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.     

Weyl’s Theorems for Some Classes of Operators

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 T_μ of a group algebra L¹(G), G a locally compact abelian group, satisfies a-Weyl’s theorem. Similar results are given for multipliers of other important commutative Banach algebras.     

Commutative Banach algebras and decomposable operators

For a multiplication operator on a semi-simple commutative Banach algebra, it is shown that the decomposability in the sense of Foia is equivalent to weak and to super-decomposability. Moreover, it can also be characterized by a convenient continuity condition for the Gelfand transform on the spectrum of the underlying Banach algebra. This result implies various permanence properties for decomposable multiplication operators and leads also to a useful characterization of the regularity for a semi-simple commutative Banach algebra. Finally, the greatest regular closed subalgebra of a commutative Banach algebra is investigated, and some applications to decomposable convolution operators on locally compact abelian groups are given.Research partially supported by NSF Grant DMS 90-96108.     

On a Banach Algebra Generated in its Multiplier Space and Applications to Locally Compact Groups

In this paper, we present a general version of the algebra A_M(G) which was introduced by B. Forrest. Indeed, for a faithful commutative Banach algebra A, we embed it in ?(A), the multiplier algebra of A, and obtain Banach algebra A_M. Then, we study the spaceability of A_M? A and A_M (G) ? ?A(G). These results give some characterizations of compactness and discreteness of locally compact groups. Also, we show that A_M(G) is an ideal in its second dual if and only if G is discrete. Finally, we study the BSE-property of A_M(G).

     

Theorems of Gelfand-Mazur type and continuity of epimorphisms from b(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? [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 $\text{b}$(K) onto a Banach algebra is continuous.     

Weak spectral synthesis in commutative Banach algebras. II

Let A be a semisimple and regular commutative Banach algebra with structure space Δ(A). Continuing our investigation in [E. Kaniuth, Weak spectral synthesis in commutative Banach algebras, J. Funct. Anal. 254 (2008) 987-1002], we establish various results on intersections and unions of weak spectral sets and weak Ditkin sets in Δ(A). As an important example, the algebra of n-times continuously differentiable functions is studied in detail. In addition, we prove a theorem on spectral synthesis for projective tensor products of commutative Banach algebras which applies to Fourier algebras of locally compact groups.     

M-ideal structure in Banach algebras

It is shown that M-ideals in a Banach algebra with identity are subalgebras, and that they are ideals if the algebra is commutative. Counterexamples demonstrate that these are the strongest results available. The theory is then applied to familiar classes of Banach spaces.     

Weak Module Amenability for Semigroup Algebras

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.     

On the spectrum of a class of non-commutative Banach algebras

Archiv der Mathematik - Let $$UT_m(X)$$ be the Banach algebra of all $$m \times m$$ upper triangular matrices with entries in a unital commutative complex Banach algebra X. It is well known that...     

Feynman’s operational calculi for noncommuting operators: The monogenic calculus

In recent papers the authors presented their approach to Feynman’s operational calculi for a system of not necessarily commuting bounded linear operators acting on a Banach space. The central objects of the theory are the disentangling algebra, a commutative Banach algebra, and the disentangling map which carries this commutative structure into the noncommutative algebra of operators. Under assumptions concerning the growth of disentangled exponential expressions, the associated functional calculus for the system of operators is a distribution with compact support which we view as the joint spectrum of the operators with respect to the disentangling map. In this paper, the functional calculus is represented in terms of a higher-dimensional analogue of the Riesz-Dunford calculus using Clifford analysis.     

Feynman's Operational Calculi for Noncommuting Systems of Operators: Tensors,Ordered Supports,and Disentangling an Exponential Factor

In a recent paper, the authors presented the key ideas involved in their approach to Feynman's operational calculi for systems of not necessarily commuting bounded linear operators acting on a Banach space. The central objects of the theory are the disentangling algebra, a commutative Banach algebra, and the disentangling map which carries this commutative structure into the noncommutative algebra of operators. The study of properties of these disentangling maps will be pursued in this paper with an emphasis on (i) Feynman's formula for disentangling an exponential factor, and (ii) the effect on the disentangling map of ordered supports of some or all of the measures which govern the disentangling.     

On uniform continuity of the spectral radius in Banach algebras

We investigate relations between different forms of subadditivity and submultiplicativity of the spectral radius. In particular, we prove that if the spectral radius is uniformly continuous on a Banach algebra, then the algebra is commutative modulo the radical; this confirms a conjecture raised by the second author in [7].     

Projections in a Synaptic Algebra

A synaptic algebra is an abstract version of the partially ordered Jordan algebra of all bounded Hermitian operators on a Hilbert space. We review the basic features of a synaptic algebra and then focus on the interaction between a synaptic algebra and its orthomodular lattice of projections. Each element in a synaptic algebra determines and is determined by a one-parameter family of projections—its spectral resolution. We observe that a synaptic algebra is commutative if and only if its projection lattice is boolean, and we prove that any commutative synaptic algebra is isomorphic to a subalgebra of the Banach algebra of all continuous functions on the Stone space of its boolean algebra of projections. We study the so-called range-closed elements of a synaptic algebra, prove that (von Neumann) regular elements are range-closed, relate certain range-closed elements to modular pairs of projections, show that the projections in a synaptic algebra form an M-symmetric orthomodular lattice, and give several sufficient conditions for modularity of the projection lattice.     

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.Partially supported by NSF Grant DMS-8802593     

P-commutativity of the Banach algebra L1**(G)

Let G be a compact abelian group. It is known that the second conjugate space L₁**(G) of the group algebra L₁(G) is a noneommutative, nonsemisimple, Banach algebra. It is not known if L₁**(G) admits an involution. If it does, we show that it is P-coimnutative, and hence enjoys many of the properties of commutative Banach algebras.     

A Banach algebra of generalized functions

An algebraic isomorphism from a convolution algebra of Laplace transformable functions with support on the half-line to a complete discrete normed convolution algebra of sequences is used to construct generalized functions. The extension of this function-to-sequence map to a commutative Banach algebra of generalized functions is shown to be a Banach algebra isomorphism which can be utilized to establish a discrete formulation of a Mikusiński-type operational calculus and to construct algorithms for the numerical solution of half-line convolution equations.     

具有连续对合运算的实Banach*代数的Jordan结构

本文讨论了实Banach*代数的Jordan结构．主要结果:第一部分指出映射到 *-半单实Banach*代数上的Jordan*同态是连续的,且其核空间是闭*理想;由映射到交换实Banach*代数上的Jordan*同态诱导的因子代数也是交换的．第二部分介绍了两个不同的锥,并讨论了他们间的关系．另外,我们得到了关于实Banach*代数*- 根基的一个新的刻画．本文是Satish Shirali的工作的实化．     

Finite universal Korovkin systems in tensor products of commutative Banach algebras

We show that the tensor product of commutative Banach algebras possesses a finite universal Korovkin system, if and only if each of the factors does, and present an explicit construction of finite universal Korovkin systems for several commutative Banach algebras of vector-valued functions. The same questions are answered for the formation of direct sums and the passage to an ideal resp. its associated quotient algebra.     

QUOTIENTS OF 2-SUMMING ALGEBRAS

Abstract

A result of A.M. Mantero and A. Tonge is strengthened by giving an example of a commutative operator algebra which is semisimple but which is not the quotient algebra of a 2-summing Banach algebra by a closed ideal.