Left Regular Representation Research Articles

A Remark About the Spectral Radius

We show that for any finitely generated non-amenable group and any e > 0, there exists some finite symmetric generating set with spectral radius less than e. We give applications to percolation theory and the theory of operator spaces. 1. The spectral radius Let Γ be a finitely generated group and let S ⊂ Γ be a finite symmetric generating set. Here, a set is called symmetric if S−1 = S, i.e. g ∈ S implies g−1 ∈ S. Consider the Hilbert space `Γ with orthonormal basis {δg | g ∈ Γ} and the left-regular representation λ : Γ→ U(`Γ), which is defined by the formula λ(g)δh := δgh. We will also use λ to denote the linear extension λ : C[Γ]→ B(`Γ). Here C[Γ] denotes the complex group ring and λ is a ∗-homomorphism with respect to the natural involution on C[Γ]. There is a natural trace τ : C[Γ]→ C, defined by τ( ∑ g agg) = ae. For each a = ∑ g agg ∈ C[Γ], we denote by ‖a‖ the operator norm of the operator λ(a) ∈ B(`Γ). We also set supp(a) := {g ∈ Γ | ag 6= 0}, size(a) := |supp(a)| and ‖a‖1 := ∑ g |ag|. It is a basic property of the operator norm that ‖a‖ = ‖a‖ whenever a is a hermitean element, i.e. a∗ = a. For a, b ∈ R[Γ], we write a ≤Γ b if ag ≤ bg for all g ∈ Γ. If 0 ≤Γ b ≤Γ a and a, b ∈ R[Γ] are hermitean, then ‖b‖ ≤ ‖a‖. Indeed, this follows from the spectral radius formula (1) ‖b‖ = lim n→∞ τ(b) ≤ lim n→∞ τ(a) = ‖a‖. Clearly, if 0 ≤Γ a, then ‖a‖1 = ‖a‖1 . For any symmetric subset S ⊂ Γ, we define the Markov operator m(S) := 1 |S| ∑ s∈S s. Kesten [5] showed that the group Γ is non-amenable if and only ‖m(S)‖ < 1 for some (and hence any) finite symmetric generating set of Γ. We will also use the notation ρ(S) := ‖m(S)‖ and call it the spectral radius of the random walk associated with S. The spectral radius formula ρ(S) = lim n→∞ τ(m(S)) gives the explanation for this terminology, since the right side of this equation is the exponential growth rate of the return probability of a random walk in the Cayley graph of Γ with respect to the generating set S after 2n steps, starting at the neutral element, see [11] for definitions and references. 1

Inverse-Closed Algebras of Integral Operators on Locally Compact Groups

We construct some inverse-closed algebras of bounded integral operators with operator-valued kernels, acting in spaces of vector-valued functions on locally compact groups. To this end we make use of covariance algebras associated to C*-dynamical systems defined by the C*-algebras of right uniformly continuous functions with respect to the left regular representation.

Regular representations of time‐frequency groups

In this paper, we study the Plancherel measure of a class of non‐connected nilpotent groups which is of special interest in Gabor theory. Let G be a time‐frequency group. That is where , are translations and modulations operators acting in and B is a non‐singular matrix. We compute the Plancherel measure of the left regular representation of G which is denoted by L. The action of G on induces a representation which we call a Gabor representation. Motivated by the admissibility of this representation, we compute the decomposition of L into direct integral of irreducible representations by providing a precise description of the unitary dual and its Plancherel measure. As a result, we generalize Hartmut Führ's results which are only obtained for the restricted case where , and Even in the case where G is not type I, we are able to obtain a decomposition of the left regular representation of G into a direct integral decomposition of irreducible representations when . Some interesting applications to Gabor theory are given as well. For example, when B is an integral matrix, we are able to obtain a direct integral decomposition of the Gabor representation of G.

An [formula omitted]-version of von Neumann dimension for Banach space representations of sofic groups

In [5], A. Gournay defined a notion of lp-dimension for Γ-invariant subspaces of lq(Γ)⊕n, with Γ amenable. The number dimlqlp(Γ,V) is dimV when p=q, and is preserved by a certain class of Γ-equivariant bounded linear isomorphisms. We develop a notion of dimlp,Σ(Y,Γ) where Y is a Banach space with a uniformly bounded action of a sofic group Γ and Σ is a sofic approximation. In particular, our definition makes sense for a large class of non-amenable groups. We also develop a notion of dimSp,Σ(Y,Γ) with Γ an Rω-embeddable group and Sp the space of finite-dimensional Schatten p-class operators. These numbers are invariant under bounded Γ-equivariant linear isomorphisms and under the natural translation action of Γ, dimlp(lp(Γ,V),Γ)=dimV, and dimSp(lp(Γ,V),Γ)=dimV for 1⩽p⩽2. In particular, this shows that lp(Γ,V) is not isomorphic to lp(Γ,W) as a representation of Γ if dimV≠dimW, and Γ is Rω-embeddable. We discuss other concrete computations in a follow-up paper, including proving that our dimension agrees with von Neumann dimension for representations contained in a multiple of the left-regular representation.

Nuclearity of semigroup [formula omitted]-algebras and the connection to amenability

We study C∗-algebras associated with subsemigroups of groups. For a large class of such semigroups including positive cones in quasi-lattice ordered groups and left Ore semigroups, we describe the corresponding semigroup C∗-algebras as C∗-algebras of inverse semigroups, groupoid C∗-algebras and full corners in associated group crossed products. These descriptions allow us to characterize nuclearity of semigroup C∗-algebras in terms of faithfulness of left regular representations and amenability of group actions. Moreover, we also determine when boundary quotients of semigroup C∗-algebras are UCT Kirchberg algebras. This leads to a unified approach to Cuntz algebras and ring C∗-algebras.

Rapid Decay Property with Examples in Crystallographic and Discrete Heisenberg Groups

Rapid Decay Property (Property (RD)) provides estimates for the operator norm of those functions (in the left-regular representation) in terms of the Sobolev norm. We provide the examples of length function and related some results. We study the stability of property (RD) with respect to some constructions of groups. We show the some lemmas related for Heisenberg groups. Finally, we show that the following groups have property (RD): • The Crystallographic groups. • The Discrete Heisenberg group.

Matrix coefficients of unitary representations and associated compactifications

We study, for a locally compact group $G$, the compactifications $(\pi,G^\pi)$ associated with unitary representations $\pi$, which we call {\it $\pi$-Eberlein compactifications}. We also study the Gelfand spectra $\Phi_{\mathcal{A}}(\pi)}$ of the uniformly closed algebras $\mathcal{A}(\pi)$ generated by matrix coefficients of such $\pi$. We note that $\Phi_{\mathcal{A}(\pi)}\cup\{0\}$ is itself a semigroup and show that the Silov boundary of $\mathcal{A}(\pi)$ is $G^\pi$. We study containment relations of various uniformly closed algebras generated by matrix coefficients, and give a new characterisation of amenability: the constant function 1 can be uniformly approximated by matrix coefficients of representations weakly contained in the left regular representation if and only if $G$ is amenable. We show that for the universal representation $\omega$, the compactification $(\omega,G^\omega)$ has a certain universality property: it is universal amongst all compactifications of $G$ which may be embedded as contractions on a Hilbert space, a fact which was also recently proved by Megrelishvili. We illustrate our results with examples including various abelian and compact groups, and the $ax+b$-group. In particular, we witness algebras $\fA(\pi)$, for certain non-self-conjugate $\pi$, as being generalised algebras of analytic functions.

Harmonic analysis on Heisenberg–Clifford Lie supergroups

We define a Fourier transform and a convolution product for functions and distributions on Heisenberg--Clifford Lie supergroups. The Fourier transform exchanges the convolution and a pointwise product, and is an intertwining operator for the left regular representation. We generalize various classical theorems, including the Paley--Wiener--Schwartz theorem, and define a convolution Banach algebra.

Fixed-point free endomorphisms and Hopf Galois structures

Let L | K L|K be a Galois extension of fields with finite Galois group G G . Greither and Pareigis showed that there is a bijection between Hopf Galois structures on L | K L|K and regular subgroups of P e r m ( G ) Perm(G) normalized by G G , and Byott translated the problem into that of finding equivalence classes of embeddings of G G in the holomorph of groups N N of the same cardinality as G G . In 2007 we showed, using Byott’s translation, that fixed point free endomorphisms of G G yield Hopf Galois structures on L | K L|K . Here we show how abelian fixed point free endomorphisms yield Hopf Galois structures directly, using the Greither-Pareigis approach and, in some cases, also via the Byott translation. The Hopf Galois structures that arise are “twistings” of the Hopf Galois structure by H λ H_{\lambda } , the K K -Hopf algebra that arises from the left regular representation of G G in P e r m ( G ) Perm(G) . The paper concludes with various old and new examples of abelian fixed point free endomorphisms.

$$C^*$$ -algebras of Toeplitz type associated with algebraic number fields

We associate with the ring $$R$$ of algebraic integers in a number field a C*-algebra $${\mathfrak T }[R]$$ . It is an extension of the ring C*-algebra $${\mathfrak A }[R]$$ studied previously by the first named author in collaboration with X. Li. In contrast to $${\mathfrak A }[R]$$ , it is functorial under homomorphisms of rings. It can also be defined using the left regular representation of the $$ax+b$$ -semigroup $$R\rtimes R^\times $$ on $$\ell ^2 (R\rtimes R^\times )$$ . The algebra $${\mathfrak T }[R]$$ carries a natural one-parameter automorphism group $$(\sigma _t)_{t\in {\mathbb R }}$$ . We determine its KMS-structure. The technical difficulties that we encounter are due to the presence of the class group in the case where $$R$$ is not a principal ideal domain. In that case, for a fixed large inverse temperature, the simplex of KMS-states splits over the class group. The “partition functions” are partial Dedekind $$\zeta $$ -functions. We prove a result characterizing the asymptotic behavior of quotients of such partial $$\zeta $$ -functions, which we then use to show uniqueness of the $$\beta $$ -KMS state for each inverse temperature $$\beta \in (1,2]$$ .

Prescribed absolute values, character sums and spectrum integrality

Let G be a group and let ρreg be the complex left-regular representation of G. We consider the following problem: For which inverse-closed subsets S⊆G the spectrum of the matrix ∑g∈Sρreg(g) is integral? (that is, all of the eigenvalues of the matrix are integers). For abelian G, a complete characterization, due to Bridges and Mena, is known. Here we are interested in the case where G is the dihedral group, D2n, which turns out to be a far-reaching generalization of the abelian case. We obtain a complete characterization when n is an odd prime and present partial results when n is a prime power.

Automorphism groups of the Pancake graphs

It is well known that the Pancake graph is widely used as models for interconnection networks. In this paper we prove that the automorphism group of the Pancake graph Pn, n⩾5, is the left regular representation of the symmetric group Sn.

Operator algebras from the discrete Heisenberg semigroup

AbstractWe study reflexivity and structural properties of operator algebras generated by representations of the discrete Heisenberg semigroup. We show that the left regular representation of this semigroup gives rise to a semi-simple reflexive algebra. We exhibit an example of a representation that gives rise to a non-reflexive algebra. En route, we establish reflexivity results for subspaces of $H^{\infty}(\mathbb{T})\otimes\mathcal{B}(\mathcal{H})$.

The Cyclotomic BMW Algebra Associated with the Two String Type B Braid Group

The cyclotomic Birman–Murakami–Wenzl (or BMW) algebras , introduced by R. Häring–Oldenburg, are extensions of the cyclotomic Hecke algebras of Ariki–Koike, in the same way as the BMW algebras are extensions of the Iwahori–Hecke algebras of type A. In this article, we produce a basis of the two strand algebra and construct its left regular representation. In the process we obtain relations amongst the parameters which play an important role in the n strand case.

Spatial discretization of restricted group C^*-algebras

We consider spatial discretizations by the finite section method of the restricted group algebra of a finitely generated discrete group, which is represented as a concrete operator algebra via its left-regular representation. Special emphasis is paid to the quasicommutator ideal of the algebra generated by the finite sections sequences and to the stability of sequences in that algebra. For both problems, the sequence of the discrete boundaries plays an essential role. Finally, for commutative groups and for free non-commutative groups, the algebras of the finite sections sequences are shown to be fractal.

Analytic Dirac approximation for real linear algebraic groups

For a real linear algebraic group G let \({\mathcal{A}(G)}\) be the algebra of analytic vectors for the left regular representation of G on the space of superexponentially decreasing functions. We present an explicit Dirac sequence in \({\mathcal{A}(G)}\). Since \({\mathcal{A}(G)}\) acts on E for every Frechet-representation (π, E) of moderate growth, this yields an elementary proof of a result of Nelson that the space of analytic vectors is dense in E.

Random products of automorphisms of Heisenberg nilmanifolds and Weil’s representation

AbstractForn≥1, letHbe the (2n+1)-dimensional real Heisenberg group, and let Λ be a lattice inH. Let Γ be the group of automorphisms of the corresponding nilmanifold Λ∖HandUthe associated unitary representation of Γ onL2(Λ∖H) . Denote byTthe maximal torus factor associated to Λ∖H. Using Weil’s representation (also known as the metaplectic representation), we show that a dense set of matrix coefficients of the restriction ofUto the orthogonal complement ofL2(T) inL2(Λ∖H) belong toℓ4n+2+ε(Γ) for every ε&gt;0 . We give the following application to random walks on Λ∖Hdefined by a probability measureμon Aut (Λ∖H) . Denoting by Γ(μ) the subgroup of Aut (Λ∖H) generated by the support ofμand byU0andV0the restrictions ofUto, respectively, the subspaces ofL2(Λ∖H) andL2(T) with zero mean, we prove the following inequality:whereλis the left regular representation of Γ(μ) onℓ2(Γ(μ)) . In particular, the action of Γ(μ) on Λ∖Hhas a spectral gap if and only if the corresponding action of Γ(μ) onThas a spectral gap.

The Fell compactification and non-Hausdorff groupoids

A compactification of Fell is applied to locally compact non-Hausdorff groupoids and yields locally compact Hausdorff groupoids. In the etale case, this construction provides a geometric picture for the left-regular representations introduced by Khoshkam and Skandalis.

The isomorphism problem for Cayley ternary relational structures for some abelian groups of order [formula omitted

A ternary relational structure X is an ordered pair ( V , E ) where V is a set and E a set of ordered 3-tuples whose coordinates are chosen from V (so a ternary relational structure is a natural generalization of a 3-uniform hypergraph). A ternary relational structure is called a Cayley ternary relational structure of a group G if Aut ( X ) , the automorphism group of X , contains the left regular representation of G . We prove that two Cayley ternary relational structures of Z 2 3 × Z p , p ≥ 11 a prime, are isomorphic if and only if they are isomorphic by a group automorphism of Z 2 3 × Z p . This result then implies that any two Cayley digraphs of Z 2 3 × Z p are isomorphic if and only if they are isomorphic by a group automorphism of Z 2 3 × Z p , p ≥ 11 a prime.

PROPERTIES OF INJECTIVE HULLS OF A RING HAVING A COMPATIBLE RING STRUCTURE

AbstractIf the injective hull E = E(RR) of a ring R is a rational extension of RR, then E has a unique structure as a ring whose multiplication is compatible with R-module multiplication. We give some known examples where such a compatible ring structure exists when E is a not a rational extension of RR, and other examples where such a compatible ring structure on E cannot exist. With insights gleaned from these examples, we study compatible ring structures on E, especially in the case when ER, and hence RR ⊆ ER, has finite length. We show that for RR and ER of finite length, if ER has a ring structure compatible with R-module multiplication, then E is a quasi-Frobenius ring under that ring structure and any two compatible ring structures on E have left regular representations conjugate in Λ = EndR(ER), so the ring structure is unique up to isomorphism. We also show that if ER is of finite length, then ER has a ring structure compatible with its R-module structure and this ring structure is unique as a set of left multiplications if and only if ER is a rational extension of RR.