HOMOLOGICAL DIMENSION OF BANACH ALGEBRAS OF ANALYTIC FUNCTIONS

1970 ◽  
Vol 12 (2) ◽  
pp. 221-233 ◽  
Author(s):  
A J Helemskiĭ
Keyword(s):  
Analytic Functions ◽  
Banach Algebras ◽  
Homological Dimension

HOMOMORPHISMS OF BANACH ALGEBRAS WITH RANGE IN C1[0, 1]

1994 ◽  
Vol 05 (02) ◽  
pp. 201-212 ◽  
Author(s):  
HERBERT KAMOWITZ ◽  
STEPHEN SCHEINBERG
Keyword(s):  
Analytic Functions ◽  
Compact Hausdorff Space ◽  
Unit Disc ◽  
Hausdorff Space ◽  
Banach Algebras ◽  
Locally Compact ◽  
Disc Algebra ◽  
Uniform Algebras ◽  
Closed Subalgebra ◽  
Uniformly Closed

Many commutative semisimple Banach algebras B including B = C (X), X compact, and B = L1 (G), G locally compact, have the property that every homomorphism from B into C1[0, 1] is compact. In this paper we consider this property for uniform algebras. Several examples of homomorphisms from somewhat complicated algebras of analytic functions to C1[0, 1] are shown to be compact. This, together with the fact that every homomorphism from the disc algebra and from the algebra H∞ (∆), ∆ = unit disc, to C1[0, 1] is compact, led to the conjecture that perhaps every homomorphism from a uniform algebra into C1[0, 1] is compact. The main result to which we devote the second half of this paper, is to construct a compact Hausdorff space X, a uniformly closed subalgebra [Formula: see text] of C (X), and an arc ϕ: [0, 1] → X such that the transformation T defined by Tf = f ◦ ϕ is a (bounded) homomorphism of [Formula: see text] into C1[0, 1] which is not compact.

LIPSCHITZ-TYPE INEQUALITIES FOR ANALYTIC FUNCTIONS IN BANACH ALGEBRAS

2019 ◽  
Vol 100 (3) ◽  
pp. 489-497
Author(s):  
SILVESTRU SEVER DRAGOMIR
Keyword(s):  
Analytic Function ◽  
Banach Algebra ◽  
Analytic Functions ◽  
Banach Algebras ◽  
Logarithmic Functions

In this paper we provide some bounds for the quantity $\Vert f(y)-f(x)\Vert$, where $f:D\rightarrow \mathbb{C}$ is an analytic function on the domain $D\subset \mathbb{C}$ and $x$, $y\in {\mathcal{B}}$, a Banach algebra, with the spectra $\unicode[STIX]{x1D70E}(x)$, $\unicode[STIX]{x1D70E}(y)\subset D$. Applications for the exponential and logarithmic functions on the Banach algebra ${\mathcal{B}}$ are also given.

scholarly journals Homology and cohomology groups of commutative Banach algebras and analytic polydiscs

2000 ◽  
Vol 42 (1) ◽  
pp. 15-24
Author(s):  
L. I. Pugach ◽  
M. C. White
Keyword(s):  
Banach Algebra ◽  
Maximal Ideal ◽  
Deformation Theory ◽  
Banach Algebras ◽  
Subject Classification ◽  
Analytic Structure ◽  
Homological Dimension ◽  
Cohomology Groups ◽  
Main Method ◽  
Commutative Banach Algebras

In this paper we deduce the existence of analytic structure in a neighbourhood of a maximal ideal M in the spectrum of a commutative Banach algebra, A, from homological assumptions. We assume properties of certain of the cohomology groups H^n(A,A/M), rather than the stronger conditions on the homological dimension of the maximal ideal the first author has considered in previous papers. The conclusion is correspondingly weaker: in the previous work one deduces the existence of a Gel'fand neighbourhood with analytic structure, here we deduce only the existence of a metric neighbourhood with analytic structure. The main method is to consider products of certain co-cycles to deduce facts about the symmetric second cohomology, which is known to be related to the deformation theory of algebras.1991 Mathematics Subject Classification. 46J20, 46M20.

scholarly journals Analytic functions and logmodular Banach algebras

1962 ◽  
Vol 108 (0) ◽  
pp. 271-317 ◽  
Author(s):  
Kenneth Hoffman
Keyword(s):  
Analytic Functions ◽  
Banach Algebras

Point derivations and cohomologies of Lipschitz algebras

2019 ◽  
Vol 62 (4) ◽  
pp. 1173-1187
Author(s):  
Kazuhiro Kawamura
Keyword(s):  
Metric Spaces ◽  
Banach Algebras ◽  
Studia Math ◽  
Lipschitz Functions ◽  
Homological Dimension ◽  
Smooth Functions ◽  
Infinite Dimensional ◽  
Higher Dimensional ◽  
Geodesic Metric ◽  
Dual Module

AbstractFor a compact metric space (K, d), LipK denotes the Banach algebra of all complex-valued Lipschitz functions on (K, d). We show that the continuous Hochschild cohomology Hn(LipK, (LipK)*) and Hn(LipK, ℂe) are both infinite-dimensional vector spaces for each n ≥ 1 if the space K contains a certain infinite sequence which converges to a point e ∈ K. Here (LipK)* is the dual module of LipK and ℂe denotes the complex numbers with a LipK-bimodule structure defined by evaluations of LipK-functions at e. Examples of such metric spaces include all compact Riemannian manifolds, compact geodesic metric spaces and infinite compact subsets of ℝ. In particular, the (small) global homological dimension of LipK is infinite for every such space. Our proof uses the description of point derivations by Sherbert [‘The structure of ideals and point derivations in Banach algebras of Lipschitz functions’, Trans. Amer. Math. Soc.111 (1964), 240–272] and directly constructs non-trivial cocycles with the help of alternating cocycles of Johnson [‘Higher-dimensional weak amenability’, Studia Math.123 (1997), 117–134]. An alternating construction of cocycles on the basis of the idea of Kleshchev [‘Homological dimension of Banach algebras of smooth functions is equal to infinity’, Vest. Math. Mosk. Univ. Ser. 1. Mat. Mech.6 (1988), 57–60] is also discussed.

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