Isolated point theorems for uniform algebras on two- and three-manifolds

2016 ◽  
Vol 144 (9) ◽  
pp. 3921-3933
Author(s):  
Swarup N. Ghosh
Keyword(s):  
Uniform Algebras ◽  
Isolated Point

A SPHERICAL VERSION OF THE KOWALSKI–SŁODKOWSKI THEOREM AND ITS APPLICATIONS

2020 ◽  
pp. 1-26
Author(s):  
SHIHO OI
Keyword(s):  
Banach Algebra ◽  
Function Space ◽  
Affirmative Answer ◽  
List Type ◽  
Private Communication ◽  
Type Number ◽  
Uniform Algebras ◽  
Surjective Isometry ◽  
Complex Linear ◽  
Local Map

Abstract Li et al. [‘Weak 2-local isometries on uniform algebras and Lipschitz algebras’, Publ. Mat.63 (2019), 241–264] generalized the Kowalski–Słodkowski theorem by establishing the following spherical variant: let A be a unital complex Banach algebra and let $\Delta : A \to \mathbb {C}$ be a mapping satisfying the following properties: (a) $\Delta $ is 1-homogeneous (that is, $\Delta (\lambda x)=\lambda \Delta (x)$ for all $x \in A$ , $\lambda \in \mathbb C$ ); (b) $\Delta (x)-\Delta (y) \in \mathbb {T}\sigma (x-y), \quad x,y \in A$ . Then $\Delta $ is linear and there exists $\lambda _{0} \in \mathbb {T}$ such that $\lambda _{0}\Delta $ is multiplicative. In this note we prove that if (a) is relaxed to $\Delta (0)=0$ , then $\Delta $ is complex-linear or conjugate-linear and $\overline {\Delta (\mathbf {1})}\Delta $ is multiplicative. We extend the Kowalski–Słodkowski theorem as a conclusion. As a corollary, we prove that every 2-local map in the set of all surjective isometries (without assuming linearity) on a certain function space is in fact a surjective isometry. This gives an affirmative answer to a problem on 2-local isometries posed by Molnár [‘On 2-local *-automorphisms and 2-local isometries of B(H)', J. Math. Anal. Appl.479(1) (2019), 569–580] and also in a private communication between Molnár and O. Hatori, 2018.

Tingley's problems on uniform algebras

2021 ◽  
pp. 125346
Author(s):  
Osamu Hatori ◽  
Shiho Oi ◽  
Rumi Shindo Togashi
Keyword(s):  
Uniform Algebras

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.

scholarly journals A theorem of Helson and Sarason in uniform algebras

1979 ◽  
Vol 55 (4) ◽  
pp. 128-131
Author(s):  
Yoshiki Ohno ◽  
Kôzô Yabuta
Keyword(s):  
Uniform Algebras

scholarly journals Noncommutative uniform algebras

2004 ◽  
Vol 162 (3) ◽  
pp. 213-218 ◽  
Author(s):  
Mati Abel ◽  
Krzysztof Jarosz
Keyword(s):  
Uniform Algebras

Weighted norm inequalities and uniform algebras, II

1990 ◽  
Vol 55 (5) ◽  
pp. 475-483
Author(s):  
Takahiko Nakazi ◽  
Takanori Yamamoto
Keyword(s):  
Weighted Norm ◽  
Weighted Norm Inequalities ◽  
Norm Inequalities ◽  
Uniform Algebras

The Philosophical Enquiry, theories of the sublime and the sister arts tradition

2018 ◽  
Author(s):  
Hélène Ibata
Keyword(s):  
Point Of View ◽  
Visual Artists ◽  
Verbal Representation ◽  
The Sublime ◽  
The Arts ◽  
Sister Arts ◽  
Isolated Point

This first chapter emphasises what Burke’s Enquiry owes to the existing discourse on the sublime (to Longinus and Addison in particular), in order to highlight its innovations, more specifically its aesthetically stimulating irrationalism and sensualism. It then focuses on Burke’s unique distinction between visual and verbal representation, his rejection of their common mimetic basis, and his argument that only the non-mimetic, suggestive medium of the verbal arts, language, may impart the sublime. At a time when parallels between the arts prevailed, this was an isolated point of view, which introduced a new paragone situation, and a challenge to visual artists. The end of the chapter examines a number of competing theories of the sublime that were compatible with painting, which makes it possible to enhance the specificity of the Enquiry and the paradox of its appeal to visual artists.

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