Although the theory of complex Banach algebras is by now classical, the first systematic exposition of the theory of real Banach algebras was given by Ingelstam [5] as late as 1965. More recently, further attention to real Banach algebras was paid in 1970 [1], where, among other things, the (real) standard algebras on finite open Klein surfaces were introduced. Generalizing these considerations, real uniform algebras were studied in [7] and [6].In the present paper, an attempt is made to develop the theory of real function algebras (see Section 1 for the definition) along the lines of the complex function algebras. Although the real function algebras are not structurally different from the real uniform algebras introduced in [7], they are easier to deal with since their elements are actually (complex-valued) functions.
Let A be a complex commutative Banach algebra with unit 1 and δ > 0.We prove the following results connecting these two notions:2010 Mathematics Subject Classification: Primary 46J05.
We define and discuss properties of the class of unbounded operators which attain minimum modulus. We establish a relationship between this class and the class of norm attaining bounded operators and compare the properties of both. Also we define absolutely minimum attaining operators (possibly unbounded) and characterize injective absolutely minimum attaining operators as those with compact generalized inverse. We give several consequences, one of those is that every such operator has a non trivial hyper invariant subspace.
Abstract. The ε -pseudospectrum Λ ε (a) of an element a of an arbitrary Banach algebra A is studied. Its relationships with the spectrum and numerical range of a are given. Characterizations of scalar, Hermitian and Hermitian idempotent elements by means of their pseudospectra are given. The stability of the pseudospectrum is discussed. It is shown that the pseudospectrum has no isolated points, and has a finite number of components, each containing an element of the spectrum of a . Suppose for some ε > 0 and a,b ∈ A, Λ ε (ax) = Λ ε (bx) ∀x ∈ A . It is shown that a = b if:(ii) a is Hermitian idempotent.(iii) a is the product of a Hermitian idempotent and an invertible element.(iv) A is semisimple and a is the product of an idempotent and an invertible element.(vii) A is a commutative semisimple Banach algebra.Mathematics subject classification (2010): 47A10, 46H05, 47A12.
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