-valued positive definite functions on Fréchet–BCH–Lie groups.This is the first paper dealing with global aspects of Lie groups whose Lie algebra is an infinite rank analog of an affine Kac–Moody algebra. That positive energy representations are semibounded is a new insight, even for loops in compact Lie groups.},
author = {Neeb, Karl Hermann},
faupublication = {yes},
journal = {Annales de l'Institut Fourier},
pages = {1823 - 1892},
peerreviewed = {Yes},
title = {{Semibounded} unitary representations of double extensions of {Hilbert}-{Loop} groups},
url = {http://eudml.org/doc/275603},
volume = {64},
year = {2014}
}
@incollection{faucris.110089584,
author = {Neeb, Karl Hermann},
booktitle = {Infinite dimensional harmonic analysis {IV}},
doi = {10.1142/9789812832825{\_}0015},
editor = {J. Hilgert et al},
faupublication = {no},
pages = {209--222},
peerreviewed = {unknown},
publisher = {World Sci. Publ., Hackensack, NJ},
title = {{Semi}-bounded unitary representations of infinite-dimensional {Lie} groups},
url = {https://arxiv.org/abs/0804.3484},
year = {2009}
}
@article{faucris.123312024,
abstract = {We combine the theory of Coxeter groups, the covering theory of graphs introduced by Malnic, Nedela and Skoviera and the theory of reflections of graphs in order to obtain the following characterization of a Coxeter group: Let pi : Gamma -> ( v, D, i, -1) be a 1-covering of a monopole admitting semiedges only. The graph G is the Cayley graph of a Coxeter group if and only if pi is regular and any deck transformation in Delta(pi) that interchanges two neighboring vertices of Gamma acts as a reflection on Gamma.},
author = {Gramlich, Ralf and Hofmann, Georg W. and Neeb, Karl Hermann},
doi = {10.1090/S0002-9947-07-04040-8},
faupublication = {no},
journal = {Transactions of the American Mathematical Society},
pages = {3647-3668},
peerreviewed = {Yes},
title = {{Semi}-edges, reflections and {Coxeter} groups},
volume = {359},
year = {2007}
}
@article{faucris.282431288,
abstract = {Let V be a standard subspace in the complex Hilbert space H, and let U : G -> U(H) be a unitary representation of a finite-dimensional Lie group. We assume the existence of an element h is an element of g such that U(exp th) = delta(it)(V) is the modular group of V and that the modular involution J(V) normalizes U(G). We want to determine the semi-group S-V = {g is an element of G: U(g)V subset of V}. In previous work, we have seen that its infinitesimal generators span a Lie algebra on which ad h defines a 3-grading, and here we completely determine the semigroup SV under the assumption that ad h defines a 3-grading on g. Concretely, we show that the ad h-eigenspaces g(+/- 1) contain closed convex cones C-+/- , such that S-V = exp(C+)G(V) exp(C-) , where G(V) = {g is an element of G : U(g)V = V} is the stabilizer of V. To obtain this result, we compare several subsemigroups of G specified by the grading and the positive cone C-U of U. In particular, we show that the orbit O-V = U(G)V with the inclusion order is an ordered symmetric space covering the adjoint orbit O-h = Ad(G)h , endowed with the partial order defined by C-U.},
author = {Neeb, Karl Hermann},
doi = {10.1215/21562261-2022-0017},
faupublication = {yes},
journal = {Kyoto Journal of Mathematics},
note = {CRIS-Team WoS Importer:2022-09-30},
pages = {577-613},
peerreviewed = {Yes},
title = {{Semigroups} in 3-graded {Lie} groups and endomorphisms of standard subspaces},
volume = {62},
year = {2022}
}
@article{faucris.122029204,
author = {Neeb, Karl Hermann},
faupublication = {no},
journal = {Semigroup Forum},
pages = {33 - 43},
peerreviewed = {Yes},
title = {{Semigroups} in the universal covering group of {SL} (2)},
volume = {43},
year = {1991}
}
@article{faucris.210380912,
author = {Neeb, Karl Hermann and Salmasian, Hadi and Zellner, Christoph},
doi = {10.1142/S0129167X17500422},
faupublication = {yes},
journal = {International Journal of Mathematics},
peerreviewed = {Yes},
title = {{Smoothing} operators and {C}*-algebras for infinite dimensional {Lie} groups},
url = {https://arxiv.org/abs/1505.02659},
volume = {28},
year = {2017}
}
@inproceedings{faucris.107716664,
author = {Neeb, Karl Hermann},
booktitle = {Tagung "Harmonic Analysis on Lie Groups and Group Algebras of Locally Compact Groups"},
date = {2016-12-05/2016-12-09},
faupublication = {yes},
peerreviewed = {unknown},
title = {{Smoothing} operators and semibounded unitary representations},
venue = {Tsinghua Sanya International Mathematics Forum},
year = {2016}
}
@inproceedings{faucris.123319944,
author = {Neeb, Karl Hermann},
booktitle = {Seminar Sophus Lie},
faupublication = {yes},
peerreviewed = {unknown},
title = {{Smoothing} operators for unitary representations},
venue = {Bad Honnef},
year = {2015}
}
@inproceedings{faucris.123516624,
author = {Neeb, Karl Hermann},
faupublication = {yes},
peerreviewed = {unknown},
title = {{Smoothing} operators in representation theory},
venue = {Universität Metz},
year = {2015}
}
@article{faucris.121055044,
author = {Neeb, Karl Hermann},
faupublication = {no},
journal = {Glasgow Mathematical Journal},
pages = {469-477},
peerreviewed = {Yes},
title = {{Smooth} vectors for highest weight representations},
volume = {42},
year = {2000}
}
@incollection{faucris.116192604,
author = {Neeb, Karl Hermann},
booktitle = {Positivity in Lie theory: open problems},
faupublication = {no},
pages = {195--220},
peerreviewed = {unknown},
publisher = {de Gruyter, Berlin},
series = {de Gruyter Exp. Math.},
title = {{Some} open problems in representation theory related to complex geometry},
volume = {26},
year = {1998}
}
@article{faucris.122520024,
abstract = {In this article we compute the spherical functions which are associated to hyperbolically ordered symmetric spaces . These spaces are usually not semisimple; one prominent example is given by with the -dimensional Heisenberg group.},
author = {Krötz, Bernhardt and Neeb, Karl Hermann},
faupublication = {no},
journal = {Representation Theory},
pages = {43-92},
peerreviewed = {unknown},
title = {{Spherical} functions on mixed symmetric spaces},
volume = {5},
year = {2001}
}
@article{faucris.123992924,
author = {Hilgert, Joachim and Neeb, Karl Hermann},
doi = {10.1006/jfan.1996.0156},
faupublication = {no},
journal = {Journal of Functional Analysis},
pages = {446--493},
peerreviewed = {Yes},
title = {{Spherical} functions on {Olshanski} spaces},
volume = {142},
year = {1996}
}
@article{faucris.123056164,
author = {Krötz, Bernhard and Neeb, Karl Hermann and Olafsson, Gestur},
doi = {10.1090/S1088-4165-97-00035-6},
faupublication = {no},
journal = {Representation Theory},
pages = {424--461 (electronic)},
peerreviewed = {unknown},
title = {{Spherical} representations and mixed symmetric spaces},
volume = {1},
year = {1997}
}
@article{faucris.124106664,
author = {Krötz, Bernhardt and Neeb, Karl Hermann},
faupublication = {no},
journal = {Transactions of the American Mathematical Society},
pages = {1233-1264},
peerreviewed = {Yes},
title = {{Spherical} {Unitary} {Highest} {Weight} {Representations}},
volume = {354},
year = {2002}
}
@unpublished{faucris.318921577,
abstract = {Real standard subspaces of complex Hilbert spaces are long known to provide the right language for Tomita-Takesaki modular theory of von Neumann algebras. In recent years they have also become an object of prominent interest in mathematical quantum field theory (QFT) and unitary representation theory of Lie groups. This workshop brought together mathematicians and physicists working with standard subspaces, particularly in QFT (construction of QFT models, characterization of entropy, information-theoretic aspects), nets of standard subspaces on causal homogeneous spaces and aspects of reflection positivity and euclidean models related to standard subspaces and modular theory.

R^{h} for some endomorphism h∈ End (E) , diagonalizable with the eigenvalues 1 , 0 , - 1 (generalizing a Lorentz boost). This data specifies a wedge domain W(E, C, h) ⊆ E and one of our main results exhibits corresponding standard subspaces as being generated using test functions on these domains. We also investigate aspects of reflection positivity for the triple (E, C, e^{π}^{i}^{h}) and the support properties of distributions on E, arising as Fourier transforms of operator-valued measures defining the Hilbert spaces H. For the imaginary part of these distributions, we find similarities to the well known Huygens’ principle, relating to wedge duality in the Minkowski context. Interesting examples are the Riesz distributions associated to euclidean Jordan algebras.},
author = {Neeb, Karl Hermann and Ørsted, Bent and Ólafsson, Gestur},
doi = {10.1007/s00220-021-04144-5},
faupublication = {yes},
journal = {Communications in Mathematical Physics},
note = {CRIS-Team Scopus Importer:2021-08-06},
peerreviewed = {Yes},
title = {{Standard} {Subspaces} of {Hilbert} {Spaces} of {Holomorphic} {Functions} on {Tube} {Domains}},
year = {2021}
}
@book{faucris.120740884,
abstract = {This self-contained text is an excellent introduction to Lie groups and their actions on manifolds. The authors start with an elementary discussion of matrix groups, followed by chapters devoted to the basic structure and representation theory of finite dimensinal Lie algebras. They then turn to global issues, demonstrating the key issue of the interplay between differential geometry and Lie theory. Special emphasis is placed on homogeneous spaces and invariant geometric structures. The last section of the book is dedicated to the structure theory of Lie groups. Particularly, they focus on maximal compact subgroups, dense subgroups, complex structures, and linearity.

This text is accessible to a broad range of mathematicians and graduate students; it will be useful both as a graduate textbook and as a research reference.},
author = {Hilgert, Joachim and Neeb, Karl Hermann},
doi = {10.1007/978-0-387-84794-8},
faupublication = {yes},
isbn = {978-0-387-84793-1},
pages = {x+744},
peerreviewed = {unknown},
publisher = {Springer, New York},
series = {Springer Monographs in Mathematics},
title = {{Structure} and geometry of {Lie} groups},
year = {2012}
}
@article{faucris.122497144,
abstract = {We explain how the theorem of Banach discussed in [Ne97] can be generalized to target spaces which are not necessarily path connected. Moreover, we correct errors in the literature concerning the classification of separable unitary representations for the Banach Lie group U(H) [Pi88, Pi90].},
author = {Neeb, Karl Hermann and Pickrell, Doug},
faupublication = {no},
journal = {Journal of Lie Theory},
pages = {107-109},
peerreviewed = {Yes},
title = {{Supplements} to the papers entitled "{On} a {Theorem} of {S}. {Banach}" and "the separable representations of {U}({H})"},
volume = {10},
year = {2000}
}
@article{faucris.240772246,
abstract = {An involutive diffeomorphism σ of a connected smooth manifold M is called dissecting if the complement of its fixed point set is not connected. Dissecting involutions on a complete Riemannian manifold are closely related to constructive quantum field theory through the work of Dimock and Jaffe/Ritter on the construction of reflection positive Hilbert spaces. In this article we classify all pairs (M, σ), where M is an irreducible connected symmetric space, not necessarily Riemannian, and σ is a dissecting involutive automorphism. In particular, we show that the only irreducible, connected and simply connected Riemannian symmetric spaces with dissecting isometric involutions are S^{n} and ℍ^{n}, where the corresponding fixed point spaces are S^{n}^{−}^{1} and ℍ^{n − 1}, respectively.},
author = {Neeb, Karl Hermann and Ólafsson, G.},
doi = {10.1007/s00031-020-09595-z},
faupublication = {yes},
journal = {Transformation Groups},
note = {CRIS-Team Scopus Importer:2020-07-24},
peerreviewed = {Yes},
title = {{SYMMETRIC} {SPACES} {WITH} {DISSECTING} {INVOLUTIONS}},
year = {2020}
}
@article{faucris.124003264,
author = {Hilgert, Joachim and Neeb, Karl Hermann},
faupublication = {no},
journal = {Compositio Mathematica},
pages = {129 - 180},
peerreviewed = {Yes},
title = {{Symplectic} convexity theorems and coadjoint orbits},
volume = {94},
year = {1994}
}
@incollection{faucris.120910284,
author = {Hilgert, Joachim and Neeb, Karl Hermann},
booktitle = {Semigroups in algebra, geometry and analysis (Oberwolfach, 1993)},
editor = {K. H. Hofmann et al.},
faupublication = {no},
pages = {201--240},
peerreviewed = {unknown},
publisher = {de Gruyter, Berlin},
series = {de Gruyter Expositions in Mathematics},
title = {{Symplectic} convexity theorems, {Lie} semigroups, and unitary representations},
volume = {20},
year = {1995}
}
@article{faucris.120923044,
author = {Neeb, Karl Hermann},
faupublication = {no},
journal = {Journal of Lie Theory},
pages = {1-46},
peerreviewed = {Yes},
title = {{The} classification of {Lie} algebras with invariant cones},
volume = {4},
year = {1994}
}
@article{faucris.119170304,
author = {Hofmann, Karl H. and Neeb, Karl Hermann},
faupublication = {no},
journal = {Journal of Group Theory},
pages = {555 - 559},
peerreviewed = {Yes},
title = {{The} compact generation of closed subgroups of locally compact groups},
volume = {12},
year = {2009}
}
@article{faucris.123059464,
author = {Neeb, Karl Hermann},
faupublication = {no},
journal = {Transactions of the American Mathematical Society},
pages = {653 - 677},
peerreviewed = {Yes},
title = {{The} dualty between subsemigroups of {Lie} groups and monotone functions},
volume = {329},
year = {1992}
}
@article{faucris.122100704,
author = {Hilgert, Joachim and Neeb, Karl Hermann and Ørsted, Bent},
faupublication = {no},
journal = {Journal of Lie Theory},
pages = {185-235},
peerreviewed = {Yes},
title = {{The} geometry of nilpotent orbits of convex type in hermitian {Lie} algebras},
volume = {4},
year = {1994}
}
@article{faucris.122216644,
author = {Neeb, Karl Hermann and Wagemann, Friedrich},
doi = {10.4153/CJM-2008-038-6},
faupublication = {no},
journal = {Canadian Journal of Mathematics-Journal Canadien De Mathematiques},
pages = {892--922},
peerreviewed = {Yes},
title = {{The} second cohomology of current algebras of general {Lie} algebras},
volume = {60},
year = {2008}
}
@article{faucris.117702024,
abstract = {We identify the universal differential module Ω^{1}(*A*) for the Fréchet algebra *A* of holomorphic functions on a complex Stein manifold *X*, and more generally on a Riemannian domain *R* over *X* and for the algebra of germs of holomorphic functions on a compact subset *K*⊂ℂ^{ n }. It turns out to be isomorphic to the Fréchet space of holomorphic 1-forms on *X*, resp. *R*, resp. to the space Ω^{1}(*K*) of germs of holomorphic 1-forms in *K*. This determines the center of the universal central extension of the Lie algebra 𝒪(*R*,𝔨 of holomorphic maps from *R* to a finite-dimensional simple complex Lie algebra 𝔨.},
author = {Neeb, Karl Hermann and Wagemann, Friedrich},
doi = {10.1007/s00229-003-0409-x},
faupublication = {no},
journal = {Manuscripta Mathematica},
pages = {441-458},
peerreviewed = {Yes},
title = {{The} universal central extension of the holomorphic current algebra},
volume = {112},
year = {2003}
}
@incollection{faucris.123913064,
author = {Neeb, Karl Hermann},
booktitle = {Quantum affine algebras, extended affine Lie algebras, and their applications},
doi = {10.1090/conm/506/09943},
editor = {Y. Gao et at},
faupublication = {yes},
pages = {227--262},
peerreviewed = {unknown},
publisher = {Amer. Math. Soc., Providence, RI},
series = {Contemp. Math.},
title = {{Unitary} highest weight modules of locally affine {Lie} algebras},
url = {https://arxiv.org/abs/0904.0134},
volume = {506},
year = {2010}
}
@article{faucris.123989184,
author = {Neeb, Karl Hermann and Ørsted, Bent},
doi = {10.1006/jfan.1997.3233},
faupublication = {no},
journal = {Journal of Functional Analysis},
pages = {263--300},
peerreviewed = {Yes},
title = {{Unitary} highest weight representations in {Hilbert} spaces of holomorphic functions on infinite-dimensional domains},
volume = {156},
year = {1998}
}
@article{faucris.120900824,
author = {Neeb, Karl Hermann and Hilgert, Joachim and Ørsted, Bent},
doi = {10.1007/BF00116520},
faupublication = {no},
journal = {Acta Applicandae Mathematicae},
pages = {151--184},
peerreviewed = {Yes},
title = {{Unitary} highest weight representations via the orbit method {I}. {The} scalar case},
volume = {44},
year = {1996}
}
@incollection{faucris.116262344,
abstract = {In this paper we review and streamline some results of Kirillov, Olshanski and Pickrell on unitary representations of the unitary group U(H)" id="MathJax-Element-1-Frame" role="presentation" style="position: relative;" tabindex="0">U(H) of a real, complex or quaternionic separable Hilbert space and the subgroup U∞(H)" id="MathJax-Element-2-Frame" role="presentation" style="position: relative;" tabindex="0">U∞(H), consisting of those unitary operators *g* for which *g* −**1** is compact. The Kirillov–Olshanski theorem on the continuous unitary representations of the identity component U∞(H)0" id="MathJax-Element-3-Frame" role="presentation" style="position: relative;" tabindex="0">U∞(H)0 asserts that they are direct sums of irreducible ones which can be realized in finite tensor products of a suitable complex Hilbert space. This is proved and generalized to inseparable spaces. These results are carried over to the full unitary group by Pickrell’s theorem, asserting that the separable unitary representations of U(H)" id="MathJax-Element-4-Frame" role="presentation" style="position: relative;" tabindex="0">U(H), for a separable Hilbert space H" id="MathJax-Element-5-Frame" role="presentation" style="position: relative;" tabindex="0">H, are uniquely determined by their restriction to U∞(H)0" id="MathJax-Element-6-Frame" role="presentation" style="position: relative;" tabindex="0">U∞(H)0. For the 10 classical infinite rank symmetric pairs (*G*, *K*) of non-unitary type, such as (GL(H),U(H))" id="MathJax-Element-7-Frame" role="presentation" style="position: relative;" tabindex="0">(GL(H),U(H)), we also show that all separable unitary representations are trivia},
author = {Neeb, Karl Hermann},
booktitle = {Lie theory workshops},
doi = {10.1007/978-3-319-09934-7{\_}8},
editor = {G. Mason, I. Penkov, J. Wolf},
faupublication = {yes},
isbn = {978-3-319-09933-0},
keywords = {Unitary group; Unitary representation; Restricted group; Schur modules; Bounded representation; Separable representation},
pages = {197 - 243},
peerreviewed = {unknown},
publisher = {Springer},
series = {Developmenst in Mathematics},
title = {{Unitary} representations of {Unitary} {Groups}},
url = {https://arxiv.org/abs/1308.1500},
volume = {37},
year = {2014}
}
@article{faucris.235592563,
author = {Neeb, Karl Hermann and Yousofzadeh, Malihe},
doi = {10.1016/j.jpaa.2019.106205},
faupublication = {yes},
journal = {Journal of Pure and Applied Algebra},
peerreviewed = {Yes},
title = {{Universal} central extensions of current {Lie} superalgebras},
url = {https://arxiv.org/abs/1707.00282},
volume = {224},
year = {2020}
}
@article{faucris.122520244,
abstract = {We call a central *Z*-extension of a group *G* weakly universal for an Abelian group *A* if the correspondence assigning to a homomorphism *Z*→*A* the corresponding *A*-extension yields a bijection of extension classes. The main problem discussed in this paper is the existence of central Lie group extensions of a connected Lie group *G* which is weakly universal for all Abelian Lie groups whose identity components are quotients of vector spaces by discrete subgroups. We call these Abelian groups regular. In the first part of the paper we deal with the corresponding question in the context of topological, Fréchet, and Banach–Lie algebras, and in the second part we turn to the groups. Here we start with a discussion of the weak universality for discrete Abelian groups and then turn to regular Lie groups *A*. The main results are a Recognition and a Characterization Theorem for weakly universal central extensions.},
author = {Neeb, Karl Hermann},
doi = {10.1023/A:1019743224737},
faupublication = {no},
journal = {Acta Applicandae Mathematicae},
keywords = {central extension; Lie algebra; infinite-dimensional Lie group; universal central extension; Lie algebra cohomology; period map},
pages = {175-219},
peerreviewed = {Yes},
title = {{Universal} {Central} {Extensions} of {Lie} {Groups}},
volume = {73},
year = {2002}
}
@article{faucris.122519364,
abstract = {*Let V be a Euclidean Jordan algebra*, Г*the associated symmetric cone and G be the identity component of the linear automorphism group of* Г.*In this paper we associate to a certain class of spherical representations (ρ, ɛ) of G certain ɛ-valued Riesz distributions generalizing the classical scalar valued Riesz distributions on V. Our construction is motivated by the analytic theory of unitary highest weight representations where it permits to study certain holomorphic families of operator valued Riesz distributions whose positive definiteness corresponds to the unitarity of a representation of the automorphism group of the associated tube domain* Г +*iV*.},
author = {Hilgert, Joachim and Neeb, Karl Hermann},
doi = {10.1007/BF02921953},
faupublication = {no},
journal = {Journal of Geometric Analysis},
keywords = {Riesz distribution; Jordan algebra; symmetric cone; highest weight representations; gamma function; tube domain},
pages = {43-75},
peerreviewed = {Yes},
title = {{Vector} valued {Riesz} distributions on {Euclidian} {Jordan} algebras},
volume = {11},
year = {2001}
}
@article{faucris.123669744,
author = {Neeb, Karl Hermann},
doi = {10.1006/jabr.1996.0015},
faupublication = {no},
journal = {Journal of Algebra},
pages = {331--361},
peerreviewed = {Yes},
title = {{Weakly} exponential {Lie} groups},
volume = {179},
year = {1996}
}
@incollection{faucris.116261244,
abstract = {We introduce a notion of a weak Poisson structure on a manifold *M* modeled on a locally convex space. This is done by specifying a Poisson bracket on a subalgebra A⊆C∞(M)" id="MathJax-Element-1-Frame" role="presentation" style="position: relative;" tabindex="0">A⊆C∞(M) which has to satisfy a non-degeneracy condition (the differentials of elements of A" id="MathJax-Element-2-Frame" role="presentation" style="position: relative;" tabindex="0">A separate tangent vectors) and we postulate the existence of smooth Hamiltonian vector fields. Motivated by applications to Hamiltonian actions, we focus on affine Poisson spaces which include in particular the linear and affine Poisson structures on duals of locally convex Lie algebras. As an interesting byproduct of our approach, we can associate to an invariant symmetric bilinear form *κ* on a Lie algebra g" id="MathJax-Element-3-Frame" role="presentation" style="position: relative;" tabindex="0">g and a *κ*-skew-symmetric derivation *D* a weak affine Poisson structure on g" id="MathJax-Element-4-Frame" role="presentation" style="position: relative;" tabindex="0">g itself. This leads naturally to a concept of a Hamiltonian *G*-action on a weak Poisson manifold with a g" id="MathJax-Element-5-Frame" role="presentation" style="position: relative;" tabindex="0">g

-valued momentum map and hence to a generalization of quasi-hamiltonian group actions.

}, author = {Neeb, Karl Hermann and Thiemann, Thomas and Sahlmann, Hanno}, booktitle = {Springer Proceedings in Mathematics & Statistics}, editor = {V. Dobrev}, faupublication = {yes}, isbn = {978-4-431-55284-0}, pages = {105-136}, peerreviewed = {unknown}, publisher = {Springer Japan}, title = {{Weak} {Poisson} structures on infinite dimensional manifolds and hamiltonian actions}, url = {https://arxiv.org/abs/1402.6818}, volume = {111}, year = {2015} } @article{faucris.276877966, abstract = {Motivated by constructions in Algebraic Quantum Field Theory we introduce wedge domains in compactly causal symmetric spaces M=G/H, which includes in particular anti-de Sitter space in all dimensions and its coverings. Our wedge domains generalize Rindler wedges in Minkowski space. The key geometric structure we use is the modular flow on M defined by an Euler element in the Lie algebra of G. Our main geometric result asserts that three seemingly different characterizations of these domains coincide: the positivity domain of the modular vector field, the domain specified by a KMS-like analytic extension condition for the modular flow, and the domain specified by a polar decomposition in terms of certain cones. In the second half of the article we show that our wedge domains share important properties with wedge domains in Minkowski space. If G is semisimple, there exist unitary representations (U, H) of G and isotone covariant nets of real subspaces H(O) subset of H, defined for any open subset O subset of M, which assign to connected components of the wedge domains a standard subspace whose modular group corresponds to the modular f low on M. This corresponds to the Bisognano-Wichmann property in Quantum Field Theory. We also show that the set of G-translates of the connected components of the wedge domain provides a geometric realization of the abstract wedge space introduced by the first author and V. Morinelli.}, author = {Neeb, Karl Hermann and Olafsson, Gestur}, doi = {10.1093/imrn/rnac131}, faupublication = {yes}, journal = {International Mathematics Research Notices}, note = {CRIS-Team WoS Importer:2022-06-17}, peerreviewed = {Yes}, title = {{Wedge} {Domains} in {Compactly} {Causal} {Symmetric} {Spaces}}, year = {2022} } @article{faucris.116326584, abstract = {It is a basic fact in infinite-dimensional Lie theory that the unit group G(A) of a continuous inverse algebra A is a Lie group. We describe criteria ensuring that the Lie group G(A) is regular in Milnor's sense. Notably, G(A) is regular if A is Mackey-complete and locally m-convex.}, author = {Glöckner, Helge and Neeb, Karl Hermann}, faupublication = {yes}, journal = {Studia Mathematica}, pages = {95 - 109}, peerreviewed = {Yes}, title = {{When} unit groups of continuous inverse algebras are regular {Lie} groups}, url = {http://eudml.org/doc/285662}, volume = {211}, year = {2012} } @article{faucris.116224284, author = {Neeb, Karl Hermann and Hilgert, Joachim}, doi = {10.1006/jfan.1995.1101}, faupublication = {no}, journal = {Journal of Functional Analysis}, pages = {86--118}, peerreviewed = {Yes}, title = {{Wiener}-{Hopf} operators on ordered homogeneous spaces {I}}, volume = {132}, year = {1995} }