scholarly journals Banach algebras with one dimensional radical

1983 ◽  
Vol 27 (1) ◽  
pp. 115-119
Author(s):  
Lawrence Stedman
Keyword(s):  
Tensor Product ◽  
Banach Algebra ◽  
Banach Algebras ◽  
Quotient Algebra ◽  
One Dimensional ◽  
Natural Mapping ◽  
Algebraic Tensor Product

A Banach algebra A with radical R is said to have property (S) if the natural mapping from the algebraic tensor product A ⊗ A onto A2 is open, when A ⊗ A is given the protective norm. The purpose of this note is to provide a counterexample to Zinde's claim that when A is commutative and R is one dimensional the fulfillment of property (S) in A implies its fulfillment in the quotient algebra A/R.

Representation theory for tensor products of Banach algebras

1981 ◽  
Vol 90 (3) ◽  
pp. 445-463 ◽  
Author(s):  
T. K. Carne
Keyword(s):  
Representation Theory ◽  
Tensor Product ◽  
Banach Algebra ◽  
Banach Algebras ◽  
Tensor Products ◽  
The Subject ◽  
Algebraic Tensor Product ◽  
Natural Way

The algebraic tensor product A1⊗A2 of two Banach algebras is an algebra in a natural way. There are certain norms α on this tensor product for which the multiplication is continuous so that the completion, A1αA2, is a Banach algebra. The representation theory of such tensor products is the subject of this paper. It will be shown that, under certain simple conditions, the tensor product of two semi-simple Banach algebras is semi-simple although, without these conditions, the result fails.

scholarly journals Some results on the projective cone normed tensor product spaces over banach algebras

2018 ◽  
Vol 38 (1) ◽  
pp. 197 ◽  
Author(s):  
Dipankar Das ◽  
Nilakshi Goswami ◽  
Vishnu Narayan Mishra
Keyword(s):  
Fixed Point ◽  
Tensor Product ◽  
Banach Algebra ◽  
Fixed Point Theorems ◽  
Banach Algebras ◽  
Iteration Scheme ◽  
Tensor Product Space ◽  
The Common ◽  
The Stability ◽  
Algebraic Tensor Product

For two real Banach algebras $\mathbb{A}_1$ and $\mathbb{A}_2$, let $K_p$ be the projective cone in $\mathbb{A}_1\otimes_\gamma \mathbb{A}_2$. Using this we define a cone norm on the algebraic tensor product of two vector spaces over the Banach algebra $\mathbb{A}_1\otimes_\gamma \mathbb{A}_2$ and discuss some properties. We derive some fixed point theorems in this projective cone normed tensor product space over Banach algebra with a suitable example. For two self mappings $S$ and $T$ on a cone Banach space over Banach algebra, the stability of the iteration scheme $x_{2n+1}=Sx_{2n}$, $x_{2n+2}=Tx_{2n+1},\;n=0,1,2,...$ converging to the common fixed point of $S$ and $T$ is also discussed here.

The injective tensor product of ℒp-algebras

1978 ◽  
Vol 83 (2) ◽  
pp. 237-242 ◽  
Author(s):  
T. K. Carne ◽  
A. M. Tonge
Keyword(s):  
Tensor Product ◽  
Banach Algebra ◽  
Banach Algebras ◽  
Injective Tensor Product ◽  
Algebraic Tensor Product

When A1 and A2 are Banach algebras, their algebraic tensor product A1 ⊗ A2 has a natural multiplication. In this paper we investigate when the condition that A1 and A2 are ℒp-spaces constrains this multiplication to extend to the injective tensor product A1A2, making it a Banach algebra.

scholarly journals Identities in tensor products of Banach algebras

1970 ◽  
Vol 2 (2) ◽  
pp. 253-260 ◽  
Author(s):  
R. J. Loy
Keyword(s):  
Tensor Product ◽  
Banach Algebras ◽  
Tensor Products ◽  
Complex Field ◽  
Approximate Identities ◽  
Algebraic Tensor Product

Let A1, A2 be Banach algebras, A1 ⊗ A2 their algebraic tensor product over the complex field. If ‖ · ‖α is an algebra norm on A1 ⊗ A2 we write A1 ⊗αA2 for the ‖ · ‖α-completion of A1 ⊗ A2. In this note we study the existence of identities and approximate identities in A1 ⊗αA2 versus their existence in A1 and A2. Some of the results obtained are already known, but our method of proof appears new, though it is quite elementary.

On Maximal Regular Ideals and Identities in the Tensor Product of Commutative Banach Algebras

1969 ◽  
Vol 21 ◽  
pp. 639-647 ◽  
Author(s):  
L. J. Lardy ◽  
J. A. Lindberg
Keyword(s):  
Tensor Product ◽  
Banach Algebras ◽  
Complex Numbers ◽  
Normed Algebra ◽  
Commutative Banach Algebras ◽  
Algebraic Tensor Product

Let A1and A2be commutative Banach algebras and A1 ⊙ A2 their algebraic tensor product over the complex numbers C.There is always a t least one norm, namely the greatest cross-norm γ (2), on A1 ⊙ A2 that renders it a normed algebra. We shall write A1 ⊗αA2 for the α-completion of A1⊙ A2when αis an algebra norm on A1⊙ A2.Gelbaum (2; 3), Tomiyama (9), and Gil de Lamadrid (4) have shown that for certain algebra norms α on A1⊙ A2 every complex homomorphism on A1 ⊙ A2 is α-continuous. In § 3 of this paper, we present a condition on an algebra norm α which is equivalent to the α-continuity of every complex homomorphism on A1⊙ A2.

scholarly journals Commutativity of Banach algebras characterized by primitive ideals and spectra

Filomat ◽  
2017 ◽  
Vol 31 (7) ◽  
pp. 2053-2060
Author(s):  
Amin Hosseini
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Quotient Algebra ◽  
Linear Functionals ◽  
Primitive Ideals

This study is an attempt to prove the following main results. Let A be a Banach algebra and U = A ? C be its unitization. By ?c(U), we denote the set of all primitive ideals P of U such that the quotient algebra U/P is commutative. We prove that if A is semi-prime and dim(?P??c(U)P)? 1, then A is commutative. Moreover, we prove the following: Let A be a semi-simple Banach algebra. Then, A is commutative if and only if G(a) = {?(a)? ? ?A} ? {0} or G(a) = {?(a)| ? ? ?A} for every a ? A, where G(a) and ?A denote the spectrum of an element a ? A, and the set of all non-zero multiplicative linear functionals on A, respectively.

ON ALGEBRA ISOMORPHISMS BETWEEN p-BANACH BEURLING ALGEBRAS

2021 ◽  
pp. 1-12
Author(s):  
PRAKASH A. DABHI ◽  
DARSHANA B. LIKHADA
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Discrete Groups ◽  
Discrete Semigroups ◽  
Beurling Algebras

Abstract Let $(G_1,\omega _1)$ and $(G_2,\omega _2)$ be weighted discrete groups and $0\lt p\leq 1$ . We characterise biseparating bicontinuous algebra isomorphisms on the p-Banach algebra $\ell ^p(G_1,\omega _1)$ . We also characterise bipositive and isometric algebra isomorphisms between the p-Banach algebras $\ell ^p(G_1,\omega _1)$ and $\ell ^p(G_2,\omega _2)$ and isometric algebra isomorphisms between $\ell ^p(S_1,\omega _1)$ and $\ell ^p(S_2,\omega _2)$ , where $(S_1,\omega _1)$ and $(S_2,\omega _2)$ are weighted discrete semigroups.

scholarly journals Nuclear subalgebras of UHF C*-algebras

1986 ◽  
Vol 29 (1) ◽  
pp. 97-100 ◽  
Author(s):  
R. J. Archbold ◽  
Alexander Kumjian
Keyword(s):  
Tensor Product ◽  
Inductive Limit ◽  
Inductive Limits ◽  
C Algebra ◽  
C Algebras ◽  
Finite Dimensional ◽  
The Family ◽  
Algebraic Tensor Product

A C*-algebra A is said to be approximately finite dimensional (AF) if it is the inductive limit of a sequence of finite dimensional C*-algebras(see [2], [5]). It is said to be nuclear if, for each C*-algebra B, there is a unique C*-norm on the *-algebraic tensor product A ⊗B [11]. Since finite dimensional C*-algebras are nuclear, and inductive limits of nuclear C*-algebras are nuclear [16];,every AF C*-algebra is nuclear. The family of nuclear C*-algebras is a large and well-behaved class (see [12]). The AF C*-algebras for a particularly tractable sub-class which has been completely classified in terms of the invariant K0 [7], [5].

A characterization of Jordan and 5-Jordan homomorphisms between Banach algebras

2018 ◽  
Vol 11 (02) ◽  
pp. 1850021 ◽  
Author(s):  
A. Zivari-Kazempour
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Commutative Banach Algebra ◽  
Jordan Homomorphism ◽  
Jordan Homomorphisms ◽  
Unital Banach Algebra

We prove that each surjective Jordan homomorphism from a Banach algebra [Formula: see text] onto a semiprime commutative Banach algebra [Formula: see text] is a homomorphism, and each 5-Jordan homomorphism from a unital Banach algebra [Formula: see text] into a semisimple commutative Banach algebra [Formula: see text] is a 5-homomorphism.

scholarly journals Compact linear operators from an algebraic standpoint

1967 ◽  
Vol 8 (1) ◽  
pp. 41-49 ◽  
Author(s):  
F. F. Bonsall
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Linear Operators ◽  
Identity Operator ◽  
Algebra Structure ◽  
Natural Setting ◽  
Norm Topology ◽  
Bounded Linear ◽  
Underlying Space ◽  
Closed Subalgebra

Let B(X) denote the Banach algebra of all bounded linear operators on a Banach space X. Let t be an element of B(X), and let edenote the identity operator on X. Since the earliest days of the theory of Banach algebras, ithas been understood that the natural setting within which to study spectral properties of t is the Banach algebra B(X), or perhaps a closed subalgebra of B(X) containing t and e. The effective application of this method to a given class of operators depends upon first translating the data into terms involving only the Banach algebra structure of B(X) without reference to the underlying space X. In particular, the appropriate topology is the norm topology in B(X) given by the usual operator norm. Theorem 1 carries out this translation for the class of compact operators t. It is proved that if t is compact, then multiplication by t is a compact linear operator on the closed subalgebra of B(X) consisting of operators that commute with t.

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