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AbstractIn this paper, we investigate the approximation property for the predual of the weighted space of holomorphic mappings, where is a countable family of weights defined on a balanced open subset U of a Banach space E. We introduce a locally convex topology on and show that is topologically isomorphic to for a suitably restricted family . As a consequence, certain characterizations of the approximation property for the space are proved. [ABSTRACT FROM AUTHOR]

We prove thet a Banach space X has the λ -bounded compact approximation property (respectively, weak λ -bounded approximation property) if and only if X has the λ -Lipschitz bounded compact approximation property (respectively, weak λ -Lipschitz bounded approximation property). Also, it is shown that the dual space of X has the approximation property if and only if for every separable reflexive Banach space Y, the space of finite rank Lipschitz maps from X to Y is dense in the Lipschitz norm in the space of compact operators from X to Y. [ABSTRACT FROM AUTHOR]

In this current article, we show that a non-zero closed nilpotent ideal in a Johnson pseudo-contractible Banach algebra cannot be L1-predual. We prove also that a nilpotent ideal in a Johnson pseudo-contractible Banach algebras with the approximation property is forced to be zero. Among other things, we characterize the property F of some semigroup algebras associated with inverse semigroups. More especially, for an inverse semigroup S such that (E(S),≤) is uniformly locally finite, l1(S) has the property F if and only if each maximal subgroup of S is amenable, where E(S) is the set of idempotents of S. Furthermore, we study the property F of some 휃-Lau product structures. [ABSTRACT FROM AUTHOR]

We study the convergence of Fourier series in the polynomial system orthonormal in the sense of Sobolev and generated by the system of modified Meixner polynomials. In particular, we show that the Fourier series of in this system converges to pointwise on the grid as . In addition, we study the approximation properties of partial sums of Fourier series in the system . [ABSTRACT FROM AUTHOR]

The notion of von Neumann equivalence (vNE), which encapsulates both measure equivalence and W*-equivalence, was introduced recently by Ishan, Peterson, and Ruth (2019). They have shown that many analytic properties, such as amenability, property (T), the Haagerup property, and proper proximality are preserved under von Neumann equivalence. In this article, we expand on the list of properties that are stable under von Neumann equivalence, and prove that weak amenability, weak Haagerup property, and the approximation property (AP) are von Neumann equivalence invariants. In particular, we get that AP is stable under measure equivalence. Furthermore, our techniques give an alternate proof for vNE-invariance of the Haagerup property. [ABSTRACT FROM AUTHOR]

Let K = 2 N be the Cantor set, let M be the set of all metrics d on K that give its usual (product) topology, and equip M with the topology of uniform convergence, where the metrics are regarded as functions on K 2 . We prove that the set of metrics d ∈ M for which the Lipschitz-free space F (K , d) has the metric approximation property is a residual F σ δ set in M , and that the set of metrics d ∈ M for which F (K , d) fails the approximation property is a dense meager set in M . This answers a question posed by G. Godefroy. [ABSTRACT FROM AUTHOR]

For a given Banach space E with an 1-unconditional basis, we introduce the (E, E)-approximation property ((E, E)-AP) which is a natural generalization of the classical approximation property (AP) and the (p, p)-AP. It is shown that the AP always implies the (E, E)-AP. If E has an 1-unconditional and shrinking basis, then the (E*, E*)-AP for X* implies the (E, E)-AP for X. If in addition that the unconditional basis for E is subsymmetric, then X* has the (E*, E*)-AP implies that X has the duality (E, E)-AP. [ABSTRACT FROM AUTHOR]

In this paper, we study the σ-tensor norm (ασ), the absolutely τ-summing operator and the σ-nuclear operator. We characterize the ασ-approximation property in terms of some density of the space of absolutely τ-summing operators. When X* or Y*** has the approximation property, we prove that an operator T from X to Y is σ-nuclear if the adjoint of T is σ-nuclear.

Let K p , q ( 1 ≤ p , q ≤ ∞ with 1 / p + 1 / q ≥ 1 ) be the ideal of (p , q) -compact operators. This paper investigates the compactness and null sequences via K p , q , and an approximation property of the ideal of K p , q -compact operators. [ABSTRACT FROM AUTHOR]

We begin by describing the unit ball of the free p -convex Banach lattice over a Banach space E (denoted by FBL (p) [ E ]) as a closed solid convex hull of an appropriate set. Based on it, we show that, if a Banach space E has the λ -Approximation Property, then FBL (p) [ E ] has the λ -Positive Approximation Property. Further, we show that operators u ∈ B (E , F) (where E and F are Banach spaces) which extend to lattice homomorphisms from FBL (q) [ E ] to FBL (p) [ F ] are precisely those whose adjoints are (q , p) -mixing. Related results are also obtained for free lattices with an upper p -estimate. [ABSTRACT FROM AUTHOR]

Building on previous papers by Anantharaman-Delaroche (AD) we introduce and study the notion of AD-amenability for partial actions and Fell bundles over discrete groups. We prove that the cross-sectional C ∗ -algebra of a Fell bundle is nuclear if and only if the underlying unit fibre is nuclear and the Fell bundle is AD-amenable. If a partial action is globalisable, then it is AD-amenable if and only if its globalisation is AD-amenable. Moreover, we prove that AD-amenability is preserved by (weak) equivalence of Fell bundles and, using a very recent idea of Ozawa and Suzuki, we show that AD-amenability is equivalent to an approximation property introduced by Exel. [ABSTRACT FROM AUTHOR]

For a finitely generated tensor norm α\alpha , we investigate the α\alpha -approximation property (α\alpha -AP) and the bounded α\alpha -approximation property (bounded α\alpha -AP) in terms of some approximation properties of operator ideals. We prove that a Banach space X has the λ\lambda -bounded αp,q{\alpha }_{p,q}-AP (1≤p,q≤∞,1/p+1/q≥1)(1\le p,q\le \infty ,1/p+1/q\ge 1) if it has the λ\lambda -bounded gp{g}_{p}-AP. As a consequence, it follows that if a Banach space X has the λ\lambda -bounded gp{g}_{p}-AP, then X has the λ\lambda -bounded wp{w}_{p}-AP.

We prove a general principle satisfied by weakly precompact sets of Lipschitz-free spaces. By this principle, certain infinite dimensional phenomena in Lipschitz-free spaces over general metric spaces may be reduced to the same phenomena in free spaces over their compact subsets. As easy consequences we derive several new and some known results. The main new results are: F(X) is weakly sequentially complete for every superreflexive Banach space X, and F(M) has the Schur property and the approximation property for every scattered complete metric space M. [ABSTRACT FROM AUTHOR]

We study the Bishop–Phelps–Bollobás property for numerical radius restricted to the case of compact operators (BPBp-nu for compact operators in short). We show that C 0 (L) spaces have the BPBp-nu for compact operators for every Hausdorff topological locally compact space L. To this end, on the one hand, we provide some techniques allowing to pass the BPBp-nu for compact operators from subspaces to the whole space and, on the other hand, we prove some strong approximation property of C 0 (L) spaces and their duals. Besides, we also show that real Hilbert spaces and isometric preduals of ℓ 1 have the BPBp-nu for compact operators. [ABSTRACT FROM AUTHOR]

We study the Haagerup–Kraus approximation property for locally compact quantum groups, generalising and unifying previous work by Kraus–Ruan and Crann. Along the way we discuss how multipliers of quantum groups interact with the C ⁎ -algebraic theory of locally compact quantum groups. Several inheritance properties of the approximation property are established in this setting, including passage to quantum subgroups, free products of discrete quantum groups, and duals of double crossed products. We also discuss a relation to the weak⁎ operator approximation property. For discrete quantum groups, we introduce a central variant of the approximation property, and relate this to a version of the approximation property for rigid C ⁎ -tensor categories, building on work of Arano–De Laat–Wahl. [ABSTRACT FROM AUTHOR]

It is known due to R. Aron, B. Cole and T. Gamelin that every complex homomorphism of the algebra of entire functions of bounded type on a Banach space $X$ can be approximated in some sense by a net of point valued homomorphism. In this paper we consider the question about a generalization of this result for the case of homomorphisms to any commutative Banach algebra $A.$ We obtained some positive results if $A$ is the algebra of uniformly continuous analytic functions on the unit ball of $X.$

Let Kp,q (1≤p,q≤∞ with 1/p+1/q≥1) be the ideal of (p,q)-compact operators. This paper investigates the compactness and null sequences via Kp,q, and an approximation property of the ideal of Kp,q-compact operators.

In this article, some properties such as the approximation property of non-zero nilpotent ideals in generalized amenable Banach algebras are investigated. Also, we show that in pseudo-contractible Banach algebras, non-zero nilpotent ideals cannot have the approximation property. [ABSTRACT FROM AUTHOR]

Biological brain stores massive amount of information. Inspired by features of the biological memory, we propose an algorithm to efficiently store different classes of spatio-temporal information in a Recurrent Neural Network (RNN). A given spatio-temporal input triggers a neuron firing pattern, known as an attractor, and it conveys information about the class to which the input belongs. These attractors are the basic elements of the memory in our RNN. Preparing a set of good attractors is the key to efficiently storing temporal information in an RNN. We achieve this by means of enhancing the “separation” and “approximation” properties associated with the attractors, during the RNN training. We furthermore elaborate how these attractors can trigger an action via the readout in the RNN, similar to the sensory motor action processing in the cerebellum cortex. We show how different voice commands by different speakers trigger hand drawn impressions of the spoken words, by means of our separation and approximation based learning. The method further recognizes the gender of the speaker. The method is evaluated on the TI-46 speech data corpus, and we have achieved 98.6% classification accuracy on the TI-46 digit corpus.

We give criteria for the spectra of some nuclear operators in subspaces of quotients of Lp-spaces to be central-symmetric, as well as for the spectra of linear operators in Banach spaces to be Zd-symmetric in the sense of B. Mityagin. Also, we present a short proof of a corresponding Mityagin’s theorem.

Biological brain stores massive amount of information. Inspired by features of the biological memory, we propose an algorithm to efficiently store different classes of spatio-temporal information in a Recurrent Neural Network (RNN). A given spatio-temporal input triggers a neuron firing pattern, known as an attractor, and it conveys information about the class to which the input belongs. These attractors are the basic elements of the memory in our RNN. Preparing a set of good attractors is the key to efficiently storing temporal information in an RNN. We achieve this by means of enhancing the "separation" and "approximation" properties associated with the attractors, during the RNN training. We furthermore elaborate how these attractors can trigger an action via the readout in the RNN, similar to the sensory motor action processing in the cerebellum cortex. We show how different voice commands by different speakers trigger hand drawn impressions of the spoken words, by means of our separation and approximation based learning. The method further recognizes the gender of the speaker. The method is evaluated on the TI-46 speech data corpus, and we have achieved 98.6% classification accuracy on the TI-46 digit corpus. [ABSTRACT FROM AUTHOR]

Let Γ be a crystal group in R d . A function φ : R d ⟶ C is said to be crystal-refinable (or Γ -refinable) if it is a linear combination of finitely many of the rescaled and translated functions φ (γ - 1 (a x)) , where the translations γ are taken on a crystal group Γ , and a is an expansive dilation matrix such that a Γ a - 1 ⊂ Γ. A Γ -refinable function φ : R d → C satisfies a refinement equation φ (x) = ∑ γ ∈ Γ d γ φ (γ - 1 (a x)) with d γ ∈ C . Let S (φ) be the linear span of { φ (γ - 1 (x)) : γ ∈ Γ } and S h = { f (x / h) : f ∈ S (φ) } . One important property of S (φ) is, how well it approximates functions in L 2 (R d) . This property is very closely related to the crystal-accuracy of S (φ) , which is the highest degree p such that all multivariate polynomials q(x) of degree (q) < p are exactly reproduced from elements in S (φ) . In this paper, we determine the accuracy p from the coefficients d γ . Moreover, we obtain from our conditions, a characterization of accuracy for a particular lattice refinable vector function, which simplifies the classical conditions. [ABSTRACT FROM AUTHOR]

The first example of a Banach space with the approximation property but without the bounded approximation property was given by Figiel and Johnson in 1973. We give the first example of a Banach lattice with the approximation property but without the bounded approximation property. As a consequence, we prove the existence of an integral operator (in the sense of Grothendieck) on a Banach lattice which is not strictly integral. [ABSTRACT FROM AUTHOR]

In this article we prove three main theorems: (1) guessing models are internally unbounded, (2) for any regular cardinal κ ≥ ω2, ISP(κ) implies that SCH holds above κ, and (3) forcing posets which have the ω1- approximation property also have the countable covering property. These results solve open problems of Viale [Ann. Pure Appl. Logic 163 (2012), no. 11, 1660-1678] and Hachtman and Sinapova [J. Symb. Log. 84 (2019), no. 2, 713-725]. [ABSTRACT FROM AUTHOR]

Let Cbk,ω (Rn) be the Banach space of Ck functions on Rn bounded together with all derivatives of order ≤ k and with derivatives of order k having moduli of continuity O(ω) for some ω ∈ C(R+). Let Cbk,ω (S) := Cbk,ω (Rn)|S be the trace space to a closed subset S ⊂ Rn. The geometric predual Gbk,ω(S) of Cbk,ω (S) is the minimal closed subspace of the dual (Cbk,ω (Rn))∗ containing evaluation functionals of points in S. We study geometric properties of spaces Gbk,ω(S) and their relations to the classical Whitney problems on the characterization of trace spaces of Ck functions on Rn. In particular, we show that each Gbk,ω (S) is a complemented subspace of Gbk,ω(Rn), describe the structure of bounded linear operators on Gbk,ω (Rn), prove that Gbk,ω (S) has the bounded approximation property and that in some cases space Cbk,ω (S)is isomorphic to the second dual of its subspace consisting of restrictions to S of C∞(Rn) functions with compact supports. [ABSTRACT FROM AUTHOR]

We investigate the multilinear variants of the quantities which measure the noncompactness of multilinear operators taking values in Banach spaces with the uniform approximation property. We show applications to multilinear variant of the Hilbert and Riesz transform on rearrangement invariant spaces. We derive lower estimates of the essential norm of these transforms. Along the way we obtain also as a by-product the lower estimates of the measure noncompactness of these transforms. As a consequence we conclude that the Hilbert and Riesz transforms are not compact multilinear transforms. [ABSTRACT FROM AUTHOR]

In this paper we study the nuclearity and weak containment property of reduced cross-sectional C ⁎ -algebras of Fell bundles over inverse semigroups. In order to develop the theory, we first prove an analogue of Fell's absorption principle in the context of Fell bundles over inverse semigroups. In parallel, the approximation property of Exel can be reformulated in this context, and Fell's absorption principle can be used to prove that the approximation property, as defined here, implies that the full and reduced cross-sectional C ⁎ -algebras are isomorphic via the left regular representation, i.e., the Fell bundle has the weak containment property. We then use this machinery to prove that a Fell bundle with the approximation property and nuclear unit fiber has a nuclear cross-sectional C ⁎ -algebra. This result gives nuclearity of a large class of C ⁎ -algebras as, remarkably, all the machinery in this paper works for étale non-Hausdorff groupoids just as well. [ABSTRACT FROM AUTHOR]

In this paper, spaces of sequences in fuzzy normed spaces are considered. These spaces are a new concept in fuzzy normed spaces. We develop fuzzy norms for spaces of sequences in fuzzy normed spaces. Especially, we study the representation of the dual of a space of sequences in a fuzzy normed space. The approximation property in our context is investigated.

Liquid state machine (LSM), a bio-inspired computing model consisting of the input sparsely connected to a randomly interlinked reservoir (or liquid) of spiking neurons followed by a readout layer, finds utility in a range of applications varying from robot control and sequence generation to action, speech, and image recognition. LSMs stand out among other Recurrent Neural Network (RNN) architectures due to their simplistic structure and lower training complexity. Plethora of recent efforts have been focused toward mimicking certain characteristics of biological systems to enhance the performance of modern artificial neural networks. It has been shown that biological neurons are more likely to be connected to other neurons in the close proximity, and tend to be disconnected as the neurons are spatially far apart. Inspired by this, we propose a group of locally connected neuron reservoirs, or an ensemble of liquids approach, for LSMs. We analyze how the segmentation of a single large liquid to create an ensemble of multiple smaller liquids affects the latency and accuracy of an LSM. In our analysis, we quantify the ability of the proposed ensemble approach to provide an improved representation of the input using the Separation Property (SP) and Approximation Property (AP). Our results illustrate that the ensemble approach enhances class discrimination (quantified as the ratio between the SP and AP), leading to better accuracy in speech and image recognition tasks, when compared to a single large liquid. Furthermore, we obtain performance benefits in terms of improved inference time and reduced memory requirements, due to lowered number of connections and the freedom to parallelize the liquid evaluation process.