Sebastian, Geethika and D, Sukumar
(2018)
A study of Banach algebras and maps on it through invertible elements.
PhD thesis, Indian institute of technology Hyderabad.
Abstract
This thesis is divided into two parts.
In the first part of the thesis we define a new class of elements within the invertible
group of a complex unital Banach algebra, namely the B class. An invertible element
a of a Banach algebra A is said to satisfy condition B or belong to the B class if the
boundary of the open ball centered at a, with radius 1
ka−1k
, necessarily intersects the
set of non invertible elements. A Banach algebra is said to satisfy condition B if all its
invertible elements satisfy condition B. We investigate the requirements for a Banach
algebra to satisfy condition B and completely characterize commutative Banach alge
bras satisfying the same. In the process we prove that: A is a commmutative unital
Banach algebra satisfying condition B iff ka
2k = kak
2
for every invertible element a
in A and A is isomorphic to a uniform algebra. We also construct a commutative
unital Banach algebra, in which the property: ka
2k = kak
2
, is true for the invertible
elements but not true for the whole algebra.
On the whole we discuss, how Banach algebras satisfying condition B can be
characterized by observing the nature of the invertible elements. In the next part
of the thesis, we study maps on Banach algebras, which can be studied by their
behaviour on the invertible elements.
Gleason, Kahane and Zelazko([11], [13] and [36]), through the famous Gleason ̇
KahaneZelazko (GKZ) theorem, showed how the multiplicativity of a linear func ̇
tional on a Banach algebra, is totally dependent on what value the functional takes
at the invertible elements. Mashregi and Ransford [22] generalized the GKZ theorem
to modules and later used it to show that every linear functional on a Hardy space
that is nonzero on the outer functions, is a constant multiple of a point evaluation.
Kowalski and S lodkowski in [15] dropped the condition of linearity in the hypothesis
of the GKZ theorem, they also replaced the assumption of preservation of invertibility,
by a single weaker assumption, to give the same conclusion.
The second part of the thesis involves us working along the same lines as Kowalski
and S lodkowski. In the hypothesis of the GKZ theorem for modules, we remove the
assumption of the map being linear, tailor the hypothesis, and get a weaker Gleason
KahaneZelazko Theorem for Banach modules. With this, we further give applications ̇
to functionals on Hardy spaces.
[error in script]
IITH Creators: 
IITH Creators  ORCiD 

D, Sukumar  UNSPECIFIED 

Item Type: 
Thesis
(PhD)

Uncontrolled Keywords: 
Invertible Elements, Spectrum, Uniform Algebra, Norm attaining operator 
Subjects: 
Mathematics 
Divisions: 
Department of Mathematics 
Depositing User: 
Team Library

Date Deposited: 
29 Oct 2018 06:00 
Last Modified: 
29 Oct 2018 06:02 
URI: 
http://raiith.iith.ac.in/id/eprint/4503 
Publisher URL: 

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