arXiv subjects

Karl-Hermann Neeb

Publications and source records attributed to Karl-Hermann Neeb.

At least 91 records · Page 5Linked to original sources

arXiv2012

Crossed products of C*-algebras for singular actions

We consider group actions of topological groups on C*-algebras of the types which occur in many physics models. These are singular actions in the sense that they need not be strongly continuous, or the group need not be locally compact. We develop a "crossed product host" in analogy to the usual crossed product for strongly continuous actions of locally compact groups, in the sense that its representation theory is in a natural bijection with the covariant representation theory of the action. We prove a uniqueness theorem for crossed product hosts, and analyze existence conditions. We also present a number of examples where a crossed product host exists, but the usual crossed product does not. For actions where a crossed product host does not exist, we obtain a "maximal" invariant subalgebra for which a crossed product host exists. We further study the case of a discontinuous action of a locally compact group in detail.

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arXiv2012

Positive energy representations and continuity of projective representations for general topological groups

Let $G$ and $T$ be topological groups, $α: T \to \Aut(G)$ a homomorphism defining a continuous action of $T$ on $G$ and $G^\sharp := G \rtimes_αT$ the corresponding semidirect product group. In this paper we address several issues concerning irreducible continuous unitary representations $(π^\sharp, \cH)$ of $G^\sharp$ whose restriction to $G$ remains irreducible. First we prove that, for $T = \R$, this is the case for any irreducible positive energy representation of $G^\sharp$, i.e., for which the one-parameter group $U_t := π^\sharp(\1,t)$ has non-negative spectrum. The passage from irreducible unitary representations of $G$ to representations of $G^\sharp$ requires that certain projective unitary representations are continuous. To facilitate this verification, we derive various effective criteria for the continuity of projective unitary representations. Based on results on Borchers for $W^*$-dynamical systems, we also derive a characterization of the continuous positive definite functions on $G$ that extend to a $G^\sharp$.

math.RT↗

arXiv2012

Positive definite superfunctions and unitary representations of Lie supergroups

For a broad class of Frechet-Lie supergroups we prove that there exists a correspondence between positive definite smooth superfunctions and matrix coefficients of unitary representations. We also give a characterization of linear functionals on the corresponding universal enveloping algebra which correspond to matrix coefficients of analytic unitary representations.

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arXiv2012

Semibounded Unitary Representations of Double Extensions of Hilbert--Loop Groups

A unitary representation of a, possibly infinite dimensional, Lie group $G$ is called semibounded if the corresponding operators $i\ddπ(x)$ from the derived representation are uniformly bounded from above on some non-empty open subset of the Lie algebra $\g$ of $G$. We classify all irreducible semibounded representations of the groups $\hat\cL_ϕ(K)$ which are double extensions of the twisted loop group $\cL_ϕ(K)$, where $K$ is a simple Hilbert--Lie group (in the sense that the scalar product on its Lie algebra is invariant) and $ϕ$ is a finite order automorphism of $K$ which leads to one of the 7 irreducible locally affine root systems with their canonical $\Z$-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.

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arXiv2012

Making Lifting Obstructions Explicit

If P \to X is a topological principal K-bundle and \hat K a central extension of K by Z, then there is a natural obstruction class δ_1(P) in \check H^2(X,\uline Z) in sheaf cohomology whose vanishing is equivalent to the existence of a \hat K-bundle \hat P over X with P \cong \hat P/Z. 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 \partial_d^P \: π_d(X) \to π_{d-1}(K) 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) \to Γinduced by δ_1(P) \in \check H^2(X,\uline Z) \cong H^3_{\rm sing}(X,Γ) coincides with \partial_2^{\hat K} \circ \partial_3^P and if Z is discrete, then δ_1(P) \in \check H^2(X,\uline Z) induces the homomorphism -\partial_1^{\hat K} \circ \partial_2^P \: π_2(X) \to Z. We also obtain some information on obstruction classes defining trivial homomorphisms on homotopy groups.

math.AT↗

arXiv2012

On convex hulls of orbits of Coxeter groups and Weyl groups

The notion of a linear Coxeter system introduced by Vinberg generalizes the geometric representation of a Coxeter group. Our main theorem asserts that if $v$ is an element of the Tits cone of a linear Coxeter system and $\cW$ is the corresponding Coxeter group, then $\cW v \subeq v - C_v,$ where $C_v$ is the convex cone generated by the coroots $\check α$, for which $α(v) > 0$. This implies that the convex hull of $\cW v$ is completely determined by the image of $v$ under the reflections in $\cW$. We also apply an analogous result for convex hulls of $\cW$-orbits in the dual space, although this action need not correspond to a linear Coxeter system. Motivated by the applications in representation theory, we further extend these results to Weyl group orbits of locally finite and locally affine root systems. In the locally affine case, we also derive some applications on minimizing linear functionals on Weyl group orbits.

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arXiv2012

When unit groups of continuous inverse algebras are regular Lie groups

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.

math.FA↗

arXiv2011

Oscillator algebras with semi-equicontinuous coadjoint orbits

A unitary representation of a, possibly infinite dimensional, Lie group G is called semi-bounded 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 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.

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arXiv2011

Categories of unitary representations of Banach-Lie supergroups and restriction functors

We prove that the categories of smooth and analytic unitary representations of Banach--Lie supergroups are well-behaved under restriction functors, in the sense that the restriction of a representation to an integral subsupergroup is well-defined. We also prove that the category of analytic representations is isomorphic to a subcategory of the category of smooth representations. These facts are needed as a crucial first step to a rigorous treatment of the analytic theory of unitary representations of Banach--Lie supergroups. They extend the known results for finite dimensional Lie supergroups. In the infinite dimensional case the proofs require several new ideas. As an application, we give an analytic realization of the oscillator representation of the restricted orthosymplectic Banach--Lie supergroup.

math.RT↗

arXiv2011

Semibounded representations of hermitian Lie groups

A unitary representation of a, possibly infinite dimensional, Lie group $G$ is called semibounded if the corresponding operators $i\ddπ(x)$ from the derived representation are uniformly bounded from above on some non-empty open subset of the Lie algebra $\g$ of $G$. A hermitian Lie group is a central extension of the identity component of the automorphism group of a hermitian Hilbert symmetric space. In the present paper we classify the irreducible semibounded unitary representations of hermitian Lie groups corresponding to infinite dimensional irreducible symmetric spaces. These groups come in three essentially different types: those corresponding to negatively curved spaces (the symmetric Hilbert domains), the unitary groups acting on the duals of Hilbert domains, such as the restricted Graßmannian, and the motion groups of flat spaces.

math.RT↗

arXiv2011

Analytic extension techniques for unitary representations of Banach-Lie groups

Let $(G,θ)$ be a Banach--Lie group with involutive automorphism $θ$, $\g = \fh \oplus \fq$ be the $θ$-eigenspaces in the Lie algebra $\g$ of $G$, and $H = (G^θ)_0$ be the identity component of its group of fixed points. An Olshanski semigroup is a semigroup $S \subeq G$ of the form $S = H \exp(W)$, where $W$ is an open $\Ad(H)$-invariant convex cone in $\fq$ and the polar map $H \times W \to S, (h,x) \mapsto h \exp x$ is a diffeomorphism. Any such semigroup carries an involution * satisfying $(h\exp x)^* = (\exp x) h^{-1}$. Our central result, generalizing the Lüscher--Mack Theorem for finite dimensional groups, asserts that any locally bounded *-representation $π\: S \to B(\cH)$ with a dense set of smooth vectors defines by "analytic continuation" a unitary representation of the simply connected Lie group $G_c$ with Lie algebra $ \g_c = \fh + i \fq$. We also characterize those unitary representations of $G_c$ obtained by this construction. With similar methods, we further show that semibounded unitary representations extend to holomorphic representations of complex Olshanski semigroups

math.RT↗

arXiv2011

Schur--Weyl Theory for $C^*$-algebras

To each irreducible infinite dimensional representation $(π,\cH)$ of a $C^*$-algebra $\cA$, we associate a collection of irreducible norm-continuous unitary representations $π_λ^\cA$ of its unitary group $\U(\cA)$, whose equivalence classes are parameterized by highest weights in the same way as the irreducible bounded unitary representations of the group $\U_\infty(\cH) = \U(\cH) \cap (\1 + K(\cH))$ are. These are precisely the representations arising in the decomposition of the tensor products $\cH^{\otimes n} \otimes (\cH^*)^{\otimes m}$ under $\U(\cA)$. We show that these representations can be realized by sections of holomorphic line bundles over homogeneous Kähler manifolds on which $\U(\cA)$ acts transitively and that the corresponding norm-closed momentum sets $I_{π_λ^\cA}^{\bf n} \subeq \fu(\cA)'$ distinguish inequivalent representations of this type.

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arXiv2010

Lie supergroups, unitary representations, and invariant cones

The main result of this article is an application of the theory of invariant convex cones of Lie algebras to the study of unitary representations of Lie supergroups. It also includes an exposition of recent results of the second author on the classification of irreducible unitary representations of nilpotent Lie supergroups using the method of coadjoint orbits.

math.RT↗

arXiv2010

On Differentiable Vectors for Representations of Infinite Dimensional Lie Groups

In this paper we develop two types of tools to deal with differentiability properties of vectors in continuous representations $π\: G \to \GL(V)$ of an infinite dimensional Lie group $G$ on a locally convex space $V$. The first class of results concerns the space $V^\infty$ of smooth vectors. If $G$ is a Banach--Lie group, we define a topology on the space $V^\infty$ of smooth vectors for which the action of $G$ on this space is smooth. If $V$ is a Banach space, then $V^\infty$ is a Fréchet space. This applies in particular to $C^*$-dynamical systems $(\cA,G, α)$, where $G$ is a Banach--Lie group. For unitary representations we show that a vector $v$ is smooth if the corresponding positive definite function $\la π(g)v,v\ra$ is smooth. The second class of results concerns criteria for $C^k$-vectors in terms of operators of the derived representation for a Banach--Lie group $G$ acting on a Banach space $V$. In particular, we provide for each $k \in \N$ examples of continuous unitary representations for which the space of $C^{k+1}$-vectors is trivial and the space of $C^k$-vectors is dense.

math.RT↗

arXiv2010

Holomorphic Realization of Unitary Representations of Banach--Lie Groups

In this paper we explore the method of holomorphic induction for unitary representations of Banach--Lie groups. First we show that the classification of complex bundle structures on homogeneous Banach bundles over complex homogeneous spaces of real Banach--Lie groups formally looks as in the finite dimensional case. We then turn to a suitable concept of holomorphic unitary induction and show that this process preserves commutants. In particular, holomorphic induction from irreducible representations leads to irreducible ones. Finally we develop criteria to identify representations as holomorphically induced ones and apply these to the class of so-called positive energy representations. All this is based on extensions of Arveson's concept of spectral subspaces to representations on Fréchet spaces, in particular on spaces of smooth vectors.

math.RT↗

arXiv2010

On Analytic Vectors for Unitary Representations of Infinite Dimensional Lie Groups

Let $G$ be a 1-connected Banach-Lie group or, more generally, a BCH--Lie group. On the complex enveloping algebra $U_\C(\g)$ of its Lie algebra $\g$ we define the concept of an analytic functional and show that every positive analytic functional $λ$ is integrable in the sense that it is of the form $λ(D) = \la \ddπ(D)v, v\ra$ 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 $π^{v,v}(g) = \la π(g)v,v\ra$ 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 $π^{v,v}$ 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 1-connected Fréchet--BCH--Lie group $G$ extends to a global analytic function.

math.RT↗

arXiv2010

Infinite Tensor Products of C_0(R): Towards a Group Algebra for R^\infty

The construction of an infinite tensor product of the C*-algebra C_0(R) is not obvious, because it is nonunital, and it has no nonzero projection. Based on a choice of an approximate identity, we construct here an infinite tensor product of C_0(R), denoted L_V. We use this to construct (partial) group algebras for the full continuous unitary representation theory of the group R^(N) = the infinite sequences with real entries, of which only finitely many entries are nonzero. We obtain an interpretation of the Bochner-Minlos theorem in R^(N) as the pure state space decomposition of the partial group algebras which generate L_V. We analyze the representation theory of L_V, and show that there is a bijection between a natural set of representations of L_V and the continuous unitary representations of R^(N), but that there is an extra part which essentially consists of the representation theory of a multiplicative semigroup which depends on the initial choice of approximate identity.

math.OA↗

arXiv2010

Localization via Automorphisms of the CARs. Local gauge invariance

The classical matter fields are sections of a vector bundle E with base manifold M. The space L^2(E) of square integrable matter fields w.r.t. a locally Lebesgue measure on M, has an important module action of C_b^\infty(M) on it. This module action defines restriction maps and encodes the local structure of the classical fields. For the quantum context, we show that this module action defines an automorphism group on the algebra A, of the canonical anticommutation relations on L^2(E), with which we can perform the analogous localization. That is, the net structure of the CAR, A, w.r.t. appropriate subsets of M can be obtained simply from the invariance algebras of appropriate subgroups. We also identify the quantum analogues of restriction maps. As a corollary, we prove a well-known "folk theorem," that the algebra A contains only trivial gauge invariant observables w.r.t. a local gauge group acting on E.

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