Left Invariant Haar Measure Research Articles

Outer invariance entropy for discrete-time linear systems on Lie groups

We introduce discrete-time linear control systems on connected Lie groups and present an upper bound for the outer invariance entropy of admissible pairs (K, Q). If the stable subgroup of the uncontrolled system is closed and K has positive measure for a left invariant Haar measure, the upper bound coincides with the outer invariance entropy.

Amenability and harmonic $L^p$-functions on hypergroups

Let $K$ be a locally compact hypergroup with a left invariant Haar measure. We show that the Liouville property and amenability are equivalent for $K$ when it is second countable. Suppose that $\sigma$ is a non-degenerate probability measure on $K$, we show that there is no non-trivial $\sigma$-harmonic function which is continuous and vanishing at infinity. Using this, we prove that the space $H_\sigma^p(K)$ of all $\sigma$-harmonic $L^p$-functions, is trivial for all $1\leq p<\infty$. Further, it is shown that $H_\sigma^\infty(K)$ contains only constant functions if and only if it is a subalgebra of $L^\infty(K)$. In the case where $\sigma$ is adapted and $K$ is compact, we show that $H_\sigma^p(K)={\mathbb C}1$ for all $1\leq p\leq\infty$.

Volumes for $${\mathrm{SL}}_N({\mathbb {R}})$$ SL N ( R ) , the Selberg Integral and Random Lattices

There is a natural left and right invariant Haar measure associated with the matrix groups GL $${}_N(\mathbb {R})$$ and SL $${}_N(\mathbb {R})$$ due to Siegel. For the associated volume to be finite it is necessary to truncate the groups by imposing a bound on the norm, or in the case of SL $${}_N(\mathbb {R})$$ , by restricting to a fundamental domain. We compute the asymptotic volumes associated with the Haar measure for GL $${}_N(\mathbb {R})$$ and SL $${}_N(\mathbb {R})$$ matrices in the case that the singular values lie between $$R_1$$ and $$1/R_2$$ in the former, and that the 2-norm, or alternatively the Frobenius norm, is bounded by R in the latter. By a result of Duke, Rudnick and Sarnak, such asymptotic formulas in the case of SL $${}_N(\mathbb {R})$$ imply an asymptotic counting formula for matrices in SL $${}_N(\mathbb {Z})$$ . We discuss too the sampling of SL $${}_N(\mathbb {R})$$ matrices from the truncated sets. By then using lattice reduction to a fundamental domain, we obtain histograms approximating the probability density functions of the lengths and pairwise angles of shortest length bases vectors in the case $$N=2$$ and 3, or equivalently of shortest linearly independent vectors in the corresponding random lattice. In the case $$N=2$$ these distributions are evaluated explicitly.

Equipment of Sets with Cardinality of the Continuum by Structures of Polish Groups with Haar Measures

It is introduced a certain approach for equipment of sets with cardinality of the continuum by structures of Polish groups with two-sided (left or right) invariant Haar measures. By using this approach we answer positively Maleki’s certain question (2012) what are the real k-dimensional manifolds with at least two different Lie group structures that have the same Haar measure. It is demonstrated that for each diffused Borel probability measure defined in a Polish space (G;ρ;Bρ(G)) without isolated points there exist a metric ρ1and a group operation ⊙ in G such that Bρ(G) = Bρ1(G) and (G;ρ1;Bρ1(G);⊙) stands a compact Polish group with a two-sided (left or right) invariant Haar measure μ , where Bρ(G) and Bρ1(G) denote Borel σ-algebras of subsets of G generated by metrics ρ and ρ1, respectively. Similar approach is used for a construction of locally compact non-compact or non-locally compact Polish groups equipped with two-sided (left or right) invariant quasi-finite Borel measures.

Induced representations of the affine group and intertwining operators: I. Analytical approach

We analyze the construction and origin of unitary operators describing the structure of the space of continuous wavelet transforms inside the space of all square-integrable functions on the affine group with respect to the left-invariant Haar measure from the viewpoint of induced representations of . We show that these operators are, in fact, intertwining operators among pairs of induced representations of the affine group . A characterization of the space of wavelet transforms using the Cauchy–Riemann-type equations is given.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Coherent states: mathematical and physical aspects’.

On Toeplitz-type Operators Related to Wavelets

Let G be the “ax + b”-group with the left invariant Haar measure dν and ψ be a fixed real-valued admissible wavelet on \(L_{2}({\mathbb{R}})\). The structure of the space of Calderon (wavelet) transforms \(W_{\psi} (L_{2}({\mathbb{R}}))\) inside \(L_{2}(G, d\nu)\) is described. Using this result some representations, properties and the Wick calculus of the Calderon-Toeplitz operators T α acting on \(W_{\psi}(L_{2}({\mathbb{R}}))\) whose symbols a = a(ζ) depend on \(v = \Im\zeta\) for \(\zeta \in G\) are investigated.

A note on wavelet subspaces

The wavelet subspaces of the space of square integrable functions on the affine group with respect to the left invariant Haar measure dν are studied using the techniques from Vasilevski (Integral Equ. Operator Theory 33:471-488, 1999) with respect to wavelets whose Fourier transforms are related to Laguerre polynomials. The orthogonal projections onto each of these wavelet subspaces are described and explicit forms of reproducing kernels are established. Isomorphisms between wavelet subspaces are given.

On the structure of the space of wavelet transforms

Let G be the “ a x + b ”-group with the left invariant Haar measure d ν and ψ be a fixed real-valued admissible wavelet on L 2 ( R ) . The complete decomposition of L 2 ( G , d ν ) onto the space of wavelet transforms W ψ ( L 2 ( R ) ) is obtained after identifying the group G with the upper half-plane Π in C . To cite this article: O. Hutník, C. R. Acad. Sci. Paris, Ser. I 346 (2008).

Estimation of multivariate normal covariance and precision matrices in a star-shape model with missing data

In this paper, we study the problem of estimating the covariance matrix Σ and the precision matrix Ω (the inverse of the covariance matrix) in a star-shape model with missing data. By considering a type of Cholesky decomposition of the precision matrix Ω = Ψ ′ Ψ , where Ψ is a lower triangular matrix with positive diagonal elements, we get the MLEs of the covariance matrix and precision matrix and prove that both of them are biased. Based on the MLEs, unbiased estimators of the covariance matrix and precision matrix are obtained. A special group G , which is a subgroup of the group consisting all lower triangular matrices, is introduced. By choosing the left invariant Haar measure on G as a prior, we obtain the closed forms of the best equivariant estimates of Ω under any of the Stein loss, the entropy loss, and the symmetric loss. Consequently, the MLE of the precision matrix (covariance matrix) is inadmissible under any of the above three loss functions. Some simulation results are given for illustration.

Sub-Laplacians of Holomorphic Lp -Type on Exponential Solvable Groups

Let L denote a right-invariant sub-Laplacian on an exponential, hence solvable Lie group G, endowed with a left-invariant Haar measure. Depending on the structure of G, and possibly also that of L, L may admit differentiable Lp-functional calculi, or may be of holomorphic Lp-type for a given p≠ 2. ‘Holomorphic Lp-type’ means that every Lp-spectral multiplier for L is necessarily holomorphic in a complex neighbourhood of some non-isolated point of the L2-spectrum of L. This can in fact only arise if the group algebra L1(G) is non-symmetric. Assume that p≠ 2. For a point ℓ in the dual g* of the Lie algebra g of G, denote by Ω(ℓ)=Ad*(G)ℓ the corresponding coadjoint orbit. It is proved that every sub-Laplacian on G is of holomorphic Lp-type, provided that there exists a point ℓ∈ g* satisfying Boidol's condition (which is equivalent to the non-symmetry of L1(G)), such that the restriction of Ω(ℓ) to the nilradical of g is closed. This work improves on results in previous work by Christ and Müller and Ludwig and Müller in twofold ways: on the one hand, no restriction is imposed on the structure of the exponential group G, and on the other hand, for the case p>1, the conditions need to hold for a single coadjoint orbit only, and not for an open set of orbits. It seems likely that the condition that the restriction of Ω(ℓ) to the nilradical of g is closed could be replaced by the weaker condition that the orbit Ω(ℓ) itself is closed. This would then prove one implication of a conjecture by Ludwig and Müller, according to which there exists a sub-Laplacian of holomorphic L1 (or, more generally, Lp) type on G if and only if there exists a point ℓ∈ g* whose orbit is closed and which satisfies Boidol's condition.

Infinite convolution products and refinable distributions on Lie groups

Sufficient conditions for the convergence in distribution of an infinite convolution product p1 * P2 * ... of measures on a connected Lie group 5 with respect to left invariant Haar measure are derived. These conditions are used to construct distributions / that satisfy To = / where T is a refinement operator constructed from a measure t and a dilation automorphism A. The existence of A implies 5 is nilpotent and simply connected and the exponential map is an analytic homeomorphism. Furthermore, there exists a unique minimal compact subset IC C 5 such that for any open set U containing IC, and for any distribution f on G with compact support, there exists an integer n(U, f) such that n > n(U, f) implies supp(Tnf) C U. If p is supported on an A-invariant uniform subgroup r, then T is related, by an intertwining operator, to a transition operator W on C((r). Necessary and sufficient conditions for Tnf to converge to / E L2, and for the F-translates of / to be orthogonal or to form a Riesz basis, are characterized in terms of the spectrum of the restriction of W to functions supported on Q K:= C 'Cn r.

Sub-Laplacians of Holomorphic Lp-type on Rank One AN-Groups and Related Solvable Groups

Consider a right-invariant sub-Laplacian L on an exponential solvable Lie group G, endowed with a left-invariant Haar measure. Depending on the structure of G and possibly also that of L, L may admit differentiable Lp-functional calculi, or may be of holomorphic Lp-type for a given p≠2, as recent studies of specific classes of groups G and sub-Laplacians L have revealed. By “holomorphic Lp-type” we mean that every Lp-spectral multiplier for L is necessarily holomorphic in a complex neighborhood of some point in the L2-spectrum of L. This can only arise if the group algebra L1(G) is non-symmetric. In this article we prove that, for large classes of exponential groups, including all rank one AN-groups, a certain Lie algebraic condition, which characterizes the non-symmetry of L1(G) [37], also suffices for L to be of holomorphic L1-type. Moreover, if this condition, which was first introduced by J. Boidol [6] in a different context, holds for generic points in the dual g* of the Lie algebra of G, then L is of holomorphic Lp-type for every p≠2. Besides the non-symmetry of L1(G), also the closedness of coadjoint orbits plays a crucial role. We also discuss an example of a higher rank AN-group. This example and our results in the rank one case suggest that sub-Laplacians on exponential Lie groups may be of holomorphic L1-type if and only if there exists a closed coadjoint orbit Ω⊂g* such that the points of Ω satisfy Boidol's condition. In the course of the proof of our main results, whose principal strategy is similar as in [8], we develop various tools which may be of independent interest and largely apply to more general Lie groups. Some of them are certainly known as “folklore” results. For instance, we study subelliptic estimates on representation spaces, the relation between spectral multipliers and unitary representations, and develop some “holomorphic” and “continuous” perturbation theory for images of sub-Laplacians under “smoothly varying” families of irreducible unitary representations.

Calculus on the multivector manifold spin+(p,q)

Coordinate-free differential and integral calculus is formulated on the spin groups, regarded as multivector manifolds embedded in Geometric Algebra. The derivative with respect to a point on the manifold is shown to be in the tangent algebra at the inverse point. The curvature of spin(p,q) is calculated and related to the structure constants of its Lie algebra. Volume elements on spin(p,q) are introduced naturally as completely antisymmetrized geometric products of bivectors. They are left invariant like Haar measure and are elements of a multivector-valued Grassman algebra. It is shown that integration in this sense is forbidden for many spin groups. A version of Stokes’s theorem is formulated on spin(p,q).

The Haar measure on a compact quantum group

Let A be a C ∗ {{\text {C}}^ \ast } -algebra with an identity. Consider the completed tensor product A ⊗ ¯ A A\bar \otimes A of A with itself with respect to the minimal or the maximal C ∗ {{\text {C}}^ \ast } -tensor product norm. Assume that Δ : A → A ⊗ ¯ A \Delta :A \to A\bar \otimes A is a non-zero ∗ ^ \ast -homomorphism such that ( Δ ⊗ ι ) Δ = ( ι ⊗ Δ ) Δ (\Delta \otimes \iota )\Delta = (\iota \otimes \Delta )\Delta where ι \iota is the identity map. Then Δ \Delta is called a comultiplication on A. The pair ( A , Δ ) (A,\Delta ) can be thought of as a ’compact quantum semi-group’. A left invariant Haar measure on the pair ( A , Δ ) (A,\Delta ) is a state φ \varphi on A such that ( ι ⊗ φ ) Δ ( a ) = φ ( a ) 1 (\iota \otimes \varphi )\Delta (a) = \varphi (a)1 for all a ∈ A a \in A . We show in this paper that a left invariant Haar measure exists if the set Δ ( A ) ( A ⊗ 1 ) \Delta (A) (A \otimes 1) is dense in A ⊗ ¯ A A\bar \otimes A . It is not hard to see that, if also Δ ( A ) ( 1 ⊗ A ) \Delta (A) (1 \otimes A) is dense, this Haar measure is unique and also right invariant in the sense that ( φ ⊗ ι ) Δ ( a ) = φ ( a ) 1 (\varphi \otimes \iota )\Delta (a) = \varphi (a)1 . The existence of a Haar measure when these two sets are dense was first proved by Woronowicz under the extra assumption that A has a faithful state (in particular when A is separable).

A concavity property for the measure of product sets in groups

Let G be a connected locally compact group with a left invariant Haar measure μ. We prove that the function ξ(x) = inf{μ(AB) : μ(A) = x} is concave for any fixed bounded set B ⊂ G. This is used to give a new proof of Kemperman’s inequality μ(AB) ≥ min(μ(A) + μ(B), μ(G)) for unimodular G.

Heat kernels on Lie groups

Let G be a Lie group with a fixed left invariant Haar measure and Λ be a right or left invariant, real, formally negative, elliptic differential operator of second order and without constant term. Then we call the “fundamental solution of the Cauchy problem” for the equation ∂u ∂t = Λu heat kernel. We prove that these heat kernels have most of the useful properties of their special case, the Gauss kernel (2 √ πt ) − n exp ( −∥x∥ 2 4t ) .

A characterization of certain weak ∗-closed subalgebras of L∞( G)

Let G be an arbitrary locally compact Hausdorff group, and let L ∞( G) be the set of essentially bounded measurable functions on G with respect to left invariant Haar measure. Let S be a linear subspace of L ∞( G) which is (i) left and right translation invariant; (ii) weak ∗-closed; (iii) self-adjoint, i.e., ƒ ϵ S implies ƒ ϵ S; and (iv) an algebra containing the constant functions. Then there exists a unique closed normal subgroup H of G such that S = {ƒ ϵ L ∞(G): aƒ = ƒ a = ƒ, ∀a ϵ H} . This extends to arbitrary locally compact groups, a result known only for Abelian groups.

On minimal vanishing algebras

I. Let G be a locally compact group with left invariant Haar measure m. For any measurable subset S of G, define Ls to be the closed subspace of LI(G) consisting of all functions which vanish (a. e.) on the complement of S. When L s forms an algebra, it is called a vanishing algebra. A vanishing algebra L s is said to be minimal if L s ~ 0 and if, whenever L T is a vanishing algebra such that L r c Ls, then L r = L s or L r --~ 0. In this note we shall prove the following theorems.

A rigidity theorem for discrete subgroups

0. Introduction. Let G denote an analytic group and let P c G denote a discrete subgroup. A left invariant Haar measure on G then induces a measure on G/r, which is determined up to a positive constant. We let v denote one such induced measure on G/P. DEFINITION 0.1. F is called a lattice, in case v(G/f) < oo. Unless specified otherwise, F will denote a lattice in the analytic group G. Let W be a simply connected Cr manifold; let M denote the set of all one-one homomorphisms

Ergodic properties of automorphisms of a locally compact group

The following remark is made by Halmos in his book [2, p. 29]. Can an automorphism of a locally compact but noncompact group be an ergodic measure preserving transformation? Nothing is known about this subject. Only in the compact case has anything ever been done. The aim of this paper is to give an answer to this question of Halmos. We prove the following in this paper: (i) A continuous automorphism T of a locally compact group G which is not bicontinuous cannot be ergodic. (ii) A continuous automorphism T of a locally compact group G with a left invariant Haar measure pi (pi need not be right invariant) and which is not measure preserving cannot be ergodic. (iii) A continuous automorphism of a locally compact, noncompact abelian group is not ergodic. NOTATIONS. We follow [2 ] and [3 ] for notions in ergodic theory and measure theory. We follow [9] and [4] for notions in topological groups. All topological spaces are assumed to be Hausdorff in this paper. Unless otherwise stated the groups occurring in this paper are not assumed to be abelian. DEFINITION 1. Let G be a locally compact group. A continuous automorphism T of G is a one-to-one map from G onto itself which is continuous and is also an algebraic isomorphism. An autohomeomorphism or a bicontinuous automorphism T of G is an algebraic isomorphism from G onto itself such that both T and T-1 are continuous. REMARK. Some examples of locally compact groups G and continuous automorphisms of G which are not autohomeomorphisms are given by Robertson [8] in his thesis. To our best knowledge this is the first place where such an example is written. We give below an example of a continuous automorphism of a locally compact group which is not bicontinuous, based on an idea of K. H. Hoffman. This example shows the need of Lemma 1 below. EXAMPLE OF A CONTINUOUS AUTOMORPHISM OF A LOCALLY COMPACT GROUP WHICH IS NOT BICONTINUOUS. Let G1 be the circle group with the usual topology. Let F1 be the circle group with discrete topology. Let F1 = F2= ... = n = ... and let G1 = G2=G3= ... =Gn =