Ilja Gogić
We consider derivations from the image of the canonical contraction $θ_A$ from the Haagerup tensor product of a C*-algebra A with itself to the space of completely bounded maps on A. We show that such derivations are necessarily inner if A is prime or if A is quasicentral with Hausdorff primitive spectrum. We also provide an example of a C*-algebra which has outer elementary derivations.
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We define a noncommutative differential calculus constructed from the inner derivation, then several relevant examples are showed. It is of interest to note that for certain $C^*$-algebra, this calculus is closely related to the classical one when the algebra associates a deformation parameter.
It is established that every derivation continuous with respect to the local measure topology acting on the *-algebra $LS(\mathcal{M})$ of all locally measurable operators affiliated with a von Neumann algebra $\mathcal{M}$ is necessary inner. If $\mathcal{M}$ is a properly infinite von Neumann algebra, then every derivation on $LS(\mathcal{M})$ is inner. In addition, it is proved that any derivation on $\mathcal{M}$ with values in Banach $\mathcal{M}$-bimodule of locally measurable operators is inner.
It is well known that every derivation of a von Neumann algebra into itself is an inner derivation and that every derivation of a von Neumann algebra into its predual is inner. It is less well known that every triple derivation (defined below) of a von Neumann algebra into itself is an inner triple derivation.We examine to what extent all triple derivations of a von Neumann algebra into its predual are inner. This rarely happens but it comes close. We prove a (triple) cohomological characterization of finite factors and a zero-one law for factors. Namely, we show that for any factor, the linea…
This paper is devoted to derivations on the algebra $S(M)$ of all measurable operators affiliated with a finite von Neumann algebra $M.$ We prove that if $M$ is a finite von Neumann algebra with a faithful normal semi-finite trace $τ$, equipped with the locally measure topology $t,$ then every $t$-continuous derivation $D:S(M)\rightarrow S(M)$ is inner. A similar result is valid for derivation on the algebra $S(M,τ)$ of $τ$-measurable operators equipped with the measure topology $t_τ$.
In the present paper we prove that every additive (not necessarily homogenous) local inner derivation on the algebra of matrices over an arbitrary field is an inner derivation, and every local inner derivation on the ring of matrices over a finite ring generated by the identity element or the ring of integers is an inner derivation. We also prove that every additive local inner derivation on the Jordan algebra of symmetric matrices over an arbitrary field is a derivation, and every local inner derivation on the Jordan ring of symmetric matrices over a finite ring generated by the identity elem…
We prove a Jordan version of Dorofeev's boundedness theorem for completely additive measues and use it to show that every (not necessarily linear nor continuous) 2-local triple derivation on a continuous JBW*-triple is a triple derivation.
Let $L$ be a locally compact Hausdorff space. Suppose $A$ is a C$^*$-algebra with the property that every weak-2-local derivation on $A$ is a {\rm(}linear{\rm)} derivation. We prove that every weak-2-local derivation on $C_0(L,A)$ is a {\rm(}linear{\rm)} derivation. Among the consequences we establish that if $B$ is an atomic von Neumann algebra or on a compact C$^*$-algebra, then every weak-2-local derivation on $C_0(L,B)$ is a linear derivation. We further show that, for a general von Neumann algebra $M$, every 2-local derivation on $C_0(L,M)$ is a linear derivation. We also prove several re…
Given a von Neumann algebra $M$ with a faithful normal semi-finite trace $τ,$ let $L(M, τ)$ be the algebra of all $τ$-measurable operators affiliated with $M.$ We prove that if $A$ is a locally convex reflexive complete metrizable solid $\ast$-subalgebra in $L(M, τ),$ which can be embedded into a locally bounded weak Fréchet $M$-bimodule, then any derivation on $A$ is inner.
Let $E=E(0,\infty)$ be a symmetric function space and $E(\mathcal{M},τ)$ be a symmetric operator space associated with a semifinite von Neumann algebra with a faithful normal semifinite trace. Our main result identifies the class of spaces $E$ for which every derivation $δ:\mathcal{A}\to E(\mathcal{M},τ)$ is necessarily inner for each $C^*$-subalgebra $\mathcal{A}$ in the class of all semifinite von Neumann algebras $\mathcal{M}$ as those with the Levi property.
Let $\mathcal{M}$ be a von Neumann algebra equipped with a faithful normal semi-finite trace $τ$ and let $S_0(τ)$ be the algebra of all $τ$-compact operators affiliated with $\mathcal{M}$. Let $E(τ)\subseteq S_0(τ)$ be a symmetric operator space (on $\mathcal{M}$) and let $\mathcal{E}$ be a symmetrically-normed Banach ideal of $τ$-compact operators in $\mathcal{M}$. We study (i) derivations $δ$ on $\mathcal{M}$ with the range in $E(τ)$ and (ii) derivations on the Banach algebra $\mathcal{E}$. In the first case our main results assert that such derivations are continuous (with respect to the no…
In the present paper we prove that every 2-local inner derivation on the matrix ring over a commutative ring is an inner derivation and every derivation on an associative ring has an extension to a derivation on the matrix ring over this associative ring. We also develop a Jordan analog of the above method and prove that every 2-local inner derivation on the Jordan matrix ring over a commutative ring is a derivation.
Let $α$ be an approximately inner flow on a $C^*$ algebra $A$ with generator $δ$ and let $δ_n$ denote the bounded generators of the approximating flows $α^{(n)}$. We analyze the structure of the set \cd=\{x\in D(δ): \lim_{n\to\infty}δ_n(x)=δ(x)\} of pointwise convergence of the generators. In particular we examine the relationship of $\cd$ and various cores related to spectral subspaces.
Modular Completely Dirichlet forms as Squares of Derivations
本文证明了冯·诺依曼代数上非迹权对应的GNS希尔伯特空间中，特定可闭导子可生成GNS对称的收缩完全正映射半群。
We show that if $C_u^*(X)$ is a uniform Roe algebra associated to a bounded geometry metric space X, then all bounded derivations on $C^*_u(X)$ are inner.
We look for an effective description of the algebra D_{Lie}(X,B) of operators on a bimodule X over an algebra B, generated by inner derivations. It is shown that in some important examples D_{Lie}(X,B) consists of all elementary operators x\to \sum_i a_ixb_i satisfying the conditions $\sum_i a_ib_i = \sum_i b_ia_i = 0. The Banach algebraic versions of these results are also obtained and applied to the description of closed Lie ideals in some Banach algebras, and to the proof of a density theorem for Lie algebras of operators on Hilbert space.
Suppose $A$ is a pro-C*-algebra. Let $L_{A}(E)$ be the pro-C*-algebra of adjointable operators on a Hilbert $A$-module $E$ and let $K_{A}(E)$ be the closed two sided $*$-ideal of all compact operators on $E$. We prove that if $E$ be a full Hilbert $A$-module, the innerness of derivations on $K_{A}(E)$ implies the innerness of derivations on $L_{A}(E)$. We show that if $A$ is a commutative pro-C*-algebra and $E$ is a Hilbert $A$-bimodule then every derivation on $K_{A}(E)$ is zero. Moreover, if $A$ is a commutative $σ$-C*-algebra and $E$ is a Hilbert $A$-bimodule then every derivation on $L_{A}…
We characterize derivations and 2-local derivations from $M_{n}(\mathcal{A})$ into $M_{n}(\mathcal{M})$, $n \ge 2$, where $\mathcal{A}$ is a unital algebra over $\mathbb{C}$ and $\mathcal{M}$ is a unital $\mathcal{A}$-bimodule. We show that every derivation $D: M_{n}(\mathcal{A}) \to M_{n}(\mathcal{M})$, $n \ge 2,$ is the sum of an inner derivation and a derivation induced by a derivation from $\mathcal{A}$ to $\mathcal{M}$. We say that $\mathcal{A}$ commutes with $\mathcal{M}$ if $am=ma$ for every $a\in\mathcal{A}$ and $m\in\mathcal{M}$. If $\mathcal{A}$ commutes with $\mathcal{M}$ we prove t…
In the present paper we prove that every 2-local inner derivation on the Lie ring of skew-symmetric matrices over a commutative ring is an inner derivation. We also apply our technique to various Lie algebras of infinite dimensional skew-adjoint matrix-valued maps on a set and prove that every 2-local spatial derivation on such algebras is a spatial derivation.
In the present paper it is proved that every inner 2-local derivation on the matrix ring $M_n(\Re)$ of $n\times n$ matrices over a commutative associative ring $\Re$ is an inner derivation. Also, it is proved that, every derivation on an associative ring $\Re$ has an extension to a derivation on the matrix ring $M_n(\Re)$ of $n\times n$ matrices over $\Re$.
This paper focuses on certain finite dimensional point derivations for the non-selfadjoint operator algebras corresponding to directed graphs. We begin by analyzing the derivations corresponding to full matrix representations of the tensor algebra of a directed graph. We determine when such a derivation is inner, and describe situations that give rise to non-inner derivations. We also analyze the situation when the derivation corresponds to a multiplicative linear functional.
We prove that every derivation acting on a von Neumann algebra $\mathcal{M}$ with values in a quasi-normed bimodule of locally measurable operators affiliated with $\mathcal{M}$ is necessarily inner.
The paper is devoted to 2-local derivations on matrix algebras over unital semi-prime Banach algebras. For a unital semi-prime Banach algebra $\mathcal{A}$ with the inner derivation property we prove that any 2-local derivation on the algebra $M_{2^n}(\mathcal{A}),$ $n\geq 2,$ is a derivation. We apply this result to $AW^\ast$-algebras and show that any 2-local derivation on an arbitrary $AW^\ast$-algebra is a derivation.
The celebrated Kadison--Sakai theorem states that every derivation on a von Neumann algebra is inner. In this paper, we prove this theorem for ultraweakly continuous *-σ-derivations, where σis an ultraweakly continuous surjective *-linear mapping. We decompose a $σ$-derivation into a sum of an inner σ-derivation and a multiple of a homomorphism. The so-called *-(σ,τ)-derivations are also discussed.
Given a von Neumann algebra $M$ with a faithful normal finite trace $τ$ denote by $L^Λ(M, τ)$ the generalized Arens algebra with respect to $M.$ We give a complete description of all additive derivations on the algebra $L^Λ(M, τ).$ In particular each additive derivation on the algebra $L^Λ(M, τ),$ where $M$ is a type II von Neumann algebra, is inner.
The paper is devoted to so-called local and 2-local derivations on the noncommutative Arens algebra $L^ω(M, τ)$ associated with a von Neumann algebra $M$ and a faithful normal semi-finite trace $τ.$ We prove that every 2-local derivation on $L^ω(M, τ)$ is a spatial derivation, and if $M$ is a finite von Neumann algebra, then each local derivation on $L^ω(M, τ)$ is also a spatial derivation and every 2-local derivation on $M$ is in fact an inner derivation.
We prove that every 2-local derivation from the algebra $M_n(\mathcal{A})(n>2)$ into its bimodule $M_n(\mathcal{M})$ is a derivation, where $\mathcal{A}$ is a unital Banach algebra and $\mathcal{M}$ is a unital $\mathcal{A}$-bimodule such that each Jordan derivation from $\mathcal{A}$ into $\mathcal{M}$ is an inner derivation, and that every 2-local derivation on a C*-algebra with a faithful traceable representation is a derivation. We also characterize local and 2-local Lie derivations on some algebras such as von Neumann algebras, nest algebras, Jiang-Su algebra and UHF algebras.
In this paper we investigate in details derivations on trivial extension algebras. We obtain generalizations of both known results on derivations on triangular matrix algebras and a known result on first cohomology group of trivial extension algebras. As a consequence we get the characterization of trivial extension algebras on which every derivation is inner. We show that, under some conditions, a trivial extension algebra on which every derivation is inner has necessarily a triangular matrix representation. The paper starts with detailed study (with examples) of the relation between the triv…
In this paper, we introduce the concept of trace-open projections in the second dual A** of a C*-algebra A, and we show that if there is a faithful normal semi-finite trace T on A**, and 1 is a T-open projection, then each 2-local derivation D from A to A** is an inner derivartion. We also show that this theorem hold for approximately 2-local derivations when A** is finite as a von neumann algebra.
Given a von Neumann algebra $M$ we introduce so called central extension $mix(M)$ of $M$. We show that $mix(M)$ is a *-subalgebra in the algebra $LS(M)$ of all locally measurable operators with respect to $M,$ and this algebra coincides with $LS(M)$ if and only if $M $ does not admit type II direct summands. We prove that if $M$ is a properly infinite von Neumann algebra then every additive derivation on the algebra $mix(M)$ is inner. This implies that on the algebra $LS(M)$, where $M$ is a type I$_\infty$ or a type III von Neumann algebra, all additive derivations are inner derivations.
In this paper we study derivations in subalgebras of $L_{0}^{wo}(ν;% \mathcal{L}(X)) $, the algebra of all weak operator measurable funtions $f:S\to \mathcal{L}(X) $, where $% \mathcal{L}(X) $ is the Banach algebra of all bounded linear operators on a Banach space $X$. It is shown, in particular, that all derivations on $L_{0}^{wo}(ν;\mathcal{L}(X)) $ are inner whenever $X$ is separable and infinite dimensional. This contrasts strongly with the fact that $L_{0}^{wo}(ν;\mathcal{L}(X)) $ admits non-trivial non-inner derivations whenever $X$ is finite dimensional and the measure $ν$ is non-atomic…
We make some observations about Rosenberg's Levi-Civita connections on noncommutative tori, noting the non-uniqueness of torsion-free metric-compatible connections without prescribed connection operator for the inner *-derivations, the nontrivial curvature form of the inner *-derivations, and the validity of the Gauss-Bonnet theorem for two classes of non-conformal deformations of the flat metric on the noncommutative two-tori, including the case of non-commuting scalings along the principal directions of a two-torus.
The derivation constant $K(A)\geq \frac{1}{2}$ has been extensively studied for \emph{unital} non-commutative $C^*$-algebras. In this paper, we investigate properties of $K(M(A))$ where $M(A)$ is the multiplier algebra of a non-unital $C^*$-algebra $A$. A number of general results are obtained which are then applied to the group $C^*$-algebras $A=C^*(G_N)$ where $G_N$ is the motion group $\R^N\rtimes SO(N)$. Utilising the rich topological structure of the unitary dual $\widehat{G_N}$, it is shown that, for $N\geq3$, $$K(M(C^*(G_N)))= \frac{1}{2}\left\lceil \frac{N}{2}\right\rceil.$$
Let $\mathcal{M}$ be a $W^*$-algebra and let $LS(\mathcal{M})$ be the algebra of all locally measurable operators affiliated with $\mathcal{M}$. It is shown that for any self-adjoint element $a\in LS(\mathcal{M})$ there exists a self-adjoint element $c_{_{0}}$ from the center of $LS(\mathcal{M})$, such that for any $ε>0$ there exists a unitary element $ u_ε$ from $\mathcal{M}$, satisfying $|[a,u_ε]| \geq (1-ε)|a-c_{_{0}}|$. A corollary of this result is that for any derivation $δ$ on $\mathcal{M}$ with the range in a (not necessarily norm-closed) ideal $I\subseteq\mathcal{M}$, the derivation $…
For a commutative C*-algebra $\mathcal A$ with unit $e$ and a Hilbert~$\mathcal A$-module $\mathcal M$, denote by End$_{\mathcal A}(\mathcal M)$ the algebra of all bounded $\mathcal A$-linear mappings on $\mathcal M$, and by End$^*_{\mathcal A}(\mathcal M)$ the algebra of all adjointable mappings on $\mathcal M$. We prove that if $\mathcal M$ is full, then each derivation on End$_{\mathcal A}(\mathcal M)$ is $\mathcal A$-linear, continuous, and inner, and each 2-local derivation on End$_{\mathcal A}(\mathcal M)$ or End$^{*}_{\mathcal A}(\mathcal M)$ is a derivation. If there exist $x_0$ in $\m…
Let $Ω$ be a compact Hausdorff space and let $A$ be a C$^*$-algebra. We prove that if every weak-2-local derivation on $A$ is a linear derivation and every derivation on $C(Ω,A)$ is inner, then every weak-2-local derivation $Δ:C(Ω,A)\to C(Ω,A)$ is a {\rm(}linear{\rm)} derivation. As a consequence we derive that, for every complex Hilbert space $H$, every weak-2-local derivation $Δ: C(Ω,B(H)) \to C(Ω,B(H))$ is a (linear) derivation. We actually show that the same conclusion remains true when $B(H)$ is replaced with an atomic von Neumann algebra. With a modified technique we prove that, if $B$ d…
To each projection $p$ in a $C^*$-algebra $A$ we associate a family of derivations on $A$, called $p$-derivations, and relate them to the space of triple derivations on $p A (1-p)$. We then show that every derivation on a ternary ring of operators is spatial and we investigate whether every such derivation on a weakly closed ternary ring of operators is inner.
A description of automorphisms and derivations of all two-dimensional algebras over an arbitrary field
本文对任意域上所有二维代数的自同构与导子进行了同构分类描述，彻底解决了相关的两个问题。
A well known result of Haagerup from 1983 states that every C*-algebra A is weakly amenable, that is, every (associative) derivation from A into its dual is inner. A Banach algebra B is said to be ternary weakly amenable if every continuous Jordan triple derivation from B into its dual is inner. We show that commutative C*-algebras are ternary weakly amenable, but that B(H) and K(H) are not, unless H is finite dimensional. More generally, we inaugurate the study of weak amenability for Jordan Banach triples, focussing on commutative JB*-triples and some Cartan factors.
Given a type I von Neumann algebra $M$ with a faithful normal semi-finite trace $τ,$ let $L(M, τ)$ be the algebra of all $τ$-measurable operators affiliated with $M.$ We give a complete description of all derivations on the algebra $L(M, τ).$ In particular, we prove that if $M$ is of type I$_\infty$ then every derivation on $L(M, τ)$ is inner.
We study a class of matrix function algebras, here denoted $\mathcal{T}^{+}(\mathcal{C}_n)$. We introduce a notion of point derivations, and classify the point derivations for certain finite dimensional representations of $\mathcal{T}^{+}(\mathcal{C}_n)$. We use point derivations and information about $n \times n$ matrices to show that every $\mathcal{T}^{+}(\mathcal{C}_n)$-valued derivation on $\mathcal{T}^{+}(\mathcal{C}_n)$ is inner.
Let $M$ be a type I von Neumann algebra with the center $Z,$ and let $LS(M)$ be the algebra of all locally measurable operators affiliated with $M.$ We prove that every $Z$-linear derivation on $LS(M)$ is inner. In particular all $Z$-linear derivations on the algebras of measurable and respectively totally measurable operators are spatial and implemented by elements from $LS(M).$
Suppose that $X$ is a (real or complex) Banach space, $dimX \geq 2$, and $\mathcal{N}$ is a nest on $X$, with each $N$ in $\mathcal{N}$ is complemented in $X$ whenever $N_{-}=N$. A ternary derivation of $Alg\mathcal{N}$ is a triple of linear maps $(γ, δ, τ)$ of $Alg\mathcal{N}$ such that $γ(AB)=δ(A)B +Aτ(B)$ for all $A,B \in Alg\mathcal{N}$. We show that for linear maps $δ$, $τ$ on $Alg\mathcal{N}$ there exists a unique linear map $γ$ from $Alg\mathcal{N}$ into $Alg\mathcal{N}$ defined by $γ(A)=RA+AT$ for some $R$, $T$ in $Alg\mathcal{N}$ such that $(γ, δ, τ)$ is a ternary derivation of $Alg\m…
We consider the relationship between derivations $d$ and $g$ of a Banach algebra $B$ that satisfy $\s(g(x)) \subseteq \s(d(x))$ for every $x\in B$, where $\s(\, . \,)$ stands for the spectrum. It turns out that in some basic situations, say if $B=B(X)$, the only possibilities are that $g=d$, $g=0$, and, if $d$ is an inner derivation implemented by an algebraic element of degree 2, also $g=-d$. The conclusions in more complex classes of algebras are not so simple, but are of a similar spirit. A rather definitive result is obtained for von Neumann algebras. In general $C^*$-algebras we have to m…
Given a von Neumann algebra $M$ with a faithful normal finite trace, we introduce the so called finite tracial algebra $M_f$ as the intersection of $L_p$-spaces $L_p(M, μ)$ over all $p \geq 1$ and over all faithful normal finite traces $μ$ on $M.$ Basic algebraic and topological properties of finite tracial algebras are studied. We prove that all derivations on these algebras are inner.
In the present paper derivations and *-automorphisms of algebras of unbounded operators over the ring of measurable functions are investigated and it is shown that all L^0-linear derivations and L^{0}-linear *-automorphisms are inner. Moreover, it is proved that each L^0-linear automorphism of the algebra of all linear operators on a bo-dense submodule of a Kaplansky-Hilbert module over the ring of measurable functions is spatial.
Let X be a Banach space over field F (R or C). Denote by B(X) the set of all bounded linear operators on X and by F(X) the set of all finite rank operators on X. A subalgebra A of B(X) is called a standard operator algebra if A contain F(X). We give a brief proof of a well-known result that every derivation from A into B(X) is inner. There is another classical result that every local derivation on B(X) is a derivation. We extend the result by proving that every local derivation from A into B(X) is a derivation. Based on these two results, we prove that every 2-local derivation from A into B(X)…
It was shown recently by M. Lorentz and R. Willett that all bounded derivations of the uniform Roe algebras of metric spaces of bounded geometry are inner. Here we calculate the space of outer derivations of the uniform Roe algebras with coefficients in uniform Roe bimodules related to various metrics on the two copies of the given space. We also give some results on the higher Hochschild cohomology with coefficients in uniform Roe bimodules.
2-Local and local derivations on Jordan matrix rings over commutative involutive rings
研究证明了在交换反环上的自共轭矩阵的乔丹环上，所有2-局部内导数都是导数，并应用于无限维自共轭矩阵值映射的乔丹代数，证明了所有2-局部空间导数都是空间导数。
This paper is concerned with derivations in algebras of (unbounded) operators affiliated with a von Neumann algebra $\mathcal{M}$. Let $\mathcal{% A}$ be one of the algebras of measurable operators, locally measurable operators or, $τ$-measurable operators. We present a complete description of von Neumann algebras $\mathcal{M}$ of type $I$ in terms of their central projections such that every derivation in $\mathcal{A}$ is inner. It is also shown that every derivation in the algebra $LS(\mathcal{M})$ of all locally measurable operators with respect to a properly infinite von Neumann algebra $\…
The paper is devoted to local derivations on the algebra $S(\mathcal{M},τ)$ of $τ$-measurable operators affiliated with a von Neumann algebra $\mathcal{M}$ and a faithful normal semi-finite trace $τ.$ We prove that every local derivation on $S(\mathcal{M},τ)$ which is continuous in the measure topology, is in fact a derivation. In the particular case of type I von Neumann algebras they all are inner derivations. It is proved that for type I finite von Neumann algebras without an abelian direct summand, and also for von Neumann algebras with the atomic lattice of projections, the condition of c…
The derivation problem for one type of bimodules over group algebras
本文研究一类具有良好组织共轭类的群上由 $\ell_p$ 范数生成的 Banach 双模 $\ell_p(G)$，并证明了该双模上的所有导子都是内导子。
Given a von Neumann algebra $M$ denote by $S(M)$ and $LS(M)$ respectively the algebras of all measurable and locally measurable operators affiliated with $M.$ For a faithful normal semi-finite trace $τ$ on $M$ let $S(M, τ)$ (resp. $S_0(M, τ)$) be the algebra of all $τ$-measurable (resp. $τ$-compact) operators from $S(M).$ We give a complete description of all derivations on the above algebras of operators in the case of type I von Neumann algebra $M.$ In particular, we prove that if $M$ is of type I$_\infty$ then every derivation on $LS(M)$ (resp. $S(M)$ and $S(M,τ)$) is inner, and each deriva…
Given a von Neumann algebra $M$ with a faithful normal semi-finite trace $τ,$ we consider the non commutative Arens algebra $L^ω(M, τ)=\bigcap\limits_{p\geq1}L^{p}(M, τ)$ and the related algebras $L^ω_2(M, τ)=\bigcap\limits_{p\geq2}L^{p}(M, τ)$ and $M+L^ω_2(M, τ)$ which are proved to be complete metrizable locally convex *-algebras. The main purpose of the present paper is to prove that any derivation of the algebra $M+L^ω_2(M, τ)$ is inner and all derivations of the algebras $L^ω(M,τ)$ and $L^ω_2(M, τ)$ are spatial and implemented by elements of $M+L^ω_2(M, τ).$
Suppose that ${\mathcal A}$ is an algebra, $σ,τ:{\mathcal A}\to{\mathcal A}$ are two linear mappings such that both $σ({\mathcal A})$ and $τ({\mathcal A})$ are subalgebras of ${\mathcal A}$ and ${\mathcal X}$ is a $\big(τ({\mathcal A}),σ({\mathcal A})\big)$-bimodule. A linear mapping $D:{\mathcal A}\to {\mathcal X}$ is called a $(σ,τ)$-derivation if $D(ab)=D(a)\cdotσ(b)+τ(a)\cdot D(b) (a,b\in {\mathcal A})$. A $(σ,τ)$-derivation $D$ is called a $(σ,τ)$-inner derivation if there exists an $x\in{\mathcal X}$ such that $D$ is of the form either $D_x^-(a)=x\cdot σ(a)-τ(a)\cdot x (a\in {\mathcal A}…
This paper analyzes the action δ of a Lie algebra X by derivations on a C*-algebra A. This action satisfies an "almost inner" property which ensures affiliation of the generators of the derivations δ with A, and is expressed in terms of corresponding pseudo-resolvents. In particular, for an abelian Lie algebra X acting on a primitive C*-algebra A, it is shown that there is a central extension of X which determines algebraic relations of the underlying pseudo- resolvents. If the Lie action δ is ergodic, i.e. the only elements of A on which all the derivations in δ_x vanish are multiples of the …
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