Characterizations of isomorphisms and derivations of some algebras

Let ? be a zero-product preserving bijective bounded linear map from a unital algebra A onto a unital algebra B such that ?(1)=k. We show that if A is a CSL algebra on a Hilbert space or a J-lattice algebra on a Banach space then there exists an isomorphism ψ from A onto B such that ?=kψ. For a nest algebra A in a factor von Neumann algebra, we characterize the linear maps on A such that δ(x)y+xδ(y)=0 for all x,y∈A with xy=0.     

π-Complementation in the Unitisation and Multiplication Algebras of a Semiprime Algebra

π-complemented algebras are defined as those (not necessarily associative or unital) algebras such that each annihilator ideal is complemented by other annihilator ideal. For a given semiprime algebra A, we discuss the π-complementation of the unitisation algebra A ¹ of A. Moreover, if in addition the multiplication algebra ?(A) of A is also semiprime, we study the π-complementation in the algebras ?(A) and ?^?(A) (the multiplication ideal of A). In associative setting, we prove that A is π-complemented if and only if ?^?(A) is π-complemented, and that A ¹ π-complemented if and only if ?(A) is π-complemented.     

Contractability of l 1 -Munn algebras with applications

A Banach algebra \A is called contractible if every bounded derivation from A into any Banach A bimodule is inner. In this article we show that a l ¹ -Munn algebra LM(A, P) is contractible if and only if A is contractible and LM(A, P) is unital. As a consequence, if a semigroup algebra l ¹(S) is contractible, then S is finite. March 30, 1999     

Nonlinear Lie derivations on upper triangular matrices

Let M _n (𝔸) and T _n (𝔸) be the algebra of all n?×?n matrices and the algebra of all n?×?n upper triangular matrices over a commutative unital algebra 𝔸, respectively. In this note we prove that every nonlinear Lie derivation from T _n (𝔸) into M _n (𝔸) is of the form A?→?AT???TA?+?A _??+?ξ(A)I _n , where T?∈?M _n (𝔸), ??:?𝔸?→?𝔸 is an additive derivation, ξ?:?T _n (𝔸)?→?𝔸 is a nonlinear map with ξ(AB???BA)?=?0 for all A,?B?∈?T _n (𝔸) and A _? is the image of A under???applied entrywise.     

Weak continuous states on Banach algebras

We prove that if a unital Banach algebra A is the dual of a Banach space A_? then the set of normal states is weak^∗ dense in the set of all states on A. Further, normal states linearly span A_?.     

On the Simplicity of Lie Algebras Associated to Leavitt Algebras

For any field 𝕂 and integer n ≥ 2, we consider the Leavitt algebra L _𝕂(n); for any integer d ≥ 1, we form the matrix ring S = M _d (L _𝕂(n)). S is an associative algebra, but we view S as a Lie algebra using the bracket [a, b] = ab ? ba for a, b ∈ S. We denote this Lie algebra as S ^?, and consider its Lie subalgebra [S ^?, S ^?]. In our main result, we show that [S ^?, S ^?] is a simple Lie algebra if and only if char(𝕂) divides n ? 1 and char(𝕂) does not divide d. In particular, when d = 1, we get that [L _𝕂(n)^?, L _𝕂(n)^?] is a simple Lie algebra if and only if char(𝕂) divides n ? 1.     

Characterizations of Jordan derivations and Jordan homomorphisms

Let 𝒜 be a unital Banach algebra and ? be a unital 𝒜-bimodule. We show that if δ is a linear mapping from 𝒜 into ? satisfying δ(ST)?=?δ(S)T?+Sδ(T) for any S, T?∈?𝒜 with ST?=?W, where W is a left or right separating point of ?, then δ is a Jordan derivation. Also, it is shown that every linear mapping h from 𝒜 into a unital Banach algebra ? which satisfies h(S)h(T)?=?h(ST) for any S,?T?∈?𝒜 with ST?=?W is a Jordan homomorphism if h(W) is a separating point of ?.     

Linear mappings derivable at some nontrivial elements

Suppose that A is an algebra and M is an A-bimodule. Let A be any element in A. A linear mapping δ from A into M is said to be derivable at A if δ(ST)=δ(S)T+Sδ(T) for any S,T in A with ST=A. Given an algebra A, such as a non-abelian von Neumann algebra or an irreducible CDCSL algebra on a Hilbert space H with dimH?2, we show that there exists a nontrivial idempotent P in A such that for any Q∈PAP which is invertible in PAP, every linear mapping derivable at Q from A into some unital A-bimodule (for example, A or B(H)) is derivation.     

Inverse closedness ofC *-algebras in Banach algebras

LetA denote a unital Banach algebra, and letB denote aC ^*-algebra which is contained as a unital subalgebra inA. We prove thatB is inverse closed inA if the norms ofA andB coincide. This generalizes well known result about inverse closedness ofC ^*-subalgebras inC ^*-algebras.     

Some remarks on Bridgeland’s Hall algebras

We study the functorial properties of Bridgeland’s Hall algebras. Specifically, let 𝒜 and ? be two categories satisfying certain conditions for the definitions of Bridgeland’s Hall algebras, and let F:𝒜→? be a fully faithful exact functor, which preserves projectives, then F induces an embedding of algebras from the Bridgeland’s Hall algebra of 𝒜 to the one of ?. In addition, let A be a finite-dimensional algebra over a finite field and B some special quotient algebra of A, then the Bridgeland’s Hall algebra of B is the quotient algebra of the one of A. Moreover, we consider the BGP-reflection functors on the category of 2-cyclic complexes and obtain some homomorphisms of algebras among the subalgebras of Bridgeland’s Hall algebras.     

Derivation Simple Color Algebras and Semisimple Lie Color Algebras

Let K be an algebraically closed field of arbitrary characteristic and Γ an abelian multiplicative group equipped with a bicharacter ε: Γ × Γ → K*. It is proved that, for any finite-dimensional derivation simple color algebra A over K, there exists a simple color algebra S and a color vector space V such that A? S? S^ε(V), where S^ε(V) is the ε-symmetric algebra of V. As an application of this result, a necessary and sufficient condition such that a Lie color algebra is semisimple is obtained.     

Examples of banach algebras over commutativebanach algebras of given homological dimension

We prove that, for every natural number n, there exists a unital semisimple Banach star algebra Aand a closed star subalgebra Bof the centre of A, different from C, such that the global B-homological dimension and the B-homological bidimension of Aare both equal to n. The algebras Aand Bcan be taken to be function algebras.     

A Note on Range-Inclusive Maps in Rings with Idempotents

Let A be a ring, let M be an A-bimodule, and let C be the center of M. A map F: A → M is said to be range-inclusive if [F(x), A] ? [x, M] for every x ? A. Recently, Bre?ar proved that if A is a unital ring and M a unital A-bimodule such that A contains wide idempotents, then every range-inclusive additive map F: A → M is of the form F(x) = λx + μ(x) for some λ ?C and μ: A → C. Our main purpose is to remove the assumption of unitality in the above result.     

Trace Invariants of Finite-Dimensional Algebras

With every finite-dimensional algebra A over any field k we associate an 8-tuple of linear or bilinear forms on A, all of which are defined in terms of traces. For every groupoid 𝒞 formed by a class of k-algebras of fixed finite dimension, this passage is functorial and, when composed with any map that is constant on isoclasses, gives rise to an abundance of maps f: 𝒞 → I such that the fibres of f form a block decomposition of 𝒞. We study this decomposition for specific choices of 𝒞 and f, thereby putting established results from diverse algebraic theories into a unifying perspective, but also gaining new insight into classical groupoids of algebras such as associative unital algebras, division algebras, or composition algebras over general ground fields.     

Continuity of epimorphisms and derivations on vector-valued group algebras

This paper deals with the automatic continuity theory for the convolution algebra of all Bochner integrable functions from a locally compact abelian group G into an arbitrary unital complex Banach algebra A. For non-compact G, it is shown that all epimorphisms and all derivations on this vector-valued group algebra are necessarily continuous while for compact G, such results depend heavily on the automatic continuity properties of the range algebra a. Dedicated to Heinz Konig on the occasion of his 65th birthday Research supported by Grant SNF 11-1015 from the Danish Science Research Council.     

Additivity of Lie multiplicative maps on triangular algebras

Let 𝒜 and ? be unital algebras over a commutative ring ?, and ? be a (𝒜,??)-bimodule, which is faithful as a left 𝒜-module and also as a right ?-module. Let 𝒰?=?Tri(𝒜,??,??) be the triangular algebra and 𝒱 any algebra over ?. Assume that Φ?:?𝒰?→?𝒱 is a Lie multiplicative isomorphism, that is, Φ satisfies Φ(ST???TS)?=?Φ(S)Φ(T)???Φ(T)Φ(S) for all S, T?∈?𝒰. Then Φ(S?+?T)?=?Φ(S)?+?Φ(T)?+?Z _S,T for all S, T?∈?𝒰, where Z _S,T is an element in the centre 𝒵(𝒱) of 𝒱 depending on S and T.     

Linear maps preserving Fredholm and Atkinson elements of C *-algebras

Let A be a unital C*-algebra of real rank zero and B be a unital semisimple complex Banach algebra. We characterize linear maps from A onto B preserving different essential spectral sets and quantities such as the essential spectrum, the (left, right) essential spectrum, the Weyl spectrum, the index and the essential spectral radius.     

Uniform diagonalisation of matrices over regular rings

The fundamental Separativity Problem for von Neumann regular rings is shown to be equivalent to a linear algebra problem: for a field F, is there a ``uniform formula' for diagonalising a matrix A over , independently of n? Here P and Q are required to be invertible matrices whose entries are fixed regular algebra expressions in the entries of A. Received July 10, 2000; accepted in final form September 26, 2000.     

Applications of the growth characteristics induced by the spectral distance

Let A be a complex unital Banach algebra. Using a connection between the spectral distance and the growth characteristics of a certain entire map into A, we derive a generalization of Gelfand’s famous power boundedness theorem. Elaborating on these ideas, with the help of a Phragm´en-Lindel¨of device for subharmonic functions, it is then shown, as the main result, that two normal elements of a C^?-algebra are equal if and only if they are quasinilpotent equivalent.