Commutative Unital Banach Algebra Research Articles

Generalized stable ranks for commutative Banach algebras

We introduce the notion of generalized E-stable ranks for commutative unital Banach algebras and determine these ranks for the disk-algebra \({A(\mathbb{D})}\), many of its subalgebras, and the algebra H ∞ of bounded holomorphic functions in the unit disk. Relations to L-sets and separating algebras, notions due to Csordas and Reiter, are given, too. Finally we show that the absolute stable rank of \({A(\mathbb{D})}\) and H ∞ is bigger than 2.

Zero product preserving maps on Banach algebras of Lipschitz functions

Let ( K , d ) be a non-empty, compact metric space and α ∈ ] 0 , 1 [ . Let A be either lip α ( K ) or Lip α ( K ) and let B be a commutative unital Banach algebra. We show that every continuous linear map T : A → B with the property that T ( f ) T ( g ) = 0 whenever f , g ∈ A are such that f g = 0 is of the form T = w Φ for some invertible element w in B and some continuous epimorphism Φ : A → B .

Lower bounds for homological dimensions of Banach algebras

Let be a commutative unital Banach algebra with infinite spectrum. Then by Helemskiĭ's global dimension theorem the global homological dimension of is strictly greater than one. This estimate has no analogue for abstract algebras or non-normable topological algebras. It is proved in the present paper that for every unital Banach algebra the global homological dimensions and the homological bidimensions of the Banach algebras and (assuming certain restrictions on ) are related by and . Thus, a partial extension of Helemskiĭ's theorem to tensor products is obtained.Bibliography: 28 titles.

SPECTRAL THEORY, TENSOR PRODUCTS AND INFINITE DIMENSIONAL HOLOMORPHY

In this expository article we survey some recent developments on the construction of a holomorphic functional calculus for an infinite number of elements in a commutative unital Banach algebra.

Comparison of the product structures in algebraic and in topological $K$-theory

The compatibility up to sign of the product structures in algebraic K-theory and in topological K-theory of unital Banach algebras is established in total degree ≤ 2 . This answers a question posed by Milnor. 1 Statement of the theorem and de nition of the product structures in K-theories As an application of the computations made in [7], we prove the following result. 1.1 Theorem. Let A and B be two unital Banach algebras. Then the diagram K p (A)⊗K q (B) ? K p+q(A⊗Z B) Kp(A)⊗Kq(B) φp ⊗ φq ? ×Kp+q(A⊗B) (−1)φp+q ? commutes for p, q ≥ 0 satisfying p+q ≤ 2 . In other words, the external product structures in algebraic and in topological K-theory of unital Banach algebras are compatible in total degree ≤ 2 , up to the sign (−1) . In particular, for commutative unital Banach algebras, the internal product structures are also compatible in the same range and up to the same sign. Let us explain the notations. For a unital Banach algebras A (always over C), we denote by GL(A) the in nite matrix group with the usual direct limit topology, by E(A) the group of in nite elementary matrices, which coincides with the commutator subgroup [GL(A), GL(A)] of GL(A) , and by St(A) the in nite Steinberg group of A with standard generators (xij(a))i 6=j, a∈A . The algebraic and topological K-theory groups are de ned by : 1Research supported by Swiss National Fund for Scienti c Research grant no. 20-56816.99 2Research supported by Swiss National Fund for Scienti c Research grant no. 20-50575.97

Spectra of Tensor Product Elements III : Holomorphic Properties

We show that under fairly general conditions, for sl a commutative unital Banach algebra and X a Banach space, each a E s/?X admits a compact-open continuous holomorphic functional calculus and, when X is separable and has the bounded approximation property, that this is uniquely implemented. Compactness, non-emptiness and polynomial convexity of the left spectrum, coincidence of left and right spectra, and countably generated uniform Banach algebras are also discussed.

Group inverses and Drazin inverses over Banach algebras

For a matrix over a complex commutative unital Banach algebra, necessary and sufficient conditions are given for the existence of its group inverse, and more generally, its Drazin inverses. The conditions are easy to check and explicit formulas for the inverses are provided. Some properties of the inverses and an application to operator theory are discussed. This note is a continuation of an earlier work of the author.

On nilpotency of the separating ideal of a derivation

We prove that the separating ideal S ( D ) S(D) of any derivation D D on a commutative unital algebra B B is nilpotent if and only if S ( D ) ∩ ( ⋂ R n ) S(D) \cap (\bigcap {{R^n})} is a nil ideal, where R R is the Jacobson radical of B B . Also we show that any derivation D D on a commutative unital semiprime Banach algebra B B is continuous if and only if ⋂ ( S ( D ) ) n = { 0 } \bigcap {{{(S(D))}^n} = \{ 0\} } . Further we show that the set of all nilpotent elements of S ( D ) S(D) is equal to ⋂ ( S ( D ) ∩ P ) \bigcap {(S(D) \cap P)} , where the intersection runs over all nonclosed prime ideals of B B not containing S ( D ) S(D) . As a consequence, we show that if a commutative unital Banach algebra has only countably many nonclosed prime ideals then the separating ideal of a derivation is nilpotent.

Differentiation in modules over topological ∗-algebras

In recent years, there has been a systematic development of the theory of A-bundles, that is, vector bundles where the field of coefficients is replaced by a topological algebra A, usually an involutive one. The above extension is justified by its applications in pure mathematics (differential topology [15]) as well as in relativistic quantum mechanics, via, e.g., the Serre-Swan theorem [22]. Bundles with coefficients in a commutative unital Banach algebra, or yet a C*-algebra, were considered by T. K. Kandelaki [6, 71, K. Fujii [S], and A. S. MiSEenko and Yu. P. Solov’ev [ 15, 161. Furthermore, A. Mallios studied A-bundles, with A being a locally m-convex (*-)algebra with unit [l&12]. Note also that certain results are still valid for more general topological algebras [ 13, Sect. 53, or even topological rings [21]. The results already obtained are mainly within the framework of topological A-bundles. The basic difficulty in the development of a theory of differential A-bundles is the lack of a suitable differential calculus in the libres, the latter being topological modules over A. One might, of course, use the existing methods of differentiation in the underlying topological vector spaces (cf. for instance [18]). However, besides the intrinsic weaknesses of these methods (regarding the higher order chain rule, continuity of differentiable maps, differentiability of the composition map and evaluation map, which are necessary for differential geometric considerations on bundles), the additional algebraic operations (scalar multiplication by elements of A, involution) should be taken into account, because they are essentially involved in the structure of A-bundles (cf. [ 11, 183). For the same reason (incompatibility with involution), the 255 0022-247X/92 55.00

Generalized inverses over Banach algebras

LetA be a matrix over a complex commutative unital Banach algebra. We give necessary and sufficient conditions forA to have a generalized inverse. Moreover, if the Banach algebra has a symmetric involution, these are also necessary and sufficient conditions forA to admit the Moore-Penrose inverse.

Commutative Gelfand Theory for Real Banach Algebras: Representations as Sections of Bundles

AbstractWe are concerned here with the development of a more general real case of the classical theorem of Gelfand ([5], 3.1.20), which represents a complex commutative unital Banach algebra as an algebra of continuous functions defined on a compact Hausdorff space.In § 1 we point out that when looking at real algebras there is not always a one-to-one correspondence between the maximal ideals of the algebra B, denoted ℳ, and the set of unital (real) algebra homomorphisms from B into C, denoted by ΦB. This simple point and subsequent observations lead to a theory of representations of real commutative unital Banach algebras where elements are represented as sections of a bundle of real fields associated with the algebra (Theorem 3.5). After establishing this representation theorem, we look into the question of when a real commutative Banach algebra is already complex. There is a natural topological obstruction which we delineate. Theorem 4.8 gives equivalent conditions which determine whether such an algebra is already complex.Finally, in § 5 we abstractly characterize those section algebras which appear as the target algebras for our Gelfand transform. We dub these algebras “almost complex C*- algebras” and provide a natural classification scheme.

Spectrum reducing extension for one operator on a Banach space

In this paper we show that, given an operator T T on a Banach space X X , there is an extension Y Y of X X such that T T extends in a natural way to an operator T ∼ {T^ \sim } on Y Y , and the spectrum of T ∼ {T^ \sim } is the approximate point spectrum of T T . This answers a question posed by Bollobás, and contributes to a theory investigated by Shilov, Arens, Bollobás, etc. The unusual transfinite construction is similar to that which we used earlier to find an inverse producing extension for a commutative unital Banach algebra which eliminates the residual spectrum of one element. We also give a counterexample, consisting of a Banach algebra L L containing elements g 1 {g_1} and g 2 {g_2} such that in no extension L ′ L’ of L L are the residual spectra of g 1 {g_1} and g 2 {g_{_2}} eliminated simultaneously.

Inverse Producing Extension of a Banach Algebra which Eliminates the Residual Spectrum of One Element

If $A$ is a commutative unital Banach algebra and $G \subset A$ is a collection of nontopological zero divisors, the question arises whether we can find an extension $A\prime$ of $A$ in which every element of $G$ has an inverse. Shilov [1] proved that this was the case if $G$ consisted of a single element, and Arens [2] conjectures that it might be true for any set $G$. In [3], Bollobás proved that this is not the case, and gave an example of an uncountable set $G$ for which no extension $A\prime$ can contain inverses for more than countably many elements of $G$. Bollobás proved that it was possible to find inverses for any countable $G$, and gave best possible bounds for the norms of the inverses in [4]. In this paper, it is proved that inverses can always be found if the elements of $G$ differ only by multiples of the unit; that is, we can eliminate the residual spectrum of one element of $A$. This answers the question posed by Bollobás in [5].

Inverse producing extension of a Banach algebra which eliminates the residual spectrum of one element

If A A is a commutative unital Banach algebra and G ⊂ A G \subset A is a collection of nontopological zero divisors, the question arises whether we can find an extension A ′ A\prime of A A in which every element of G G has an inverse. Shilov [1] proved that this was the case if G G consisted of a single element, and Arens [2] conjectures that it might be true for any set G G . In [3], Bollobás proved that this is not the case, and gave an example of an uncountable set G G for which no extension A ′ A\prime can contain inverses for more than countably many elements of G G . Bollobás proved that it was possible to find inverses for any countable G G , and gave best possible bounds for the norms of the inverses in [4]. In this paper, it is proved that inverses can always be found if the elements of G G differ only by multiples of the unit; that is, we can eliminate the residual spectrum of one element of A A . This answers the question posed by Bollobás in [5].

Theorems of Gelfand-Mazur type and continuity of epimorphisms from [formula omitted](K)

We prove that a commutative unital Banach algebra which is a valuation ring must reduce to the field of complex numbers, which implies that every homomorphism from l ∞ onto a Banach algebra is continuous. We show also that if b ϵ [ b Rad B] − for some nonnilpotent element b of the radical of a commutative Banach algebra B, then the set of all primes of B cannot form a chain, and we deduce from this result that every homomorphism from b (K) onto a Banach algebra is continuous.

Commutative Abelian Galois extensions of a Banach algebra

Let A be a commutative unital Banach algebra with connected maximal ideal space X. We show that the Gelfand transform induces an isomorphism between the group of commutative Galois extensions of A with given finite Abelian Galois group, and the corresponding group of extensions of C( X). This result is applied, when X is sufficiently nice, to construct a separable projective finitely generated faithful Banach A-algebra whose maximal ideal space is a given finitely fibered covering space of X.

Adjoining inverses to commutative Banach algebras

Let A A be a commutative unital Banach algebra. Suppose G ⊂ A G \subset A is such that | | a | | ⩽ | | g a | | ||a|| \leqslant ||ga|| for all g ∈ G , a ∈ A g \in G,a \in A . Two questions are considered in the paper. Does there exist a superalgebra B B of A A in which every g ∈ G g \in G is invertible? Can one always have also | | g − 1 | | ⩽ 1 ||{g^{ - 1}}|| \leqslant 1 if g ∈ G g \in G ? Arens proved that if G = { g } G = \{ g\} then there is an algebra containing g − 1 {g^{ - 1}} , with | | g − 1 | | ⩽ 1 ||{g^{ - 1}}|| \leqslant 1 . In the paper it is shown that if G G is countable B B exists, but if G G is uncountable, this is not necessarily so. The answer to the second question is negative even if G G consists of only two elements.