A study of Banach algebras and maps on it through invertible elements

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.

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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- ̇ Kahane-Zelazko (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 non-zero 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- Kahane-Zelazko Theorem for Banach modules. With this, we further give applications ̇ to functionals on Hardy spaces.

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