Structure Of Banach Algebras Research Articles

Centralizer-Like Additive Maps on the Lie Structure of Banach Algebras

Centralizer-Like Additive Maps on the Lie Structure of Banach Algebras

On some questions for the q-integration operator

We use the q-Duhamel product to provide a Banach algebra structure to some closed subspaces of the Wiener disk- algebra $$W_{+}\left( \mathbb {D}\right) $$ of analytic functions on the unit disk $$\mathbb {D}$$ of the complex plane $$\mathbb {C.}$$ We study the q-integration operator on $$W_{+}\left( \mathbb {D}\right) ,$$ namely, we characterize invariant subspaces of this operator and describe its extended eigenvalues and extended eigenvectors. Moreover, we prove an addition formula for the spectral multiplicity of the direct sum of q-integration operator on $$W_{+}\left( \mathbb {D}\right) $$ and some Banach space operator.

A Banach algebra structure on the $q$-Bergman space and related topics

A Banach algebra structure on the $q$-Bergman space and related topics

Dual Convolution for the Affine Group of the Real Line

The Fourier algebra of the affine group of the real line has a natural identification, as a Banach space, with the space of trace-class operators on L^2({{mathbb {R}}}^times , dt/ |t|). In this paper we study the “dual convolution product” of trace-class operators that corresponds to pointwise product in the Fourier algebra. Answering a question raised in work of Eymard and Terp, we provide an intrinsic description of this operation which does not rely on the identification with the Fourier algebra, and obtain a similar result for the connected component of this affine group. In both cases we construct explicit derivations on the corresponding Banach algebras, verifying the derivation identity directly without requiring the inverse Fourier transform. We also initiate the study of the analogous Banach algebra structure for trace-class operators on L^p({{mathbb {R}}}^times , dt/ |t|) for pin (1,2)cup (2,infty ).

Banach algebra structure on strongly simple extensions

Banach algebra structure on strongly simple extensions

Independent increment processes: a multilinearity preserving property

ABSTRACT We observe a multilinearity preserving property of conditional expectation for infinite-dimensional independent increment processes defined on some abstract Banach space B. It is similar in nature to the polynomial preserving property analysed greatly for finite-dimensional stochastic processes and thus offers an infinite-dimensional generalization. However, while polynomials are defined using the multiplication operator and as such require a Banach algebra structure, the multilinearity preserving property we prove here holds even for processes defined on a Banach space which is not necessarily a Banach algebra. In the special case of B being a commutative Banach algebra, we show that independent increment processes are polynomial processes in a sense that coincides with a canonical extension of polynomial processes from the finite-dimensional case. The assumption of commutativity is shown to be crucial and in a non-commutative Banach algebra the multilinearity concept arises naturally. Some of our results hold beyond independent increment processes and thus shed light on infinite-dimensional polynomial processes in general.

On the uniform Roe algebra as a Banach algebra and embeddings of ℓp uniform Roe algebras

We work on $\ell_p$ uniform Roe algebras associated to metric spaces, and on their mutual embedding. We generalize results of I. Farah and the authors to mutual embeddings of uniform Roe algebras of operators on $\ell_p$ spaces. Simultaneously, we obtain rigidity results for the classic uniform Roe $\mathrm{C}^*$-algebras which depend only on their Banach algebra structure.

On the Banach algebra structure for C(n) of the bidisc and related topics

Let C(n)=C(n)(D×D) be a Banach space of complex valued functions f(x,y) that are continuous on the closed bidisc D×D¯, where D={z∈C:|z|<1} is the unit disc in the complex plane C and has nth partial derivatives in D×D which can be extended to functions continuous on D×D¯. The Duhamel product is defined on C(n) by the formula (f⊛g)(z,w)=∂2∂z∂w∫0z∫0wf(z−u,w−v)g(u,v)dvdu. In the present paper we prove that C(n)(D×D) is a Banach algebra with respect to the Duhamel product ⊛. This result extends some known results. We also investigate the structure of the set of all extended eigenvalues and extended eigenvectors of some double integration operator Wzw. In particular, the commutant of the double integration operator Wzw is also described.

Characterization of $K$-frame vectors and $K$-frame generator multipliers

‎‎Let $\mathcal{U}$ be a unitary system and let $\mathcal{B(U)}$ be the Bessel vector space for $\mathcal{U}$‎. ‎In this article‎, ‎we give a characterization of Bessel vector spaces and local commutant spaces at different complete frame vectors‎. ‎The relation between local commutant spaces at different complete frame vectors is investigated‎. ‎Moreover‎, ‎by introducing multiplication and adjoint on the Bessel vector space for a unital semigroup of unitary operators‎, ‎we give a $C^*$-algebra structure to $\mathcal{B(U)}$‎. ‎Then‎, ‎we construct some subsets of $K$-frame vectors that have a Banach space or Banach algebra structure‎. ‎Also‎, ‎as a consequence‎, ‎the set of complete frame vectors for different unitary systems contains Banach spaces or Banach algebras‎. ‎In the end‎, ‎we give several characterizations of $K$-frame generator multipliers and Parseval $K$-frame generator multipliers‎.

Spectrally reasonable measures

In this paper we investigate the problems related to measures with a natural spectrum (equal to the closure of the set of the values of the Fourier-Stieltjes transform). Since it is known that the set of all such measures does not have a Banach algebra structure we consider the set of all suitable perturbations called spectrally reasonable measures. In particular, we exhibit a broad class of spectrally reasonable measures which contains absolutely continuous ones. On the other hand, we show that except trivial cases all discrete (purely atomic) measures do not posses this property.

Cyclic Convolution Operators on the Hardy Spaces

Using Banach algebra structure of the Hardy space, we describe all finite codimensional invariant subspaces of a cyclic convolution operator on the Hardy space $H^p$ of the unit disc for $1 \leq p \leq \infty$. We also observe that every operator in the commutant of such operators is not weakly supercyclic.

Basisity problem and weighted shift operators

We investigate a basisity problem in the space ℒAp(D) and in its invariant subspaces. Namely, let W denote a unilateral weighted shift operator acting in the space ℒAp(D), 1≤ p∞, by Wzn=λnzn+1,n≥0, with respect to the standard basis {zn}n≥0. Applying the so-called “discrete Duhamel product” technique, it is proven that for any integer k≥1 the sequence {(wi+nk)−1(W|Ei)knf}n≥0 is a basic sequence in Ei:=span {zi+n:n≥0} equivalent to the basis {zi+n}n≥0 if and only if fˆ(i)≠0. We also investigate a Banach algebra structure for the subspaces Ei,i≥0.

$m$-infrabarrelledness and $m$-convexity

$m$-infrabarrelledness, in the context of locally convex algebras, is considered to prove results previously obtained for barrelled algebras. Thus, any unital commutative $m$-infrabarrelled advertibly complete and pseudo-complete locally $m$-convex algebra with bounded elements has the $Q$-property; hence, it is functionally continuous (: all characters are continuous). In the framework of commutative $GB^{\ast }$-algebras with jointly continuous multiplication and bounded elements, the notions {\em $m$-infrabarrelled algebra} and {\em $C^{\ast }$-algebra} coincide. In unital uniform locally $m$-convex algebras, $m$-infrabarrelledness is equivalent to the Banach algebra structure, modulo pseudo-completeness. Moreover, $m$-infrabarrelledness for locally $A$-convex algebras (in particular, $A$-normed ones) is also examined.

Natural symmetric tensor norms

In the spirit of the work of Grothendieck, we introduce and study natural symmetric n-fold tensor norms. These are norms obtained from the projective norm by some natural operations. We prove that there are exactly six natural symmetric tensor norms for n ⩾ 3 , a noteworthy difference with the 2-fold case in which there are four. We also describe the polynomial ideals associated to these natural symmetric tensor norms. Using a symmetric version of a result of Carne, we establish which natural symmetric tensor norms preserve the Banach algebra structure.

The Banach algebras generated by operators with one-point spectrum

In this article, we present some results related to the structure of Banach algebras generated by operators with one-point spectrum.

On the Relationship between Interpolation of Banach Algebras and Interpolation of Bilinear Operators

AbstractWe show that if the general real method (· , ·)Γ preserves the Banach-algebra structure, then a bilinear interpolation theorem holds for (· , ·)Γ.

An amalgamation of the Banach spaces associated with James and Schreier, Part II: Banach-algebra structure

We create a new family of Banach spaces, the James‐Schreier spaces, by amalgamating two important classical Banach spaces: James’ quasi-reflexive Banach space on the one hand and Schreier’s Banach space giving a counterexample to the Banach‐Saks property on the other. We then investigate the properties of these James‐Schreier spaces, paying particular attention to how key properties of their ‘ancestors’ (that is, the James space and the Schreier space) are expressed in them. Our main results include that each James‐Schreier space is c0-saturated and that no James‐Schreier space embeds in a Banach space with an unconditional basis. 2010 Mathematics Subject Classification: Primary 46B45; Secondary 46B03, 47B37.

On interpolation of Banach algebras and factorization of weakly compact homomorphisms

We show a necessary and sufficient condition on the lattice Γ for the general real method ( ⋅ , ⋅ ) Γ to preserve the Banach-algebra structure. As an application we derive factorization of weakly compact homomorphisms through interpolation properties of weakly compact operators.

On the structure of Banach algebras associated with automorphisms

In the present paper we study the structure of a Banach algebra B(A, T_g) generated by a certain Banach algebra $A$ of operators acting in a Banach space D and a group {T_g}_{g \in G} of isometries of D such that T_g A T^{-1}_g = A. We investigate the interrelations between the existence of the expectation of B(A, T_g) onto A, topological freedom of the automorphisms of A induced by T_g and the dual action of the group G on B(A, T_g). The results obtained are applied to the description of the structure of Banach algebras generated by 'weighted composition operators' acting in various spaces.

A Banach algebra structure for the Wiener algebra of the disc

We prove that the Wiener disc algebra of all holomorphic functions in the unit disc is a Banach algebra with respect to the Duhamel product We also give applications of the Duhamel product in description of cyclic vectors of the convolution operators and commutant of Volterra integration operator.