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Banach Algebras
Published in Eberhard Malkowsky, Vladimir Rakočević, Advanced Functional Analysis, 2019
Eberhard Malkowsky, Vladimir Rakočević
This chapter is dedicated to the study of Banach algebras. The introduction of the concept of a Banach algebra is followed by a great number of examples and a study of the invertibility of elements in a complex Banach algebra with identity, and of the spectrum, resolvent and spectral radius. Also, the concept of the topological divisor of zero is introduced and several results related to this concept are established. Further highlights include the study of subalgebras, one theorem by Hochwald–Morell, Theorem 6.9.1, and two theorems by Harte for regular elements in Banach algebras, Theorems 6.9.4 and 6.9.6, conditions for the invertibility of operators in the Banach algebra of bounded linear operators from a Banach space into itself in Section 6.10, and detailed studies of the spectra of adjoint, normal and compact operators in Sections 6.11 and 6.12. Finally, the concept of C∗–algebras is introduced, their most important properties are established and conditions for the invertibility of the difference of projections are given in Sections 6.13 and 6.14.
The Interplay Between Topological Algebras Theory and Algebras of Holomorphic Functions
Published in Michael Ruzhansky, Hemen Dutta, Advanced Topics in Mathematical Analysis, 2019
Many properties of the spectrum of a Banach algebra resemble basic properties of a domain D of Cn, where the holomorphic functions on the domain D are the equivalent of the elements of the Banach algebra. These properties are thus usually named analytic properties of the spectrum.
Preliminaries
Published in Hugo D. Junghenn, Principles of Analysis, 2018
A complete normed algebra is called a Banach algebra. These structures occur in many important settings, particularly in the theory of operators on Hilbert spaces. The Banach space B(X) of all bounded functions under pointwise multiplication is a simple example of a commutative unital Banach algebra. Other examples appear throughout the text. General commutative Banach algebras are discussed in detail in Chapter .
Measure-theoretic pressure and topological pressure in mean metrics
Published in Dynamical Systems, 2019
Let a triple (or pair be a topological dynamical system (TDS for short) in the sense that is a continuous map on the compact metric space X with metric d. The terms and represent the set of T-invariant Borel probability measures and the set of T-invariant ergodic Borel probability measures, respectively. Let be the Banach algebra of real-value continuous functions of X equipped with the supremum norm.