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Marcel de Jeu

Publications and source records attributed to Marcel de Jeu.

At least 37 records · Page 2Linked to original sources
arXiv2017

Disintegration of positive isometric group representations on $\mathrm{L}^p$-spaces

Let $G$ be a Polish locally compact group acting on a Polish space $X$ with a $G$-invariant probability measure $μ$. We factorize the integral with respect to $μ$ in terms of the integrals with respect to the ergodic measures on $X$, and show that $\mathrm{L}^p(X,μ)$ ($1\leq p<\infty$) is $G$-equivariantly isometrically lattice isomorphic to an $\mathrm{L}^p$-direct integral of the spaces $\mathrm{L}^{p}(X,λ)$, where $λ$ ranges over the ergodic measures on $X$. This yields a disintegration of the canonical representation of $G$ as isometric lattice automorphisms of $\mathrm{L}^p(X,μ)$ as an $\mathrm{L}^p$-direct integral of order indecomposable representations. If $(X^\prime,μ^\prime)$ is a probability space, and, for some $1\leq q<\infty$, $G$ acts in a strongly continuous manner on $\mathrm{L}^q(X^\prime,μ^\prime)$ as isometric lattice automorphisms that leave the constants fixed, then $G$ acts on $\mathrm{L}^{p}(X^{\prime},μ^{\prime})$ in a similar fashion for all $1\leq p<\infty$. Moreover, there exists an alternative model in which these representations originate from a continuous action of $G$ on a compact Hausdorff space. If $(X^\prime,μ^\prime)$ is separable, the representation of $G$ on $\mathrm{L}^p(X^\prime,μ^\prime)$ can then be disintegrated into order indecomposable representations. The notions of $\mathrm{L}^p$-direct integrals of Banach spaces and representations that are developed extend those in the literature.

math.RT↗
arXiv2017

The closure of ideals of $\boldsymbol{\ell^1(Σ)}$ in its enveloping $\boldsymbol{\mathrm{C}^\ast}$-algebra

If $X$ is a compact Hausdorff space and $σ$ is a homeomorphism of $X$, then an involutive Banach algebra $\ell^1(Σ)$ of crossed product type is naturally associated with the topological dynamical system $Σ=(X,σ)$. We initiate the study of the relation between two-sided ideals of $\ell^1(Σ)$ and ${\mathrm C}^\ast(Σ)$, the enveloping $\mathrm{C}^\ast$-algebra ${\mathrm C}(X)\rtimes_σ\mathbb Z$ of $\ell^1(Σ)$. Among others, we prove that the closure of a proper two-sided ideal of $\ell^1(Σ)$ in ${\mathrm C}^\ast(Σ)$ is again a proper two-sided ideal of ${\mathrm C}^\ast(Σ)$.

math.OA↗
arXiv2016

Simultaneous power factorization in modules over Banach algebras

Let $A$ be a Banach algebra with a bounded left approximate identity $\{e_λ\}_{λ\inΛ}$, let $π$ be a continuous representation of $A$ on a Banach space $X$, and let $S$ be a non-empty subset of $X$ such that $\lim_λπ(e_λ)s=s$ uniformly on $S$. If $S$ is bounded, or if $\{e_λ\}_{λ\inΛ}$ is commutative, then we show that there exist $a\in A$ and maps $x_n: S\to X$ for $n\geq 1$ such that $s=π(a^n)x_n(s)$ for all $n\geq 1$ and $s\in S$. The properties of $a\in A$ and the maps $x_n$, as produced by the constructive proof, are studied in some detail. The results generalize previous simultaneous factorization theorems as well as Allan and Sinclair's power factorization theorem. In an ordered context, we also consider the existence of a positive factorization for a subset of the positive cone of an ordered Banach space that is a positive module over an ordered Banach algebra with a positive bounded left approximate identity. Such factorizations are not always possible. In certain cases, including those for positive modules over ordered Banach algebras of bounded functions, such positive factorizations exist, but the general picture is still unclear. Furthermore, simultaneous pointwise power factorizations for sets of bounded maps with values in a Banach module (such as sets of bounded convergent nets) are obtained. A worked example for the left regular representation of $\mathrm{C}_0({\mathbb R})$ and unbounded $S$ is included.

math.FA↗
arXiv2016

Nonsimplicity of certain universal $\mathrm{C}^\ast$-algebras

Given $n\geq 2$, $z_{ij}\in\mathbb{T}$ such that $z_{ij}=\overline z_{ji}$ for $1\leq i,j\leq n$ and $z_{ii}=1$ for $1\leq i\leq n$, and integers $p_1,...,p_n\geq 1$, we show that the universal $\mathrm{C}^*$-algebra generated by unitaries $u_1,...,u_n$ such that $u_i^{p_i}u_j^{p_j}=z_{ij}u_j^{p_j}u_i^{p_i}$ for $1\leq i,j \leq n$ is not simple if at least one exponent $p_i$ is at least two. We indicate how the method of proof by `working with various quotients' can be used to establish nonsimplicity of universal $\mathrm{C}^*$-algebras in other cases.

math.OA↗
arXiv2016

Algebraically irreducible representations and structure space of the Banach algebra associated with a topological dynamical system

If $X$ is a compact Hausdorff space and $σ$ is a homeomorphism of $X$, then a Banach algebra $\ell^1(Σ)$ of crossed product type is naturally associated with this topological dynamical system $Σ=(X,σ)$. If $X$ consists of one point, then $\ell^1(Σ)$ is the group algebra of the integers. We study the algebraically irreducible representations of $\ell^1(Σ)$ on complex vector spaces, its primitive ideals and its structure space. The finite dimensional algebraically irreducible representations are determined up to algebraic equivalence, and a sufficiently rich family of infinite dimensional algebraically irreducible representations is constructed to be able to conclude that $\ell^1(Σ)$ is semisimple. All primitive ideals of $\ell^1(Σ)$ are selfadjoint, and $\ell^1(Σ)$ is Hermitian if there are only periodic points in $X$. If $X$ is metrisable or all points are periodic, then all primitive ideals arise as in our construction. A part of the structure space of $\ell^1(Σ)$ is conditionally shown to be homeomorphic to the product of a space of finite orbits and $\mathbb T$. If $X$ is a finite set, then the structure space is the topological disjoint union of a number of tori, one for each orbit in $X$. If all points of $X$ have the same finite period, then it is the product of the orbit space $X/\mathbb Z$ and $\mathbb T$. For rational rotations of $\mathbb T$, this implies that the structure space is homeomorphic to $\mathbb T^2$.

math.OA↗
arXiv2015

Positive representations of $C_0(X)$. I

We introduce the notion of a positive spectral measure on a $σ$-algebra, taking values in the positive projections on a Banach lattice. Such a measure generates a bounded positive representation of the bounded measurable functions. If $X$ is a locally compact Hausdorff space, and $π$ is a positive representation of $C_0(X)$ on a KB-space, then $π$ is the restriction to $C_0(X)$ of such a representation generated by a unique regular positive spectral measure on the Borel $σ$-algebra of $X$. The relation between a positive representation of $C_0(X)$ on a Banach lattice and -- if it exists -- a generating positive spectral measure on the Borel $σ$-algebra is further investigated; here and elsewhere phenomena occur that are specific for the ordered context.

math.FA↗
arXiv2014

Rapid polynomial approximation in $L_2$-spaces with Freud weights on the real line

The weights $W_α(x)=\exp{(-|x|^α)}$ $(α>1)$ form a subclass of Freud weights on the real line. Primarily from a functional analytic angle, we investigate the subspace of $L_2(\mathbb R, W_α^2(x)\,dx)$ consisting of those elements that can be rapidly approximated by polynomials. This subspace has a natural Fréchet topology, in which it is isomorphic to the space of rapidly decreasing sequences. We show that it consists of smooth functions and obtain concrete results on its topology. For $α=2$ there is a complete and elementary description of this topological vector space in terms of the Schwartz functions.

math.CA↗
arXiv2014

A strong open mapping theorem for surjections from cones onto Banach spaces

We show that a continuous additive positively homogeneous map from a closed not necessarily proper cone in a Banach space onto a Banach space is an open map precisely when it is surjective. This generalization of the usual Open Mapping Theorem for Banach spaces is then combined with Michael's Selection Theorem to yield the existence of a continuous bounded positively homogeneous right inverse of such a surjective map; a strong version of the usual Open Mapping Theorem is then a special case. As another consequence, an improved version of the analogue of Andô's Theorem for an ordered Banach space is obtained for a Banach space that is, more generally than in Andô's Theorem, a sum of possibly uncountably many closed not necessarily proper cones. Applications are given for a (pre)-ordered Banach space and for various spaces of continuous functions taking values in such a Banach space or, more generally, taking values in an arbitrary Banach space that is a finite sum of closed not necessarily proper cones.

math.FA↗
arXiv2013

Crossed products of Banach algebras. III

In earlier work a crossed product of a Banach algebra was constructed from a Banach algebra dynamical system $(A,G,α)$ and a class $\mathcal{R}$ of continuous covariant representations, and its representations were determined. In this paper we adapt the theory to the ordered context. We construct a pre-ordered crossed product of a Banach algebra from a pre-ordered Banach algebra dynamical system $(A,G,α)$ and a given uniformly bounded class $\mathcal{R}$ of continuous covariant representations of $(A,G,α)$. If $A$ has a positive bounded approximate left identity and $\mathcal{R}$ consists of non-degenerate continuous covariant representations, we establish a bijection between the positive non-degenerate bounded representations of the pre-ordered crossed product on pre-ordered Banach spaces with closed cones and the positive non-degenerate $\mathcal{R}$-continuous covariant representations of $(A,G,α)$ on such spaces. Under mild conditions, we show that this pre-ordered crossed product is the essentially unique pre-ordered Banach algebra for which such a bijection exists. Finally, we study pre-ordered generalized Beurling algebras. We show that they are bipositively topologically isomorphic to pre-ordered crossed products of Banach algebras associated with pre-ordered Banach algebra dynamical systems, and hence the general theory allows us to describe their positive representations on pre-ordered Banach spaces with closed cones.

math.FA↗
arXiv2013

Crossed products of Banach algebras. II

In earlier work a crossed product of a Banach algebra was constructed from a Banach algebra dynamical system $(A,G,α)$ and a class $\mathcal{R}$ of continuous covariant representations, and its representations were determined. In this paper the theory is developed further. We consider the dependence of the crossed product on the class $\mathcal{R}$ and its essential uniqueness. Next we study generalized Beurling algebras: weighted Bochner spaces of $A$-valued functions on $G$ with a continuous multiplication. Though not Banach algebras in general, they are isomorphic to a crossed product of a Banach algebra, and the earlier work therefore predicts the structure of their representations. Classical results for the usual Beurling (Banach) algebras of scalar valued functions are then retrieved as special cases. We also show how, e.g., an anti-covariant pair of anti-representations of $A$ and $G$ can be viewed as a covariant pair for a related Banach algebra dynamical system, so that the earlier work becomes applicable to classes of such other pairs. After including material on the representations of the projective tensor product of Banach algebras, we combine this idea with the results already obtained and describe the two-sided modules over the generalized Beurling algebras, where again specializing to the scalars gives a classical result.

math.FA↗
arXiv2012

Noncommutative spectral synthesis for the involutive Banach algebra associated with a topological dynamical system

If X is a compact Hausdorff space, supplied with a homeomorphism, then a crossed product involutive Banach algebra is naturally associated with these data. If X consists of one point, then this algebra is the group algebra of the integers. In this paper, we study spectral synthesis for the closed ideals of this associated algebra in two versions, one modeled after C(X), and one modeled after the group algebra of the integers. We identify the closed ideals which are equal to (what is the analogue of) the kernel of their hull, and determine when this holds for all closed ideals, i.e., when spectral synthesis holds. In both models, this is the case precisely when the homeomorphism has no periodic points.

math.FA↗
arXiv2012

Compact groups of positive operators on Banach lattices

In this paper we study groups of positive operators on Banach lattices. If a certain factorization property holds for the elements of such a group, the group has a homomorphic image in the isometric positive operators which has the same invariant ideals as the original group. If the group is compact in the strong operator topology, it equals a group of isometric positive operators conjugated by a single central lattice automorphism, provided an additional technical assumption is satisfied, for which we again have only examples. We obtain a characterization of positive representations of a group with compact image in the strong operator topology, and use this for normalized symmetric Banach sequence spaces to prove an ordered version of the decomposition theorem for unitary representations of compact groups. Applications concerning spaces of continuous functions are also considered.

math.FA↗
arXiv2012

On the Banach $*$-algebra crossed product associated with a topological dynamical system

Given a topological dynamical system $Σ= (X, σ)$, where $X$ is a compact Hausdorff space and $σ$ a homeomorphism of $X$, we introduce the associated Banach $^*$-algebra crossed product $\ell^1 (Σ)$ and analyse its ideal structure. This algebra is the Banach algebra most naturally associated with the dynamical system, and it has a richer structure than its well studied $C^*$-envelope, as becomes evident from the possible existence of non-self-adjoint closed ideals. This paper initiates the study of these algebras and links their ideal structure to the topological dynamics. It is determined when exactly the algebra is simple, or prime, and when there exists a non-self-adjoint closed ideal. In addition, a structure theorem is obtained for the case when $X$ consists of one finite orbit, and the algebra is shown to be Hermitian if $X$ is finite. The key to these results lies in analysing the commutant of $C(X)$ in the algebra, which can be shown to be a maximal abelian subalgebra with non-zero intersection with each non-zero closed ideal.

math.OA↗
arXiv2012

Positive representations of finite groups in Riesz spaces

In this paper, which is part of a study of positive representations of locally compact groups in Banach lattices, we initiate the theory of positive representations of finite groups in Riesz spaces. If such a representation has only the zero subspace and possibly the space itself as invariant principal bands, then the space is Archimedean and finite dimensional. Various notions of irreducibility of a positive representation are introduced and, for a finite group acting positively in a space with sufficiently many projections, these are shown to be equal. We describe the finite dimensional positive Archimedean representations of a finite group and establish that, up to order equivalence, these are order direct sums, with unique multiplicities, of the order indecomposable positive representations naturally associated with transitive $G$-spaces. Character theory is shown to break down for positive representations. Induction and systems of imprimitivity are introduced in an ordered context, where the multiplicity formulation of Frobenius reciprocity turns out not to hold.

math.FA↗
arXiv2012

Maximal abelian subalgebras and projections in two Banach algebras associated with a topological dynamical system

If $Σ=(X,σ)$ is a topological dynamical system, where $X$ is a compact Hausdorff space and $σ$ is a homeomorphism of $X$, then a crossed product Banach $\sp{*}$-algebra $\ell^1(Σ)$ is naturally associated with these data. If $X$ consists of one point, then $\ell^1(Σ)$ is the group algebra of the integers. The commutant $C(X)'_1$ of $C(X)$ in $\ell^1(Σ)$ is known to be a maximal abelian subalgebra which has non-zero intersection with each non-zero closed ideal, and the same holds for the commutant $C(X)'_*$ of $C(X)$ in $C^*(Σ)$, the enveloping $C^*$-algebra of $\ell^1(Σ)$. This intersection property has proven to be a valuable tool in investigating these algebras. Motivated by this pivotal role, we study $C(X)'_1$ and $C(X)'_*$ in detail in the present paper. The maximal ideal space of $C(X)'_1$ is described explicitly, and is seen to coincide with its pure state space and to be a topological quotient of $X\times\mathbb{T}$. We show that $C(X)'_1$ is hermitian and semisimple, and that its enveloping $C^*$-algebra is $C(X)'_*$. Furthermore, we establish necessary and sufficient conditions for projections onto $C(X)'_1$ and $C(X)'_*$ to exist, and give explicit formulas for such projections, which we show to be unique. In the appendix, topological results for the periodic points of a homeomorphism of a locally compact Hausdorff space are given.

math.OA↗
arXiv2011

Connections between dynamical systems and crossed products of Banach algebras by $\mathbb{Z}$

Starting with a complex commutative semi-simple regular Banach algebra $A$ and an automorphism $σ$ of $A$, we form the crossed product of $A$ with the integers, where the latter act on $A$ via iterations of $σ$. The automorphism induces a topological dynamical system on the character space $Δ(A)$ of $A$ in a natural way. We prove an equivalence between the property that every non-zero ideal in the crossed product has non-zero intersection with the subalgebra $A$, maximal commutativity of $A$ in the crossed product, and density of the non-periodic points of the induced system on the character space. We also prove that every non-trivial ideal in the crossed product always intersects the commutant of $A$ non-trivially. Furthermore, under the assumption that $A$ is unital and such that $Δ(A)$ consists of infinitely many points, we show equivalence between simplicity of the crossed product and minimality of the induced system, and between primeness of the crossed product and topological transitivity of the system.

math.DS↗