scholarly journals 2-normed algebras-II

2011 ◽  
Vol 90 (104) ◽  
pp. 135-143
Author(s):  
Neeraj Srivastava ◽  
S. Bhattacharya ◽  
S.N. Lal
Keyword(s):  
Banach Space ◽  
Banach Algebra ◽  
Spectral Radius ◽  
Banach Algebras ◽  
Basic Properties ◽  
Definition Of ◽  
Spectral Radius Formula

In the first part of the paper [5], we gave a new definition of real or complex 2-normed algebras and 2-Banach algebras. Here we give two examples which establish that not all 2-normed algebras are normable and a 2-Banach algebra need not be a 2-Banach space. We conclude by deriving a new and interesting spectral radius formula for 1-Banach algebras from the basic properties of 2-Banach algebras and thus vindicating our definitions of 2-normed and 2-Banach algebras given in [5].

scholarly journals The spectral theorem in Banach algebras

1972 ◽  
Vol 13 (1) ◽  
pp. 49-55
Author(s):  
Stephen Plafker
Keyword(s):  
Banach Space ◽  
Banach Algebra ◽  
Banach Algebras ◽  
Hermitian Operator ◽  
Spectral Theorem ◽  
Continuous Linear ◽  
Hermitian Element ◽  
Definition Of ◽  
Original Norm ◽  
Original Theorem

The concept of a hermitian element of a Banach algebra was first introduced by Vidav [21] who proved that, if a Banach algebra 𝒜 has “enough” hermitian elements, then 𝒜 can be renormed and given an involution to make it a stellar algebra. (Following Bourbaki [5] we shall use the expression “stellar algebra” in place of the term “C*-algebra”.) This theorem was improved by Berkson [2], Glickfeld [10] and Palmer [17]. The improvements consist of removing hypotheses from Vidav's original theorem and in showing that Vidav's new norm is in fact the original norm of the algebra. Lumer [13] gave a spatial definition of a hermitian operator on a Banach space E and proved it to be equivalent to Vidav's definition when one considers the Banach algebra 𝓛(E) of continuous linear mappings of E into E.

scholarly journals The Linearity of Riemann Integral on Functions from ℝ into Real Banach Space

2013 ◽  
Vol 21 (3) ◽  
pp. 185-191
Author(s):  
Keiko Narita ◽  
Noboru Endou ◽  
Yasunari Shidama
Keyword(s):  
Banach Space ◽  
Integral Operator ◽  
Real Banach Space ◽  
Closed Interval ◽  
Continuous Functions ◽  
Real Numbers ◽  
Riemann Integral ◽  
Basic Properties ◽  
Bounded Functions ◽  
Definition Of

Summary In this article, we described basic properties of Riemann integral on functions from R into Real Banach Space. We proved mainly the linearity of integral operator about the integral of continuous functions on closed interval of the set of real numbers. These theorems were based on the article [10] and we referred to the former articles about Riemann integral. We applied definitions and theorems introduced in the article [9] and the article [11] to the proof. Using the definition of the article [10], we also proved some theorems on bounded functions.

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 ◽  
Image Position

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 A formula for the inner spectral radius

2004 ◽  
Vol 2004 (61) ◽  
pp. 3285-3290
Author(s):  
S. Mahmoud Manjegani
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Asymptotic Formula ◽  
Spectral Radius ◽  
Bounded Linear Operator ◽  
Bounded Linear ◽  
Spectral Radius Formula

This note presents an asymptotic formula for the minimum of the moduli of the elements in the spectrum of a bounded linear operator acting on Banach spaceX. This minimum moduli is called the inner spectral radius, and the formula established herein is an analogue of Gelfand's spectral radius formula.

scholarly journals Approximate complementation and its applications in studying ideals of Banach algebras

2003 ◽  
Vol 92 (2) ◽  
pp. 301 ◽  
Author(s):  
Yong Zhang
Keyword(s):  
Banach Space ◽  
Banach Algebra ◽  
Compact Subset ◽  
Approximation Property ◽  
Banach Algebras ◽  
Approximate Identity ◽  
Amenable Banach Algebra

We show that a subspace of a Banach space having the approximation property inherits this property if and only if it is approximately complemented in the space. For an amenable Banach algebra a closed left, right or two-sided ideal admits a bounded right, left or two-sided approximate identity if and only if it is bounded approximately complemented in the algebra. If an amenable Banach algebra has a symmetric diagonal, then a closed left (right) ideal $J$ has a right (resp. left) approximate identity $(p_{\alpha})$ such that, for every compact subset $K$ of $J$, the net $(a\cdot p_{\alpha})$ (resp. $(p_{\alpha}\cdot a)$) converges to $a$ uniformly for $a \in K$ if and only if $J$ is approximately complemented in the algebra.

Amenability of Lipschitz algebras

1992 ◽  
Vol 112 (3) ◽  
pp. 581-588 ◽  
Author(s):  
Frédéric Gourdeau
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Alternative Definition ◽  
Definition Of

In this article, we study the amenability of Banach algebras in general, and that of Lipschitz algebras in particular. After introducing an alternative definition of amenability, we extend a result of [5], thereby proving a new characterization of amenability for Banach algebras. This characterization relates the amenability of a Banach algebra A to the space of bounded homomorphisms from A into another Banach algebra B (Theorem 4). This result allows us to solve the problem of amenability for virtually all Lipschitz algebras (of complex or Banach algebra valued functions), a class of algebras which has been studied in [2], [4] and [5].

scholarly journals Completely continuous elements of Banach algebras related to locally compact groups

2007 ◽  
Vol 76 (1) ◽  
pp. 49-54 ◽  
Author(s):  
M. J. Mehdipour ◽  
R. Nasr-Isfahani
Keyword(s):  
Banach Space ◽  
Compact Group ◽  
Banach Algebra ◽  
Banach Algebras ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Completely Continuous ◽  
Measurable Functions

Let G be a locally compact group and be the Banach space of all essentially bounded measurable functions on G vansihing an infinity. Here, we study some families of right completely continuous elements in the Banach algebra equipped with an Arens type product. As the main result, we show that has a certain right completely continuous element if and only if G is compact.

scholarly journals ZERO JORDAN PRODUCT DETERMINED BANACH ALGEBRAS

2020 ◽  
pp. 1-14
Author(s):  
J. ALAMINOS ◽  
M. BREŠAR ◽  
J. EXTREMERA ◽  
A. R. VILLENA
Keyword(s):  
Banach Space ◽  
Banach Algebra ◽  
Banach Algebras ◽  
Group Algebras ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Continuous Linear ◽  
Bilinear Map ◽  
Locally Compact ◽  
Jordan Product

A Banach algebra $A$ is said to be a zero Jordan product determined Banach algebra if, for every Banach space $X$ , every bilinear map $\unicode[STIX]{x1D711}:A\times A\rightarrow X$ satisfying $\unicode[STIX]{x1D711}(a,b)=0$ whenever $a$ , $b\in A$ are such that $ab+ba=0$ , is of the form $\unicode[STIX]{x1D711}(a,b)=\unicode[STIX]{x1D70E}(ab+ba)$ for some continuous linear map $\unicode[STIX]{x1D70E}$ . We show that all $C^{\ast }$ -algebras and all group algebras $L^{1}(G)$ of amenable locally compact groups have this property and also discuss some applications.

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

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