Isometric Representation Research Articles

Extensions of Semigroups and Morphisms of Semigroup $ C^{*} $-Algebras

The paper is devoted to the normal extensions of discrete semigroups and $ * $ -homomorphisms of semigroup $ C^{*} $ -algebras. We study the normal extensions of abelian semigroups by arbitrary groups. Considering numerical semigroups, we prove that they are normal extensions of the semigroup of nonnegative integers by finite cyclic groups. On the other hand, we prove that if a semigroup is a normal extension of the semigroup of nonnegative integers by a finite cyclic group generated by a single element then this semigroup is isomorphic to a numerical semigroup. As regard a normal extension with a generating set, we consider two reduced semigroup $ C^{*} $ -algebras defined by this extension. We show that there exists an embedding of the semigroup $ C^{*} $ -algebras which is generated by an injective homomorphism of the semigroups and the natural isometric representations of these semigroups.

Arveson's characterisation of CCR flows: The multiparameter case

In this paper, we revisit Arveson's characterisation of CCR flows in terms of the decomposability of the product system in the multiparameter context. We show that a multiparameter E0-semigroup is a CCR flow if and only if it is decomposable and admits a unit. In contrast to the one parameter situation, we exhibit uncountably many examples of decomposable E0-semigroups which do not admit any unit. As applications, we show that for a pure isometric representation V, the associated CCR flow αV remembers the unitary equivalence class of V, compute the positive contractive as well as the projective local cocycles of a CCR flow and derive a necessary and a sufficient condition for a CCR flow to be prime.

E-semigroups over closed convex cones

In this paper, we study E-semigroups over convex cones. We prove a structure theorem for E-semigroups which leave the algebra of compact operators invariant. Then we study in detail the CCR flows, E0-semigroups constructed from isometric representations, by describing their units and gauge groups. We exhibit an uncountable family of 2-parameter CCR flows, containing mutually non-cocycle-conjugate E0-semigroups.

Blaschke $$\boldsymbol{C}^{\boldsymbol{*}}$$-algebras

In the paper we introduce the notion of the Blaschke $$C^{*}$$ -algebra and we consider the isometric representations of uniform Blaschke algebra. We extend the Coburn’s Theorem for a family of non-unitary isometries connected by a family of finite Blaschke products. We show that the Blaschke $$C^{*}$$ -algebra is isomorphic to the inductive limit of Toeplitz algebras, as well as the (non commutative) $$C^{*}$$ -algebra generated by a unital isometric representation of the uniform Blaschke algebra by multiplications in the respective Hardy space.

Isometric tensor network representation of string-net liquids

Recently, a class of tensor networks called isometric tensor network states (isoTNS) was proposed which generalizes the canonical form of matrix product states to tensor networks in higher dimensions. While this ansatz allows for efficient numerical computations, it remained unclear which phases admit an isoTNS representation. In this work, we show that two-dimensional string-net liquids, which represent a wide variety of topological phases including discrete gauge theories, admit an exact isoTNS representation. We further show that the isometric form can be preserved after applying a finite depth local quantum circuit. Taken together, these results show that long-range entanglement by itself is not an obstruction to isoTNS representation and suggest that all two-dimensional gapped phases with gappable edges admit an isoTNS representation.

Tensor algebras of product systems and their C⁎-envelopes

Let (G,P) be an abelian, lattice ordered group and let X be a compactly aligned product system over P with coefficients in A. We show that the C*-envelope of the Nica tensor algebra NTX+ coincides with both Sehnem's covariance algebra A×XP and the co-universal C⁎-algebra NOXr for injective, gauge-compatible, Nica-covariant representations of Carlsen, Larsen, Sims and Vittadello. We give several applications of this result on both the selfadjoint and non-selfadjoint operator algebra theory. First we guarantee the existence of NOXr, thus settling a problem of Carlsen, Larsen, Sims and Vittadello which was open even for abelian, lattice ordered groups. As a second application, we resolve a problem posed by Skalski and Zacharias on dilating isometric representations of product systems to unitary representations. As a third application we characterize the C⁎-envelope of the tensor algebra of a finitely aligned higher-rank graph which also holds for topological higher-rank graphs. As a final application we prove reduced Hao-Ng isomorphisms for generalized gauge actions of discrete groups on C⁎-algebras of product systems. This generalizes recent results that were obtained by various authors in the case where (G,P)=(Z,N).

Extending representations of Banach algebras to their biduals

We show that a representation of a Banach algebra A on a Banach space X can be extended to a canonical representation of $$A^{**}$$ on X if and only if certain orbit maps $$A\rightarrow X$$ are weakly compact. When this is the case, we show that the essential space of the representation is complemented if A has a bounded left approximate identity. This provides a tool to disregard the difference between degenerate and nondegenerate representations. Our results have interesting consequences both in $$C^*$$-algebras and in abstract harmonic analysis. For example, a $$C^*$$-algebra A has an isometric representation on an $$L^p$$-space, for $$p\in [1,\infty ){\setminus }\{2\}$$, if and only if A is commutative. Moreover, the $$L^p$$-operator algebra of a locally compact group is universal with respect to arbitrary representations on $$L^p$$-spaces.

C*-algebras generated by isometries and true representations

Let (G; P) be a quasi-lattice ordered group. In this paper we present a modied proof of Laca and Raeburn's theorem about the covariant isometric representations of amenable quasi-lattice ordered groups [7, Theorem 3.7], by following a two stage strategy. First, we construct a universal covariant representation for a given quasi-lattice ordered group (G; P) and show that it is unique. The construction of this object is new; we have not followed either Nica's approach in [10] or Laca and Raeburn's approach in [7], although all three objects are essentially the same. Our approach is a very natural one and avoids some of the intricacies of the other approaches. Then we show if (G; P) is amenable, true representations of (G; P) generate C-algebras which are canonically isomorphic to the universal object.

Fourier Spaces and Completely Isometric Representations of Arens Product Algebras

AbstractMotivated by the definition of a semigroup compactication of a locally compact group and a large collection of examples, we introduce the notion of an (operator) homogeneous left dual Banach algebra (HLDBA) over a (completely contractive) Banach algebra $A$ . We prove a Gelfand-type representation theorem showing that every HLDBA over A has a concrete realization as an (operator) homogeneous left Arens product algebra: the dual of a subspace of $A^{\ast }$ with a compatible (matrix) norm and a type of left Arens product $\Box$ . Examples include all left Arens product algebras over $A$ , but also, when $A$ is the group algebra of a locally compact group, the dual of its Fourier algebra. Beginning with any (completely) contractive (operator) $A$ -module action $Q$ on a space $X$ , we introduce the (operator) Fourier space $({\mathcal{F}}_{Q}(A^{\ast }),\Vert \cdot \Vert _{Q})$ and prove that $({\mathcal{F}}_{Q}(A^{\ast })^{\ast },\Box )$ is the unique (operator) HLDBA over $A$ for which there is a weak $^{\ast }$ -continuous completely isometric representation as completely bounded operators on $X^{\ast }$ extending the dual module representation. Applying our theory to several examples of (completely contractive) Banach algebras $A$ and module operations, we provide new characterizations of familiar HLDBAs over A and we recover, and often extend, some (completely) isometric representation theorems concerning these HLDBAs.

Similarity Problems, Følner Sets and Isometric Representations of Amenable Semigroups

We revisit Sz.-Nagy's criteria for similarity of Hilbert space bounded linear operators to isometries or unitaries and present new ones. We also discuss counterparts of the Dixmier-Day theorem concerning bounded representations of amenable groups and semigroups. We highlight the role of Folner sets in similarity problems in both settings of unimodular, $\sigma$-compact, amenable groups and in discrete semigroups possessing the Strong Folner condition (SFC).

Isometric Shape Representation by Integrating Shape Function Maps and Deep Learning

With the wide application of isometric 3D models, the representation and recognition of them receive increasing attention. Most existing methods for shape representation and analysis either focus on using the constrained hand-craft models or purely devote efforts on developing complicated deep learning methods. In order to make full use of the advantages of both, this paper presents a novel mixture modeling approach by integrating both the shape function map (SFM) and the deep convolutional neural network (CNN). First, multiple SFM maps are constructed to grasp the rigid and non-rigid information that are usually considered separately for shape representation. Then, to fully characterize the low-level information existing in SFM, diverse sets of deep features are learned on different SFM maps during the training process of classifiers. Finally, both the rigid and non-rigid deep features are integrated for more discriminative feature abstraction. The experimental results on standard shape benchmark datasets have validated the superior performance of our proposed approach on feature extraction, classification and retrieval. Besides, our evaluation results on extensive shape datasets (e.g. noisy, CAD and protein shapes) have again verified the effectiveness of the proposed algorithm.

On complements of Kazhdan projections in semisimple groups

We prove that for an isometric representation of some groups on certain Banach spaces, the complement of the subspace of invariant vectors is 1-complemented.

The injective and surjective hulls of the ideal of (p,q)-compact operators and their approximation properties

Let 1 ≤ p , q ≤ ∞ with 1 / p + 1 / q ≥ 1 . We establish the isometric representations of the injective hull K i n j p , q and surjective hull K p , q s u r of the ideal of ( p , q ) -compact operators. We obtain some factorizations of K p , q i n j and K p , q s u r , and prove that if a Banach space X has the K p , p ⁎ i n j -approximation property ( K p , p ⁎ i n j -AP) (respectively, K p , p ⁎ s u r -AP), then X has the K p , q i n j -AP (respectively, K p , q s u r -AP). It also follows that if a Banach space X has the AP, then X has the K p , q i n j -AP and the K s u r p , q -AP, and that every Banach space has the K i n j 2 , q -AP, the K p , 2 i n j -AP, K s u r 2 , q -AP and the K p , 2 s u r -AP. Finally, we find the dual space of ( K p , q i n j ( X , Y ) , ‖ ⋅ ‖ K p , q i n j ) (respectively, ( K p , q s u r ( X , Y ) , ‖ ⋅ ‖ K p , q s u r ) ) when X ⁎ has the AP or Y has the K p , q i n j -AP (respectively, K p , q s u r -AP).

Isometric representations of weighted spaces of little Lipschitz functions

Given a compact pointed metric space X and a weight v on the complement of the diagonal set in $$X\times X$$ , we prove that the Banach space $$\mathrm {lip}_v(X)$$ of all weighted little Lipschitz scalar-valued functions on X vanishing at the basepoint, equipped with the weighted Lipschitz norm, embeds almost isometrically into $$c_0$$ . This result has many consequences on the structure of those Banach spaces and their duals. Moreover, we prove that this isomorphism can never be an isometric embedding whenever X is a $$\mathbb {T}$$ -balanced subset containing 0 and compact for some metrizable topology of a complex Banach space and v is a radial 0-weight.

Limit Automorphisms of the C*-Algebras Generated by Isometric Representations for Semigroups of Rationals

We consider inductive sequences of Toeplitz algebras whose connecting homomorphisms are defined by collections of primes. The inductive limits of these sequences are C*-algebras generated by representations for semigroups of rationals. We study the limit endomorphisms of these C*-algebras induced by morphisms between copies of the same inductive sequences of Toeplitz algebras. We establish necessary and sufficient conditions for these endomorphisms to be automorphisms of the algebras.

Group 1-cohomology is complemented

We show a structural property of cohomology with coefficients in an isometric representation on a uniformly convex Banach space: if the cohomology group $H^1(G,\pi)$ is reduced, then, up to an isomorphism, it is a closed complemented, subspace of the space of cocycles and its complement is the subspace of coboundaries.

Property [formula omitted] and property [formula omitted] for Orlicz spaces LΦ

An Orlicz space LΦ(Ω) is a Banach function space defined by using a Young function Φ, which generalizes the Lp spaces. We show, for an Orlicz space LΦ([0,1]) which is not isomorphic to L∞([0,1]), if a locally compact second countable group has property (TLΦ([0,1])), which is a generalization of Kazhdan's property (T) for linear isometric representations on LΦ([0,1]), then it has Kazhdan's property (T). We also show, for a separable complex Orlicz space LΦ(Ω) with gauge norm, Ω=R,[0,1],N, if a locally compact second countable group has Kazhdan's property (T), then it has property (TLΦ(Ω)). We prove, for a finitely generated group Γ and a Banach space B whose modulus of convexity is sufficiently large, if Γ has Kazhdan's property (T), then it has property (FB), which is a fixed point property for affine isometric actions on B. Moreover, we see that, for a hyperbolic group Γ (which may have Kazhdan's property (T)) and an Orlicz sequence space ℓΦΨ with gauge norm such that the Young function Ψ sufficiently rapidly increases near 0, Γ doesn't have property (FℓΦΨ). These results are generalizations of the results for Lp-spaces.

Property Fℓq implies property Fℓp for 1 < p < q < ∞

It is known that for σ-compact groups Kazhdan's Property (T) is equivalent to Serre's Property FH. Generalized versions of those properties, called properties (TB) and FB, can be defined in terms of the isometric representations of a group on an arbitrary Banach space B. Property FB implies (TB).It is known that a group with Property (Tℓp) shares some properties with Kazhdan's groups, for example compact generation and compact abelianization. Moreover in the case of discrete groups, Property (Tℓp) implies Lubotzky's Property (τ).In this paper we prove that in the case of discrete groups and ℓp(N) spaces, for 1<p<q<∞,p≠2, Property Fℓq implies Property Fℓp.

Complementation of the subspace of G-invariant vectors

Let [Formula: see text] be an isometric representation of a group [Formula: see text] in a Banach space [Formula: see text] over a normalizing non-discrete absolute valued division ring [Formula: see text]. If [Formula: see text] and [Formula: see text] are supportive and [Formula: see text] verifies the separation property, then [Formula: see text] is 1-complemented in [Formula: see text] along [Formula: see text]. As an immediate consequence, in an isometric representation of a group in a smooth Banach space whose dual is also smooth, the subspace of [Formula: see text]-invariant vectors is [Formula: see text]-complemented.

On strong property (T) and fixed point properties for Lie groups

We consider certain strengthenings of property (T) relative to Banach spaces that are satisfied by high rank Lie groups. Let X be a Banach space for which, for all k, the Banach--Mazur distance to a Hilbert space of all k-dimensional subspaces is bounded above by a power of k strictly less than one half. We prove that every connected simple Lie group of sufficiently large real rank depending on X has strong property (T) of Lafforgue with respect to X. As a consequence, we obtain that every continuous affine isometric action of such a high rank group (or a lattice in such a group) on X has a fixed point. This result corroborates a conjecture of Bader, Furman, Gelander and Monod. For the special linear Lie groups, we also present a more direct approach to fixed point properties, or, more precisely, to the boundedness of quasi-cocycles. Without appealing to strong property (T), we prove that given a Banach space X as above, every special linear group of sufficiently large rank satisfies the following property: every quasi-1-cocycle with values in an isometric representation on X is bounded.