Derivations of Commutative Banach Algebras
نویسنده
چکیده
In [2] Singer and Wermer showed that a bounded derivation in a commutative Banach algebra 21 necessarily maps 21 into the radical 91. They conjectured at this time that the assumption of boundedness could be dropped. It is a corollary of results proved below that if 21 is in addition regular and semi-simple, this is indeed the case. What is actually proved here is that under the above hypotheses, if D is a derivation of 2Ï into C($%), <£» the structure space of 2Ï, then D is a bounded operator from 21 to C(^a). The topologies are the norm topology in 21 and the sup norm topology in C($a). An application of the closed graph theorem shows that if D maps 2Ï into itself, D must be a bounded operator in 21, hence by the Singer, Wermer theorem, D = 0. If 21 is regular but not semi-simple, then it follows from the above that D will map 21 into 9t provided that D maps 9Î into 9î. This the author can verify only if 91 is nilpotent. In what follows 2Ï will always denote a regular, commutative, semisimple Banach algebra with norm ||-||. Applying the Gelfand isomorphism we will identify 2Ï and the corresponding subalgebra of C($a). For convenience we also will assume 2Ï possesses an identity. I t is easily seen that this doesn't affect the generality of the results. Let ffît be a maximal ideal of 21, and 4> the corresponding point in $a. It is noted in [2 ] that there exists a derivation D of 21 into some semi-simple extension 33 of 21 iff SJî^SR^ for some maximal ideal SDÎ0. In fact S3 may be taken to be J5(a), the ring of bounded complex functions on $#. For if this condition is satisfied, following Singer and Wermer, we define by Zorn's Lemma a nontrivial linear functional f$ on 2Ï which annihilates $0$ and the identity. If we define D by
منابع مشابه
POINT DERIVATIONS ON BANACH ALGEBRAS OF α-LIPSCHITZ VECTOR-VALUED OPERATORS
The Lipschitz function algebras were first defined in the 1960s by some mathematicians, including Schubert. Initially, the Lipschitz real-value and complex-value functions are defined and quantitative properties of these algebras are investigated. Over time these algebras have been studied and generalized by many mathematicians such as Cao, Zhang, Xu, Weaver, and others. Let be a non-emp...
متن کاملDerivations in semiprime rings and Banach algebras
Let $R$ be a 2-torsion free semiprime ring with extended centroid $C$, $U$ the Utumi quotient ring of $R$ and $m,n>0$ are fixed integers. We show that if $R$ admits derivation $d$ such that $b[[d(x), x]_n,[y,d(y)]_m]=0$ for all $x,yin R$ where $0neq bin R$, then there exists a central idempotent element $e$ of $U$ such that $eU$ is commutative ring and $d$ induce a zero derivation on $(1-e)U$. ...
متن کاملDerivations on dual triangular Banach algebras
Ideal Connes-amenability of dual Banach algebras was investigated in [17] by A. Minapoor, A. Bodaghi and D. Ebrahimi Bagha. They studied weak∗continuous derivations from dual Banach algebras into their weak∗-closed two- sided ideals. This work considers weak∗-continuous derivations of dual triangular Banach algebras into their weak∗-closed two- sided ideals . We investigate when weak∗continuous...
متن کاملAlgebras defined by homomorphisms
Let $mathcal{R}$ be a commutative ring with identity, let $A$ and $B$ be two $mathcal{R}$-algebras and $varphi:Blongrightarrow A$ be an $mathcal{R}$-additive algebra homomorphism. We introduce a new algebra $Atimes_varphi B$, and give some basic properties of this algebra. Generalized $2$-cocycle derivations on $Atimes_varphi B$ are studied. Accordingly, $Atimes_varphi B$ is considered from th...
متن کاملHomomorphism Weak amenability of certain Banach algebras
In this paper we introduce the notion of $varphi$-commutativity for a Banach algebra $A$, where $varphi$ is a continuous homomorphism on $A$ and study the concept of $varphi$-weak amenability for $varphi$-commutative Banach algebras. We give an example to show that the class of $varphi$-weakly amenable Banach algebras is larger than that of weakly amenable commutative Banach algebras. We charac...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2007