Banach Algebra

The main theme of this paper can be described as a study of the Drazin inverse for bounded linear operators in a Banach space X when 0 is an isolated spectral point ofthe operator. This inverse is useful for instance in the solution of differential equations formulated in a Banach space X. Since the elements of X rarely enter into our considerations, the exposition seems to gain in clarity when the operators are regarded as elements of the Banach algebra L(X).

Additive perturbation results for the generalized Drazin inverse of Banach space operators are presented. Precisely, if A d denotes the generalized Drazin inverse of a bounded linear operator A on an arbitrary complex Banach space, then in some special cases (A + B) d is computed in terms of A d and B d . Thus, recent results of R. E. Hartwig, G. Wang and Y. Wei are extended to infinite dimensional settings with simplified proof. 2000 Mathematics subject classification: 47A05, 47A55, 15A09.

This paper studies additive properties of the generalized Drazin inverse (g-Drazin inverse) in a Banach algebra and finds an explicit expression for the g-Drazin inverse of the sum a + b in terms of a and b and their g-Drazin inverses under fairly mild conditions on a and b.

We generalize the well-known Baker's superstability result for exponential mappings with values in the field of complex numbers to the case of an arbitrary commutative complex semisimple Banach algebra. It was shown by Ger that the superstability phenomenon disappears if we formulate the stability question for exponential complex-valued functions in a more natural way. We improve his result by showing that the maximal possible distance of an ε-approximately exponential function to the set of all exponential functions tends to zero as ε tends to zero. In order to get this result we have to prove a stability theorem for real-valued functions additive modulo the set of all integers Z.

Several features of an analytic (infinite-dimensional) Grassmannian of (commensurable) subspaces of a Hilbert space were developed in the context of integrable PDEs (KP hierarchy). We extended some of those features when polarized separable Hilbert spaces are generalized to a class of polarized Hilbert modules, in particular the Baker and τ-functions, which become operator-valued. Following from Part I we produce a pre-determinant structure for a class of τ-functions defined in the setting of the similarity class of projections of a certain Banach *algebra. This structure is explicitly derived from the transition map of a corresponding principal bundle. The determinant of this map gives a generalized, operator-valued τ-function that takes values in a commutative C*-algebra. We extend to this setting the operator cross-ratio which had been used to produce the scalar-valued τ-function, as well as the associated notion of a Schwarzian derivative along curves inside the space of similarity classes. We link directly this cross-ratio with Fay's trisecant identity for the τ-function (equivalent to the KP hierarchy). By * Partial research support under grant NSF-DMS-0808708 is very gratefully acknowledged. 1 restriction to the image of the Krichever map, we use the Schwarzian to introduce the notion of operator-valued projective structure on a compact Riemann surface: this allows a deformation inside the Grassmannian (as it varies its complex structure). Lastly, we use our identification of the Jacobian of the Riemann surface in terms of extensions of the Burchnall-Chaundy C*-algebra (Part I) to describe the KP hierarchy.

Groups of unbounded operators are approached in the setting of the Esterle quasimultiplier theory. We introduce groups of regular quasimultipliers of growth ω, or ω-groups for short, where ω is a continuous weight on the real line. We study the relationship of ω-groups with families of operators and homomorphisms such as regularized, distribution and integrated groups, holomorphic semigroups, and functional calculi. Some convolution Banach algebras of functions with derivatives to fractional order are needed, which we construct using the Weyl fractional calculus.

We study the Arens regularity of module actions of Banach left or right modules over Banach algebras. We prove that if A has a brai (blai), then the right (left) module action of A on A * is Arens regular if and only if A is reflexive. We find that Arens regularity is implied by the factorization of A * or A * * when A is a left or a right ideal in A * *. The Arens regularity and strong irregularity of A are related to those of the module actions of A on the nth dual A (n) of A. Banach algebras A for which Z(A * *) = A but A Z t (A * *) are found (here Z(A * *) and Z t (A * *) are the topological centres of A * * with respect to the first and second Arens product, respectively). This also gives examples of Banach algebras such that A Z(A * *) A * *. Finally, the triangular Banach algebras T are used to find Banach algebras having the following properties: (i) T * T = T T * but Z(T * *) = Z t (T * *); (ii) Z(T * *) = Z t (T * *) and T * T = T * but T T * = T * ; (iii) Z(T * *) = T but T is not weakly sequentially complete. The results (ii) and (iii) are new examples answering questions asked by Lau andÜlger.

We will show that an uniform treatment yields Wiener–Tauberian type results for various Banach algebras and modules consisting of radial sections of some homogenous vector bundles on rank one Riemannian symmetric spaces G/K of noncompact type. One example of such a vector bundle is the spinor bundle. The algebras and modules we consider are natural generalizations of the commutative Banach

We consider the topological space of all composition operators on the Banach algebra of bounded analytic functions on the unit disk. We obtain a function theoretic characterization of isolated points and show that each isolated composition operator is essentially isolated.

In the present investigation we link noncommutative geometry over noncommutative tori with Gabor analysis, where the first has its roots in operator algebras and the second in time-frequency analysis. We are therefore in the position to invoke modern methods of operator algebras, e.g. topological stable rank of Banach algebras, to display the deeper properties of Gabor frames. Furthermore, we are able to extend results due to Connes and Rieffel on projective modules over noncommutative tori to Banach algebras, which arise in a natural manner in Gabor analysis. The main goal of this investigation is twofold: (i) an interpretation of projective modules over noncommutative tori in terms of Gabor analysis and (ii) to show that the Morita-Rieffel equivalence between noncommutative tori is the natural framework for the duality theory of Gabor frames. More concretely, we interpret generators of projective modules over noncommutative tori as the Gabor atoms of multi-window Gabor frames for modulation spaces. Moreover, we show that this implies the existence of good multi-window Gabor frames for modulation spaces with Gabor atoms in e.g. Feichtinger's algebra or in Schwartz space.

We investigate the stability of Pexiderized mappings in Banach modules over a unital Banach algebra. As a consequence, we establish the Hyers-Ulam stability of the orthogonal Cauchy functional equation of Pexider type f 1 (x + y) = f 2 (x) + f 3 (y), x ⊥ y in which ⊥ is the orthogonality in the sense of Rätz. * 2000 Mathematics Subject Classification. Primary 39B52, secondary 39B82, 46H25.

We present a new method for constructing C 0 -semigroups for which properties of the resolvent of the generator and continuity properties in operator topology are controlled simultaneously. It allows us to show that a) there exists a C 0 -semigroup continuous in operator topology for no t ∈ [0, 1] such that the resolvent of its generator has a logarithmic decay at infinity along vertical lines; b) there exists a C 0 -semigroup continuous in operator topology for no t ∈ R + such that the resolvent of its generator has a decay along vertical lines arbitrarily close to a logarithmic one. These examples rule out any possibility of characterizing norm-continuity of semigroups on arbitrary Banach spaces in terms of resolvent decay.

We investigate the extent to which the study of quasimultipliers can be made beyond Banach algebras. We will focus mainly on the class of F-algebras, in particular on complete k-normed algebras, 0 < k ≤ 1, not necessarily locally convex. We include a few counterexamples to demonstrate that some of our results do not carry over to general F-algebras. The bilinearity and joint continuity of quasimultipliers on an F-algebra A are obtained under the assumption of strong factorability. Further, we establish several properties of the strict and quasistrict topologies on the algebra QM A of quasimultipliers of a complete k-normed algebra A having a minimal ultraapproximate identity.

Let S be an inverse semigroup with an upward directed set of idempotents E. In this paper we define the module topological center of second dual of a Banach algebra which is a Banach module over another Banach algebra with compatible actions, and find it for ℓ 1 (S) * * (as an ℓ 1 (E)-module). We also prove that ℓ 1 (S) * * is ℓ 1 (E)-module amenable if and only if an appropriate group homomorphic image of S is finite.

For locally compact groups, Fourier algebras and Fourier-Stieltjes algebras have proved to be useful dual objects. They encode the representation theory of the group via the positive de nite functions on the group: positive de nite functions correspond to cyclic representations and span these algebras as linear spaces. They encode information about the algebra of the group in the geometry of the Banach space structure, and the group appears as a topological subspace of the maximal ideal space of the algebra W1, W2]. Because groupoids and their representations appear in studying operator algebras, ergodic theory, geometry, and the representation theory of groups, it would be useful to have a duality theory for them. This paper gives a rst step toward extending the theory of Fourier-Stieltjes algebras from groups to groupoids. If G is a locally compact (second countable) groupoid, we show that B(G), the linear span of the Borel positive de nite functions on G, is a Banach algebra when represented as an algebra of completely bounded maps on a C -algebra associated with G. This necesarily involves identifying equivalent elements of B(G). An example shows that the linear span of the continuous positive de nite functions need not be complete. For groups, B(G) is isometric to the Banach space dual of C (G). For groupoids, the best analog of that fact is to be found in a representation of B(G) as a Banach space of completely bounded maps from a C -algebra associated with G to a C -algebra associated with the equivalence relation induced by G. This paper adds weight to the clues in the earlier study of Fourier-Stieltjes algebras that there is a much more general kind of duality for Banach algebras waiting to be explored W5].

We develop a theory of almost periodic elements in Banach algebras and present an abstract version of a noncommutative Wiener's Lemma. The theory can be used, for example, to derive some of the recently obtained results in time-frequency analysis such as the spectral properties of the finite linear combinations of time-frequency shifts.

The purpose of this paper is to establish some results concerning generalized left derivations in rings and Banach algebras. In fact, we prove the following results: Let R be a 2-torsion free semiprime ring, and let G : R −→ R be a generalized Jordan left derivation with associated Jordan left derivation δ : R −→ R. Then every generalized Jordan left derivation is a generalized left derivation on R. This result gives an affirmative answer to the question posed as a remark in Ashraf and Ali (Bull. Korean Math. Soc. 45:253-261, 2008). Also, the study of generalized left derivation has been made which acts as a homomorphism or as an anti-homomorphism on some appropriate subset of the ring R. Further, we introduce the notion of generalized left bi-derivation and prove that if a prime ring R admits a generalized left bi-derivation G with associated left bi-derivation B then either R is commutative or G is a right bi-centralizer(or bi-multiplier) on R. Finally, it is shown that every generalized Jordan left derivation on a semisimple Banach algebra is continuous.

The work of the Brussels-Austin groups on irreversibility over the last years has shown that Quantum Large Poincard systems with diagonal s#lgulariO, lead to an extension of the conventional formulation of dynamics at the level of mixtures which is manifestly time asymmetric. States with diagonal singularity acquire meaning as linear fractionals over the involutive Banach algebra of operators with diagonal shlgularity. We show in this paper that the logic of quantum systems with diagonal singularity is not the conventional logic of Hilbert space, because only finite combinations of prepositions are allowed.

The paper introduces and studies the weighted g-Drazin inverse for bounded linear operators between Banach spaces, extending the concept of the weighted Drazin inverse of Rakočević and Wei (Linear Algebra Appl. 350 (2002), 25–39) and of Cline and Greville (Linear Algebra Appl. 29 (1980), 53–62). We use the Mbekhta decomposition to study the structure of an operator possessing the weighted g-Drazin inverse, give an operator matrix representation for the inverse, and study its continuity. An open problem of Rakočević and Wei is solved.

We study multiplier algebras for a large class of Banach algebras which contains the group algebra L 1 (G), the Beurling algebras L 1 (G, ω), and the Fourier algebra A(G) of a locally compact group G. This study yields numerous new results and unifies some existing theorems on L 1 (G) and A(G) through an abstract Banach algebraic approach. Applications are obtained on representations of multipliers over locally compact quantum groups and on topological centre problems. In particular, five open problems in abstract harmonic analysis are solved.

We study the set S = {(a, b) ∈ A × A : aba = a, bab = b} which pairs the relatively regular elements of a Banach algebra A with their pseudoinverses, and prove that it is an analytic submanifold of A × A. If A is a C * -algebra, inside S lies a copy the set I of partial isometries, we prove that this set is a C ∞ submanifold of S (as well as a submanifold of A). These manifolds carry actions from, respectively, GA × GA and UA × UA, where GA is the group of invertibles of A and UA is the subgroup of unitary elements. These actions define homogeneous reductive structures for S and I (in the differential geometric sense). Certain topological and homotopical properties of these sets are derived. In particular, it is shown that if A is a von Neumann algebra and p is a purely infinite projection of A, then the connected component Ip of p in I is simply connected. If 1 − p is also purely infinite, then Ip is contractible. This paper contains a study of the differential geometry of the set S of all pairs (a, b) of elements of a Banach algebra A such that b (resp. a) is a generalized inverse of a (resp. b), and of the set I of all partial isometries of a C*-algebra A. It can be seen as a continuation of for the Banach algebra setting, and of [25] and [26] for the C*-algebra setting. The first appearences of generalized inverse methods in analysis go back to Fredholm, Hilbert and Hurwitz. In an explicit form, it was E. H. Moore who first considered what today is called de Moore-Penrose inverse. It was only after 1950, when Bjerhammar [10], and Penrose [29]-[30] rediscovered the Moore-Penrose pseudoinverse of a matrix, that the subject took great impulse (see the recent paper [9] by A. Ben Israel for an account of this story). Its popularity came from the extremely vast and diversified field of its applications in many scientific disciplines. The reader is referred to the book [27], edited by M. Z. Nashed, which contains many developments, applications and a list of 1776 references related to generalized inverses.

Substituting the usual growth condition by an assumption that a specific initial value problem has a maximal solution, we obtain existence results for functional nonlinear integral equations with variable delay. Application of the technique to initial value problems for differential equations as well as to integrodifferential equations are given.

It is known that a Banach algebra A inherits amenability from its second Banach dual A * *. No example is yet known whether this fails if one considers the weak amenability instead, but the property is known to hold for the group algebra L 1 (G), the Fourier algebra A(G) when G is amenable, the Banach algebras A which are left ideals in A * * , the dual Banach algebras, and the Banach algebras A which are Arens regular and have every derivation from A into A * weakly compact. In this paper, we extend this class of algebras to the Banach algebras for which the second adjoint of each derivation D : A → A * satisfies D ′′ (A * *) ⊆ WAP(A), the Banach algebras A which are right ideals in A * * and satisfy A * * A = A * * , and to the Figà-Talamanca-Herz algebra A p (G) for G amenable. We also provide a short proof of the interesting recent criterion on when the second adjoint of a derivation is again a derivation.

Suppose A is a dual Banach algebra, and a representation π : A → B( 2 ) is unital, weak * continuous, and contractive. We use a "Hilbert-Schmidt version" of Arveson Distance Formula to construct an operator space X, isometric to 2 ⊗ 2 , such that the space of completely bounded maps on X consists of Hilbert-Schmidt perturbations of π(A) ⊗ I 2 . This allows us to establish the existence of operator spaces with various interesting properties. For instance, we construct an operator space X for which the group K 1 (CB(X)) contains Z 2 as a subgroup, and a completely indecomposable operator space containing an infinite dimensional homogeneous Hilbertian subspace.

For an open subset U of a locally convex space E; let ðHðUÞ; t 0 Þ denote the vector space of all holomorphic functions on U; with the compact-open topology. If E is a separable Fre´chet space with the bounded approximation property, or if E is a (DFC)-space with the approximation property, we show that ðHðUÞ; t 0 Þ has the approximation property for every open subset U of E: These theorems extend classical results of Aron and Schottenloher. As applications of these approximation theorems we characterize the spectra of certain topological algebras of holomorphic mappings with values in a Banach algebra. r

In this paper, two results concerning the global attractivity and global asymptotic attractivity of the solutions for a nonlinear functional integral equation are proved via a variant of the Krasnoselskii fixed point theorem due to Dhage [B.C. Dhage, A fixed point theorem in Banach algebras with applications to functional integral equations, Kyungpook Math. J. 44 (2004) 145-155]. The investigations are placed in the Banach space of real functions defined, continuous and bounded on an unbounded interval. A couple of examples are indicated for demonstrating the natural realizations of the abstract results presented in the paper. Our results generalize the attractivity results of Banas and Rzepka [J. Banas, B. Rzepka, An application of measures of noncompactness in the study of asymptotic stability,

We extend the concept of Arens regularity of a Banach algebra A to the case that there is an O-module structure on A, and show that when S is an inverse semigroup with totally ordered subsemigroup E of idempotents, then A = 1 (S) is module Arens regular if and only if an appropriate group homomorphic image of S is finite. When S is a discrete group, this is just Young's theorem which asserts that 1 (S) is Arens regular if and only if S is finite.

We introduce the class of operators on Banach spaces having property (H) and study Weyl's theorems, and related results for operators which satisfy this property. We show that a-Weyl's theorem holds for every decomposable operator having property (H). We also show that a-Weyl's theorem holds for every multiplier T of a commutative semi-simple regular Tauberian Banach algebra. In particular every convolution operator Tµ of a group algebra L 1 (G), G a locally compact abelian group, satisfies a-Weyl's theorem. Similar results are given for multipliers of other important commutative Banach algebras. . Primary 47A10, 47A11; Secondary 47A53, 47A55.

We study the weak module amenability of Banach algebras which are Banach module over another Banach algebra with compatible actions. As an example we show that the semigroup algebra of an inverse semigroup is always weakly amenable as a module over the semigroup algebra of its subsemigroup of idempotents.

In this paper, some new fixed point theorems concerning the nonlinear alternative of Leray-Schauder type are proved in a Banach algebra. Applications are given to nonlinear functional integral equations in Banach algebras for proving the existence results. Our results of this paper comple- ment the results that appear in Granas et. al. (12) and Dhage and Regan (10).

The purpose of this paper is to derive an integral representation of the Drazin inverse of an element of a Banach algebra in a more general situation than previously obtained by the second author, and to give an application to the Moore–Penrose inverse in a $C^*$-algebra.AMS 2000 Mathematics subject classification:Primary 46H05; 46L05