scholarly journals Commutators in Banach algebras

1979 ◽  
Vol 22 (3) ◽  
pp. 207-211 ◽  
Author(s):  
Vlastimil Pták
Keyword(s):  
Banach Algebra ◽  
Present Author ◽  
Present Note ◽  
Banach Algebras ◽  
Related Condition ◽  
Image Position

In a recent paper (6) the present author has shown that, for an element a of a Banach algebra A, the conditionfor all x∈A and some constant α is equivalent to [x, a]∈Rad a for all x∈A; it turns out that α may be replaced by |α|σ It is the purpose of the present note to investigate a related condition

scholarly journals Maximal ideal spaces of Banach algebras of derivable elements

1970 ◽  
Vol 11 (3) ◽  
pp. 310-312 ◽  
Author(s):  
R. J. Loy
Keyword(s):  
Banach Algebra ◽  
Maximal Ideal ◽  
Present Note ◽  
Banach Algebras ◽  
Commutative Banach Algebra ◽  
Supremum Norm ◽  
Simple Consequence ◽  
Maximal Ideals ◽  
Image Position

Let A be a commutative Banach algebra, D a closed derivation defined on a subalgebra Δ of A, and with range in A. The elements of Δ may be called derivable in the obvious sense. For each integer k ≦.l, denote by Δk the domain of Dk (so that Dgr;1 = Δ); it is a simple consequence of Leibniz's formula that each Δk is an algebra. The classical example of this situation is A = C(O, 1) under the supremum norm with D ordinary differentiation, and here Δk = Ck(0, 1) is a Banach algebra under the norm ∥.∥k: Furthermore, the maximal ideals of Ak are precisely those subsets of Δk of the form M ∩ Δk where M is a maximal ideal of A, and = M, the bar denoting closure in A. In the present note we show how this extends to the general case.

On the Definition of C*-Algebras II

1985 ◽  
Vol 37 (4) ◽  
pp. 664-681 ◽  
Author(s):  
Zoltán Magyar ◽  
Zoltán Sebestyén
Keyword(s):  
Hilbert Space ◽  
Banach Algebra ◽  
Representation Theorem ◽  
Banach Algebras ◽  
Bounded Operators ◽  
Fundamental Representation ◽  
C Algebras ◽  
Definition Of ◽  
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The theory of noncommutative involutive Banach algebras (briefly Banach *-algebras) owes its origin to Gelfand and Naimark, who proved in 1943 the fundamental representation theorem that a Banach *-algebra with C*-condition(C*)is *-isomorphic and isometric to a norm-closed self-adjoint subalgebra of all bounded operators on a suitable Hilbert space.At the same time they conjectured that the C*-condition can be replaced by the B*-condition.(B*)In other words any B*-algebra is actually a C*-algebra. This was shown by Glimm and Kadison [5] in 1960.

scholarly journals Some inequalities for norm unitaries in Banach algebras

1978 ◽  
Vol 21 (1) ◽  
pp. 17-23 ◽  
Author(s):  
M. J. Crabb ◽  
J. Duncan
Keyword(s):  
Banach Space ◽  
Banach Algebra ◽  
Unit Circle ◽  
Banach Algebras ◽  
Bounded Operators ◽  
Image Position ◽  
Unital Banach Algebra

Let A be a complex unital Banach algebra. An element u∈A is a norm unitary if(For the algebra of all bounded operators on a Banach space, the norm unitaries arethe invertible isometries.) Given a norm unitary u∈A, we have Sp(u)⊃Γ, where Sp(u) denotes the spectrum of u and Γ denotes the unit circle in C. If Sp(u)≠Γ we may suppose, by replacing eiθu, that . Then there exists h ∈ A such that

scholarly journals Higher point derivations on commutative Banach algebras, III

1981 ◽  
Vol 24 (1) ◽  
pp. 31-40 ◽  
Author(s):  
H. G. Dales ◽  
J. P. McClure
Keyword(s):  
Banach Algebra ◽  
Infinite Order ◽  
Infinite Sequence ◽  
Banach Algebras ◽  
Commutative Banach Algebra ◽  
Complex Field ◽  
Linear Functionals ◽  
Point Derivation ◽  
Commutative Banach Algebras ◽  
Image Position

Let A be a commutative Banach algebra with identity 1 over the complex field C, and let d0 be a character on A. We recall that a (higher) point derivation of order q on A at d0 is a sequence d1, …, dq of linear functionals on A such that the identitieshold for each choice of f and g in A and k in {1, …, q}. A point derivation of infinite order is an infinite sequence {dk} of linear functionals such that (1.1) holds for all k. A point derivation is continuous if each dk is continuous, totally discontinuous if dk is discontinuous for each k≧1, and degenerate if d1 = 0.

A representation theorem for partially ordered Banach algebras

1968 ◽  
Vol 64 (1) ◽  
pp. 53-59 ◽  
Author(s):  
Kung-Fu Ng
Keyword(s):  
Banach Space ◽  
Banach Algebra ◽  
Representation Theorem ◽  
Banach Algebras ◽  
Positive Cone ◽  
Partially Ordered ◽  
Real Algebra ◽  
Empty Subset ◽  
Image Position ◽  
Ordered Banach Algebra

Let be a real algebra which is also a Banach space. Then is called a partially ordered Banach algebra if there is specified a non-empty subset of , called the positive cone, such thatand(A 5) is 1-normal and closed. (We recall that is said to be 1-normal if

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•

Nonstandard Ideals in Radical Convolution Algebras on a Half-Line

1987 ◽  
Vol 39 (2) ◽  
pp. 309-321 ◽  
Author(s):  
H. G. Dales ◽  
J. P. McClure
Keyword(s):  
Banach Space ◽  
Banach Algebra ◽  
Banach Algebras ◽  
Half Line ◽  
Positive Sequence ◽  
Convolution Product ◽  
Weight Sequence ◽  
Convolution Algebras ◽  
Image Position

This note is about the interplay between two classes of radical Banach algebras, and we begin by describing the algebras in question.A weight sequence is a positive sequence w = (wn) defined on Z+ (the non-negative integers) and satisfying w0 = 1 and wm+n ≦ wmwn for all m and n in Z+. For such a sequence w, the Banach spaceis a Banach algebra with respect to the convolution product, defined by

Jordan algebras spanned by Hermitian elements of a Banach algebra

1977 ◽  
Vol 81 (1) ◽  
pp. 3-13 ◽  
Author(s):  
F. F. Bonsall
Keyword(s):  
Hilbert Space ◽  
Banach Algebra ◽  
Banach Algebras ◽  
Jordan Algebras ◽  
The Self ◽  
Image Position ◽  
Adjoint Operators ◽  
Unital Banach Algebra ◽  
Algebra Of Operators

The Vidav–Palmer theorem [(11), (5), (2) (p. 65)] characterizes C*-algebras among Banach algebras in terms of the algebra and norm structure alone, without reference to an involution, in the following way. Let B denote a complex unital Banach algebra, and let Her (B) denote the set of Hermitian elements of B, that is the elements of B with real numerical ranges. In this notation, the Vidav–Palmer theorem tells us that ifthen B is isometrically isomorphic to a C*-algebra of operators on a Hilbert space, with the Hermitian elements corresponding to the self-adjoint operators in the C*-algebra.

Central Double Centralizers on Quasi-Central Banach Algebras with Bounded Approximate Identity

1983 ◽  
Vol 35 (2) ◽  
pp. 373-384
Author(s):  
Sin-Ei Takahasi
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Approximate Identity ◽  
Central Complex ◽  
Double Centralizer ◽  
Continuous Complex ◽  
Image Position ◽  
Centralizer Algebra ◽  
Complex Valued ◽  
Canonical Image

We assume throughout this paper that A is a semi-simple, quasi-central, complex Banach algebra with a bounded approximate identity {eα}. The author [6] has shown that every central double centralizer T on A can be, under suitable conditions, represented as a bounded continuous complex-valued function ΦT on Prim A, the structure space of A with the hull-kernel topology, such thatHere x + P for P ∊ Prim A denotes the canonical image of x in A/P. This map Φ is called Dixmier's representation of Z(M(A)), the central double centralizer algebra of A. We denote by τ the canonical isomorphism of A into the Banach algebra D(A) with the restricted Arens product as defined in [6]. Also denote by μ Davenport's representation of Z(M(A)). In fact, this map μ is given byfor each T ∊ Z(M(A)).

Banach algebras and absolutely summing operators

1976 ◽  
Vol 80 (3) ◽  
pp. 465-473 ◽  
Author(s):  
Andrew M. Tonge
Keyword(s):  
Banach Algebra ◽  
Operator Algebra ◽  
Dual Space ◽  
Banach Algebras ◽  
Uniform Algebra ◽  
Bounded Linear ◽  
Linear Map ◽  
Absolutely Summing Operators ◽  
Image Position

If R is a Banach algebra and ø ∈ R′, the dual space, then we may define a bounded linear map byWe shall show that for suitable p the requirement that each be p-absolutely summing constrains R to be an operator algebra, or even, in certain cases, a uniform algebra. In this way we are able to give generalizations of results of Varopoulos (12) and Kaijser (4).

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