Banach Algebra

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.

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.

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.

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.

The Banach space $E$ has the weakly compact approximation property (W.A.P. for short) if there is a constant $C < \infty$ so that for any weakly compact set $D \subset E$ and $\epsilon > 0$ there is a weakly compact operator $V: E \to E$ satisfying $\sup_{x\in D} || x - Vx || < \epsilon$ and $|| V|| \leq C$. We give several examples of Banach spaces both with and without this approximation property. Our main results demonstrate that the James-type spaces from a general class of quasi-reflexive spaces (which contains the classical James' space $J$) have the W.A.P, but that James' tree space $JT$ fails to have the W.A.P. It is also shown that the dual $J^*$ has the W.A.P. It follows that the Banach algebras $W(J)$ and $W(J^*)$, consisting of the weakly compact operators, have bounded left approximate identities. Among the other results we obtain a concrete Banach space $Y$ so that $Y$ fails to have the W.A.P., but $Y$ has this approximation property without the uniform bound $C$.

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.

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).

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.

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 spectral sets of elements of a Banach algebra in terms of the existence of idempotent partitions of the unit of the algebra. Circularly isolated spectral sets are characterized, and a new Mbekhta decomposition for bounded linear operators is obtained.

We extend the concept of Arens regularity of a Banach algebra \(\mathcal{A}\) to the case that there is an \(\mathcal{O}\) -module structure on \(\mathcal{A}\) , and show that when S is an inverse semigroup with totally ordered subsemigroup E of idempotents, then A=ℓ 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 ℓ 1(S) is Arens regular if and only if S is finite.

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

Let $A$ be a unital commutative Banach algebra with maximal ideal space $X.$ We determine the rational H-type of the group $GL_n (A)$ of invertible n by n matrices with coefficients in A, in terms of the rational cohomology of $X.$ We also address an old problem of J. L. Taylor. Let $Lc_n (A)$ denote the space of "last columns" of $GL_n (A).$ For $n > 1 + s/2,$ we construct a natural isomorphism from the rational Cech cohomology group $H^s (X; Q)$ to the rational homotopy group $\pi_{2 n - 1 - s} (Lc_n (A)) \otimes Q,$ which shows that the rational cohomology groups of $X$ are determined by a topological invariant associated to $A.$ As part of our analysis, we determine the rational H-type of certain gauge groups $F (X, G)$ for $G$ a Lie group or, more generally, a rational H-space.

Let G be a locally compact group, A(G) its Fourier algebra and L1(G) the space of Haar integrable functions on G. We study the Segal algebra SA(G)=A(G)\cap L1(G) in A(G). It admits an operator space structure which makes it a completely contractive Banach algebra. We compute the dual space of SA(G). We use it show that restriction operator u|->u|H:SA(G)->A(H), for some non-open closed subgroups H, is a surjective complete qutient map. We also show that if N is a non-compact closed subgroup, then the averaging operator tau_N:SA(G)->L1(G/N), tau_N u(sN)=\int_N u(sn)dn is a surjective complete quotient map. This puts an operator space perspective on the philosophy that SA(G) is ``locally A(G) while globally L1''. Also, using the operator space structure we can show that SA(G) is operator amenable exactly when when G is compact; and we can show that it is always operator weakly amenable. To obtain the latter fact, we use E. Samei's theory of hyper-Tauberian Banach algebras.

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.

The paper is devoted to study of singular integral operators with fixed singularities at endpoints of contours on weighted Lebesgue spaces with general Muckenhoupt weights. Compactness of certain integral operators with fixed singularities is established. The membership of singular integral operators with fixed singularities to Banach algebras of singular integral operators on weighted Lebesgue spaces with slowly oscillating Muckenhoupt weights is proved on the basis of Balakrishnan's formula from the theory of strongly continuous semi-groups of closed linear operators. Symbol calculus for such operators, Fredholm criteria and index formulas are obtained.

The Banach space ℓ 1 (Z) admits many non-isomorphic preduals, for example, C(K) for any compact countable space K, along with many more exotic Banach spaces. In this paper, we impose an extra condition: the predual must make the bilateral shift on ℓ 1 (Z) weak *continuous. This is equivalent to making the natural convolution multiplication on ℓ 1 (Z) separately weak *-continuous and so turning ℓ 1 (Z) into a dual Banach algebra. We call such preduals shift-invariant. It is known that the only shift-invariant predual arising from the standard duality between C 0 (K) (for countable locally compact K) and ℓ 1 (Z) is c 0 (Z). We provide an explicit construction of an uncountable family of distinct preduals which do make the bilateral shift weak *-continuous. Using Szlenk index arguments, we show that merely as Banach spaces, these are all isomorphic to c 0. We then build some theory to study such preduals, showing that they arise from certain semigroup compactifications of Z. This allows us to produce a large number of other examples, including non-isometric preduals, and preduals which are not Banach space isomorphic to c 0. * Supported by NSF grants DMS0856148 and DMS0556013.

A surjective bounded homomorphism fails to preserve $n$-weak amenability, in general. We however show that it preserves the property if the involved homomorphism enjoys a right inverse. We examine this fact for certain homomorphisms on several Banach algebras.

We study weakly compact left and right multipliers on the Banach algebra L ∞ 0 (G) * of a locally compact group G. We prove that G is compact if and only if L ∞ 0 (G) * has either a non-zero weakly compact left multiplier or a certain weakly compact right multiplier on L ∞ 0 (G) *. We also give a description of weakly compact multipliers on L ∞ 0 (G) * in terms of weakly completely continuous elements of L ∞ 0 (G) *. Finally we show that G is finite if and only if there exists a multiplicative linear functional n on L ∞ 0 (G) such that n is a weakly completely continuous element of L ∞ 0 (G) * .

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.

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 a commutative inverse semigroup is always weakly amenable as a module over the semigroup algebra of its subsemigroup of idempotents.

A logarithmic residue is a contour integral of a logarithmic derivative (left or right) of an analytic Banach algebra valued function. For functions possessing a meromorphic inverse with simple poles only, the logarithmic residues are identified as the sums of idempotents. With the help of this observation, the issue of left versus right logarithmic residues is investigated, both for connected and nonconnected underlying Cauchy domains. Examples are given to elucidate the subject matter.

In this paper a concept of a generalized directional derivative, which satisfies Leibniz rule is proposed for locally Lip- schitz functions, defined on an open subset of a Banach space. Al- though Leibniz rule is of less importance for a subdifferential cal- culus, it is of course of some theoretical interest to know about the existence of generalized directional derivatives

Let A be a Banach algebra. We call a pair (G, B) a Gelfand theory for A if the following axioms are satisfied: (G 1) B is a C^∗-algebra, and G : A → B is a homomorphism; (G 2) the assignment L G^-1(L) is a bijection between the sets of maximal modularleft ideals of B and A, respectively; (G 3) for each maximal modular left ideal L of B, the linear map G_L : A / G^-1(L) → B /L induced by B has dense range. The Gelfand theory of a commutative Banach algebra is easily seen to be characterized by these axioms. Gelfand theories of arbitrary Banach algebras enjoy many of the properties of commutative Gelfand theory. We show that unital, homogeneous Banach algebras always have a Gelfand theory. For liminal C^∗-algebras with discrete spectrum, we show that the identity is the only Gelfand theory (up to an appropriate notion of equivalence).

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.

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.

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).

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.