Banach Subalgebra Research Articles

Stability of relative essential spectra involving relative demicompactness concept in Banach subalgebra

This paper develops the notion of relative demicompact elements of an algebra with respect to a Banach subalgebra as a generalization of relative demicompact linear operators acting on Banach spaces. Drawing on this novel notion, we build a new class of Fredholm perturbation regarding a given Banach subalgebra B which contains its inessential ideal kB and the set of left Fredholm perturbations suggested in [6]. The developed class of Fredholm perturbation exhibits that is a two-sided closed ideal of B that is key in the characterization of the weyl spectrum of elements affiliated with B.

Demicompactness perturbation in Banach algebras and some stability results

Abstract In this paper, we introduce and investigate a new concept that we call demicompact elements in Banach algebras as a generalization of demicompact linear operators acting on Banach spaces. Our concept is used to construct a new class of Fredholm perturbations with respect to a given Banach subalgebra B, that contains an inessential ideal k B {k_{B}} and the set of left Fredholm perturbations introduced in [A. Ben Ali and N. Moalla, Fredholm perturbation theory and some essential spectra in Banach algebra with respect to subalgebra, Indag. Math. (N. S.) 28 2017, 2, 276–286]. We show that it is a two-sided closed ideal of B playing an important role in the characterization of the Weyl spectrum of elements affiliated with B.

Duality of Gabor frames and Heisenberg modules

Given a locally compact abelian group G and a closed subgroup \Lambda in G\times\hat{G} , Rieffel associated to \Lambda a Hilbert C^* -module \mathcal E , known as a Heisenberg module. He proved that \mathcal E is an equivalence bimodule between the twisted group C^* -algebra C^*(\Lambda, \mathsf{c}) and C^*(\Lambda^\circ,\bar{\mathsf{c}}) , where \Lambda^{\circ} denotes the adjoint subgroup of \Lambda . Our main goal is to study Heisenberg modules using tools from time-frequency analysis and pointing out that Heisenberg modules provide the natural setting of the duality theory of Gabor systems. More concretely, we show that the Feichtinger algebra \mathbb S_O(G) is an equivalence bimodule between the Banach subalgebras \mathbb S_O(\Lambda,\mathsf{c}) and \mathbb S_O(\Lambda^\circ,\bar{\mathsf{c}}) of C^*(\Lambda,\mathsf{c}) and C^*(\Lambda^\circ,\bar{\mathsf{c}}) , respectively. Further, we prove that \mathbb S_O(G) is finitely generated and projective exactly for co-compact closed subgroups \Lambda . In this case the generators g_1,\ldots,g_n of the left \mathbb S_O(\Lambda) -module \mathbb S_O(G) are the Gabor atoms of a multi-window Gabor frame for L^2(G) . We prove that this is equivalent to g_1,\ldots,g_n being a Gabor super frame for the closed subspace generated by the Gabor system for \Lambda^\circ . This duality principle is of independent interest and is also studied for infinitely many Gabor atoms. We also show that for any non-rational lattice \Lambda in \mathbb R^{2m} with volume s(\Lambda) < 1 there exists a Gabor frame generated by a single atom in \Lambda^\circ(\mathbb R^m) .

Banach algebras of Fourier multipliers equivalent at infinity to nice Fourier multipliers

Let \(\mathcal {M}_{X(\mathbb {R})}\) be the Banach algebra of all Fourier multipliers on a Banach function space \(X(\mathbb {R})\) such that the Hardy–Littlewood maximal operator is bounded on \(X(\mathbb {R})\) and on its associate space \(X'(\mathbb {R})\). For two sets \(\varPsi ,\varOmega \subset \mathcal {M}_{X(\mathbb {R})}\), let \(\varPsi _\varOmega\) be the set of those \(c\in \varPsi\) for which there exists \(d\in \varOmega\) such that the multiplier norm of \(\chi _{\mathbb {R}\setminus [-N,N]}(c-d)\) tends to zero as \(N\rightarrow \infty\). In this case, we say that the Fourier multiplier c is equivalent at infinity to the Fourier multiplier d. We show that if \(\varOmega\) is a unital Banach subalgebra of \(\mathcal {M}_{X(\mathbb {R})}\) consisting of nice Fourier multipliers (for instance, continuous or slowly oscillating in certain sense) and \(\varPsi\) is an arbitrary unital Banach subalgebra of \(\mathcal {M}_{X(\mathbb {R})}\), then \(\varPsi _\varOmega\) is a also a unital Banach subalgebra of \(\mathcal {M}_{X(\mathbb {R})}\).

Decomposition of the $$(n,\epsilon )$$-pseudospectrum of an element of a Banach algebra

Let A be a complex Banach algebra with unit. For an integer $$n\ge 0$$ and $$\epsilon >0$$, the $$(n,\epsilon )$$-pseudospectrum of $$a\in A$$ is defined by $$\begin{aligned} \varLambda _{n,\epsilon } (A,a):=\left\{ \lambda \in \mathbb {C}: (\lambda -a) \text { is not invertible in } A \text { or } \Vert (\lambda -a)^{-2^{n}}\Vert ^{1/2^n} \ge \frac{1}{\epsilon }\right\} . \end{aligned}$$Let $$p\in A$$ be a nontrivial idempotent. Then $$pAp=\{pbp:b\in A\}$$ is a Banach subalgebra of A with unit p, known as a reduced Banach algebra. Suppose $$ap=pa$$. We study the relationship of $$\varLambda _{n,\epsilon }(A,a)$$ and $$\varLambda _{n,\epsilon }(pAp,pa)$$. We extend this by considering first a finite family, and then an at most countable family of idempotents satisfying some conditions. We establish that under suitable assumptions, the $$(n,\epsilon )$$-pseudospectrum of a can be decomposed into the union of the $$(n,\epsilon )$$-pseudospectra of some elements in reduced Banach algebras.

Representation Theorems for Operators on Free Banach Spaces of Countable Type

This work will be centered in commutative Banach subalgebras of the algebra of bounded linear operators defined on free Banach spaces of countable type. The main goal of this work will be to formulate a representation theorem for these operators through integrals defined by spectral measures type. In order to get this objective, we will show that, under special conditions, each one of these algebras is isometrically isomorphic to some space of continuous functions defined over a compact set. Then, we will identify such compact sets developing the Gelfand space theory in the non-Archimedean setting. This fact will allow us to define a measure which is known as spectral measure. As a second goal, we will formulate a matrix representation theorem for this class of operators in which the entries of the matrices will be integrals coming from scalar measures.

Polynomial control on stability, inversion and powers of matrices on simple graphs

Spatially distributed networks of large size arise in a variety of science and engineering problems, such as wireless sensor networks and smart power grids. Most of their features can be described by properties of their state-space matrices whose entries have indices in the vertex set of a graph. In this paper, we introduce novel algebras of Beurling type that contain matrices on a connected simple graph having polynomial off-diagonal decay, and we show that they are Banach subalgebras of B(ℓp),1≤p≤∞, the space of all bounded operators on the space ℓp of all p-summable sequences. The ℓp-stability of state-space matrices is an essential hypothesis for the robustness of spatially distributed networks. In this paper, we establish the equivalence among ℓp-stabilities of matrices in Beurling algebras for different exponents 1≤p≤∞, with quantitative analysis for the lower stability bounds. Admission of norm-control inversion plays a crucial role in some engineering practice. In this paper, we prove that matrices in Beurling subalgebras of B(ℓ2) have norm-controlled inversion and we find a norm-controlled polynomial with close to optimal degree. Polynomial estimate to powers of matrices is important for numerical implementation of spatially distributed networks. In this paper, we apply our results on norm-controlled inversion to obtain a polynomial estimate to powers of matrices in Beurling algebras. The polynomial estimate is a noncommutative extension about convolution powers of a complex function and is applicable to estimate the probability of hopping from one agent to another agent in a stationary Markov chain on a spatially distributed network.

On the Classification of p-Adic UHF Banach Algebras

For any prime number p let Ωp be the p-adic counterpart of the complex numbers C. In this paper we investigate the class of p-adic UHF Banach algebras. A p-adic UHF Banach algebra is any unital p-adic Banach algebra A of the form $$A = \overline {U{M_n}} $$ , where (Mn) is an increasing sequence of p-adic Banach subalgebras of M such that each Mn is generated over Ωp by an algebraic system of matrix units {e ( ) | 1 ≤ i, j ≤ pn }. The main result is that the supernatural number associated to a p-adic TUHF Banach algebra is an invariant of the algebra.

Some properties of $$LUC({\mathcal X},{\mathcal G})^*$$ L U C ( X , G ) ∗ as a banach left $$LUC({\mathcal G})^*$$ L U C ( G ) ∗ -module

Associated with a locally compact group $$\mathcal G$$ and a $$\mathcal G$$ -space $$\mathcal X$$ there is a Banach subspace $$LUC({\mathcal X},{\mathcal G})$$ of $$C_b({\mathcal X})$$ , which has been introduced and studied by Chu and Lau (Math Z 268:649–673, 2011). In this paper, we study some properties of the first dual space of $$LUC({\mathcal X},{\mathcal G})$$ . In particular, we introduce a left action of $$LUC({\mathcal G})^*$$ on $$LUC({\mathcal X},{\mathcal G})^*$$ to make it a Banach left module and then we investigate the Banach subalgebra $${{\mathfrak {Z}}({\mathcal X},{\mathcal G})}$$ of $$LUC({\mathcal G})^*$$ , as the topological centre related to this module action, which contains $$M({\mathcal G})$$ as a closed subalgebra. Also, we show that the faithfulness of this module action is related to the properties of the action of $$\mathcal G$$ on $$\mathcal X$$ and we prove an analogue of the main result of Lau (Math Proc Cambridge Philos Soc 99:273–283, 1986) for $${\mathcal G}$$ -spaces. Sufficient and/or necessary conditions for the equality $${{\mathfrak {Z}}({\mathcal X},{\mathcal G})}=M({\mathcal G})$$ or $$LUC({\mathcal G})^*$$ are given. Finally, we apply our results to some special cases of $$\mathcal G$$ and $$\mathcal X$$ for obtaining various examples whose topological centres $${{\mathfrak {Z}}({\mathcal X},{\mathcal G})}$$ are $$M({\mathcal G})$$ , $$LUC({\mathcal G})^*$$ or neither of them.

Character Amenability of the Intersection of Lipschitz Algebras

AbstractLet (X, d) be ametric space and letJ⊆ [0,∞) be nonempty. We study the structure of the arbitrary intersections of Lipschitz algebras and define a special Banach subalgebra of ∩y∊JLipyX, denoted by ILipJX. Mainly, we investigate theC-character amenability of ILipJX, in particular Lipschitz algebras. We address a gap in the proof of a recent result in this field. Then we remove this gap and obtain a necessary and suõcient condition forC-character amenability of ILipJX, specially Lipschitz algebras, under an additional assumption.

A Gelfand–Naimark type theorem

Let X be a completely regular space. For a non-vanishing self-adjoint Banach subalgebra H of CB(X) which has local units we construct the spectrum sp(H) of H as an open subspace of the Stone–Čech compactification of X which contains X as a dense subspace. The construction of sp(H) is simple. This enables us to study certain properties of sp(H), among them are various compactness and connectedness properties. In particular, we find necessary and sufficient conditions in terms of either H or X under which sp(H) is connected, locally connected and pseudocompact, strongly zero-dimensional, basically disconnected, extremally disconnected, or an F-space.

Seminormed ⁎-subalgebras of ℓ∞(X)

Arbitrary representations of an involutive commutative unital F-algebra A as a subalgebra of FX are considered, where F=C or R and X≠∅. The Gelfand spectrum of A is explained as a topological extension of X where a seminorm on the image of A in FX is present. It is shown that among all seminorms, the sup-norm is of special importance which reduces FX to ℓ∞(X). The Banach subalgebra of ℓ∞(X) of all Σ-measurable bounded functions on X, Mb(X,Σ), is studied for which Σ is a σ-algebra of subsets of X. In particular, we study lifting of positive measures from (X,Σ) to the Gelfand spectrum of Mb(X,Σ) and observe an unexpected shift in the support of measures. In the case that Σ is the Borel algebra of a topology, we study the relation of the underlying topology of X and the topology of the Gelfand spectrum of Mb(X,Σ).

Dense Banach subalgebras of the null sequence algebra which do not satisfy a differential seminorm condition

We construct dense Banach subalgebras A of the null sequence algebra c0 which are spectral-invariant but do not satisfy the D1-condition ‖ab‖A≤K(‖a‖∞‖b‖A+‖a‖A‖b‖∞) for all a,b∈A. The sequences in A vanish in a skewed manner with respect to an unbounded function σ:N→[1,∞).

Inverse-closed Banach subalgebras of higher-dimensional non-commutative tori

Inverse-closed Banach subalgebras of higher-dimensional non-commutative tori

Rakotch type contractive maps, square roots and uniform convexity

In this paper we introduce a family of weakly contractive maps on the space $B(S)$ of bounded real valued functions and use it to show that a fundamental step in the proof of the well-known Stone-Weierstrass approximation theorem can be achieved via Rakotch’s fixed point theorem for weakly contractive maps. With the same technique, we obtain Zemanek’s theorem on the existence of square roots in certain Banach subalgebras of $B(S)$ , and, finally, in the context of abstract Banach algebras, we exhibit some relationship between weakly contractive maps on the closed unit ball and the geometry of the spheres.

Wiener’s lemma for singular integral operators of Bessel potential type

In this paper, we introduce an algebra of singular integral operators containing Bessel potentials of positive order, and show that the corresponding unital Banach algebra is an inverse-closed Banach subalgebra of \({\mathcal {B}} (L^q_w)\), the Banach algebra of all bounded operators on the weighted space \(L_w^q\), for all \(1\le q<\infty \) and Muckenhoupt \(A_q\)-weights \(w\).

Representation theorems for Banach algebras

For a space X denote by Cb(X) the Banach algebra of all continuous bounded scalar-valued functions on X and denote by C0(X) the set of all elements in Cb(X) which vanish at infinity.We prove that certain Banach subalgebras H of Cb(X) are isometrically isomorphic to C0(Y), for some unique (up to homeomorphism) locally compact Hausdorff space Y. The space Y is explicitly constructed as a subspace of the Stone–Čech compactification of X. The known construction of Y enables us to examine certain properties of either H or Y and derive results not expected to be deducible from the standard Gelfand Theory.

On extensions of [formula omitted]-valued operators

We study pairs of Banach spaces (X,Y), with Y⊂X, for which the thesis of Sobczyk’s theorem holds, namely, such that every bounded c0-valued operator defined in Y extends to X. We are mainly concerned with the case when X is a C(K) space and Y≡C(L) is a Banach subalgebra of C(K). The main result of the article states that, if K is a compact line and L is countable, then every bounded c0-valued operator defined in C(L) extends to C(K).

FOLDING AND UNFOLDING QUANTUM STATES

The reduction of a quantum system ("folding" a quantum system) is described as the reduction of its Lie–Jordan Banach algebra of observables with respect to Lie–Jordan Banach subalgebras and Jordan ideals. The space of states of the reduced Lie–Jordan Banach algebra is described in terms of unreduced states ("unfolding" states) as well as the GNS construction. A few examples are discussed.

Boundary representations on co-invariant subspaces of Bergman space

Let M be an invariant subspace of the Bergman space L 2 a (D) and S M be the compression of the coordinate multiplication operator M z to the co-invariant subspace L 2 a (D) ⊖ M. The present paper determines when the identity representation of C* (S M ) is a boundary representation for the Banach subalgebra B(S M ). The paper also considers boundary representations on the co-invariant subspaces of L 2 a (B n ).