scholarly journals Strictly real Banach algebras

1993 ◽  
Vol 47 (3) ◽  
pp. 505-519 ◽  
Author(s):  
John Boris Miller
Keyword(s):  
Banach Algebra ◽  
Uniqueness Theorem ◽  
Banach Algebras ◽  
Complex Algebra ◽  
Conjugation Operator ◽  
Complex Banach Algebra

A complex Banach algebra is a complexification of a real Banach algebra if and only if it carries a conjugation operator. We prove a uniqueness theorem concerning strictly real selfconjugate subalgebras of a given complex algebra. An example is given of a complex Banach algebra carrying two distinct but commuting conjugations, whose selfconjugate subalgebras are both strictly real. The class of strictly real Banach algebras is shown to be a variety, and the manner of their generation by suitable elements is proved. A corollary describes some strictly real subalgebras in Hermitian Banach star algebras, including C* algebras.

scholarly journals An order-preserving representation theorem for complex Banach algebras and some examples

1973 ◽  
Vol 14 (2) ◽  
pp. 128-135 ◽  
Author(s):  
A. C. Thompson ◽  
M. S. Vijayakumar
Keyword(s):  
Banach Algebra ◽  
Representation Theorem ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Banach Algebras ◽  
Order Structure ◽  
Commutative Case ◽  
Complex Matrices ◽  
Complex Banach Algebra

Let A be a complex Banach algebra with unit e of norm one. We show that A can be represented on a compact Hausdorff space ω which arises entirely out of the algebraic and norm structures of A. This space induces an order structure on A that is preserved by the representation. In the commutative case, ω is the spectrum of A, and we have a generalization of Gelfand's representation theorem for commutative complex Banach algebras with unit. Various aspects of this representation are illustrated by considering algebras of n × n complex matrices.

Complemented Banach Algebras

1964 ◽  
Vol 16 ◽  
pp. 149-150
Author(s):  
A. Olubummo
Keyword(s):  
Banach Algebra ◽  
Left Ideal ◽  
Banach Algebras ◽  
Complex Banach Algebra ◽  
Left Annihilator

Let A be a complex Banach algebra and Lr (Ll) be the lattice of all closed right (left) ideals in A. Following Tomiuk (5), we say that A is a right complemented algebra if there exists a mapping I —> IP of Lτ into Lr such that if I ∊ Lr, then I ∩ Ip = (0), (Ip)p = I, I ⴲ Ip = A and if I1, I2 ∊ Lr with I1 ⊆ I2 then .If in a Banach algebra A every proper closed right ideal has a non-zero left annihilator, then A is called a left annihilator algebra. If, in addition, the corresponding statement holds for every proper closed left ideal and r(A) = (0) = l(A), A is called an annihilator algebra (1).

scholarly journals On the strong radical of certain Banach algebras

1978 ◽  
Vol 21 (1) ◽  
pp. 81-85 ◽  
Author(s):  
Bertram Yood
Keyword(s):  
Banach Algebra ◽  
Left Ideal ◽  
Banach Algebras ◽  
Complex Banach Algebra ◽  
Maximal Ideals

Let A be a complex Banach algebra. By an ideal in A we mean a two-sided idealunless otherwise specified. As in (7, p. 59) by the strong radical of A we mean theintersection of the modular maximal ideals of A (if there are no such ideals we set =A). Our aim is to discuss the nature of and the relation of to A for a specialclass of Banach algebras. Henceforth A will denote a semi-simple modular annihilatorBanach algebra (one for which the left (right) annihilator of each modular maximalright (left) ideal is not (0)). For the theory of such algebras see (2) and (9).

A theorem in Banach algebras and its applications

1979 ◽  
Vol 20 (2) ◽  
pp. 247-252 ◽  
Author(s):  
V.K. Srinivasan ◽  
Hu Shaing
Keyword(s):  
Banach Algebra ◽  
Principal Ideal ◽  
Banach Algebras ◽  
Negative Integer ◽  
Complex Field ◽  
Principal Ideal Domain ◽  
Bezout Domain ◽  
Ideal Domain ◽  
Complex Banach Algebra ◽  
Divisor Of Zero

If A is a complex Banach algebra which is also a Bezout domain, it is shown that for any prime p and a non-negative integer n, pn is not a topological divisor of zero. Using the above result it is shown that a complex Banach algebra which is a principal ideal domain is isomorphic to the complex field.

Spectra of approximate identities in Banach algebras

1979 ◽  
Vol 86 (2) ◽  
pp. 271-278 ◽  
Author(s):  
P. G. Dixon
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Approximate Identity ◽  
Unit Interval ◽  
Complex Banach Algebra ◽  
Approximate Identities

1. Introduction. The aim of this paper is to show that, in every complex Banach algebra with a one-sided or two-sided bounded approximate identity, there exists another bounded approximate identity of the same sort whose spectra lie close to the unit interval [0, 1].

Idempotents in Complex Banach Algebras

1987 ◽  
Vol 39 (3) ◽  
pp. 625-630
Author(s):  
G. N. Hile ◽  
W. E. Pfaffenberger
Keyword(s):  
Banach Algebra ◽  
Complex Number ◽  
Banach Algebras ◽  
Rectifiable Curve ◽  
Resolvent Set ◽  
Complex Banach Algebra ◽  
Image Position

The concept of the spectrum of A relative to Q, where A and Q commute and are elements in a complex Banach algebra with identity I, was developed in [1]. A complex number z is in the Q-resolvent set of A if and only if is invertible in otherwise, z is in the Q-spectrum of A, or spectrum of A relative to Q. One result from [1] was the following.THEOREM. Suppose no points in the ordinary spectrum of Q have unit magnitude. Let C be a simple closed rectifiable curve which lies in the Q-resolvent of A, and let*where P is defined asxs•

scholarly journals On non-negative spectrum in Banach algebras

1973 ◽  
Vol 18 (4) ◽  
pp. 295-298 ◽  
Author(s):  
Bertram Yood
Keyword(s):  
Banach Algebra ◽  
Real Number ◽  
Spectral Radius ◽  
Banach Algebras ◽  
Semi Group ◽  
Negative Spectrum ◽  
Complex Banach Algebra ◽  
Negative Real Number

Let A be a complex Banach algebra with an identity 1. In this note we study the subset Λ of A consisting of all g ∈ A such that the spectrum of g, sp(g), contains at least one non-negative real number. Clearly Λ is not, in general, a semi-group with respect to either addition or multiplication. However, Λ is an instance of a subset Q of A with the following properties, where ρ(f) denotes the spectral radius of f (4, p. 30).

Generalized Spectral Theory in Complex Banach Algebras

1985 ◽  
Vol 37 (6) ◽  
pp. 1211-1236
Author(s):  
G. N. Hile ◽  
W. E. Pfaffenberger
Keyword(s):  
Banach Algebra ◽  
Unit Circle ◽  
Compact Subset ◽  
Spectral Theory ◽  
Complex Plane ◽  
Banach Algebras ◽  
Complex Banach Algebra ◽  
Nonempty Compact ◽  
Nonempty Compact Subset

Let A be an element of a complex Banach algebra with identitI. The ordinary spectrum of A, sp(A), consists of those points z in the complex plane such that A — zI has no inverse in . If Q is any other element of , we define spQ(A), the spectrum of A relative to Q, or Q-spectrum of A, as those points z such that has no inverse in . Thus if Q = 0 the Q-spectrum of A is the same as the ordinary spectrum of A.The generalized notion of spectrum, spQ(A), retains many of the properties of the ordinary spectrum, particularly when A and Q commute and the ordinary spectrum of Q does not meet the unit circle. Under these conditions the Q-spectrum of A is a nonempty compact subset of the plane, and if both sp(A) and sp(Q) are finite (or countable), so is spQ(A).

scholarly journals On some Gelfand-Mazur like theorems in Banach algebras

1979 ◽  
Vol 20 (2) ◽  
pp. 211-215 ◽  
Author(s):  
V.K. Srinivasan
Keyword(s):  
Banach Algebra ◽  
Integral Domain ◽  
Banach Algebras ◽  
Complex Field ◽  
Principal Ideal Domain ◽  
Locally Finite ◽  
Several Variables ◽  
Complex Banach Algebra ◽  
Noetherian Domain ◽  
Formal Power

The following Gelfand-Mazur like theorems are proved in this paper:(1) A complex Banach algebra which is locally finite, and which is also an integral domain, is isomorphic to the complex field .(2) A complex Banach algebra which is a noetherian domain is isomorphic to .(3) A complex Banach algebra which is a principal ideal domain is isomorphic to .An application is given to the algebra of all complex formal power series in several variables.

scholarly journals Ring homomorphisms on real Banach algebras

2003 ◽  
Vol 2003 (48) ◽  
pp. 3025-3029
Author(s):  
Takeshi Miura ◽  
Sin-Ei Takahasi
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Ring Homomorphism ◽  
Ring Homomorphisms ◽  
Complex Banach Algebra ◽  
Carrier Space ◽  
Rho A

LetBbe a strictly real commutative real Banach algebra with the carrier spaceΦB. IfAis a commutative real Banach algebra, then we give a representation of a ring homomorphismρ:A→B, which needs not be linear nor continuous. IfAis a commutative complex Banach algebra, thenρ(A)is contained in the radical ofB.

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