Let $A$ be a $C^*$-algebra acting on a Hilbert space $H$, $σ:A\to B(H)$ be a linear mapping and $d:A\to B(H)$ be a $σ$-derivation. Generalizing the celebrated theorem of Sakai, we prove that if $σ$ is a continuous $*$-mapping then $d$ is automatically continuous. In addition, we show the converse is true in the sense that if $d$ is a continuous $*-σ$-derivation then there exists a continuous linear mapping $Σ:A\to B(H)$ such that $d$ is $*-Σ$-derivation. The continuity of the so-called $*-(σ,τ)$-derivations is also discussed.
Suppose that $\calak$ is a $C^*$-algebra acting on a Hilbert space $\calhk$, and that $ϕ, ψ$ are mappings from $\calak$ into $B(\calhk)$ which are not assumed to be necessarily linear or continuous. A $(ϕ, ψ)$-derivation is a linear mapping $d: \calak \to B(\calhk)$ such that $$d(ab)=ϕ(a)d(b)+d(a)ψ(b)\quad (a,b\in \calak).$$ We prove that if $ϕ$ is a multiplicative (not necessarily linear) $*$-mapping, then every $*$-$(ϕ,ϕ)$-derivation is automatically continuous. Using this fact, we show that every $*$-$(ϕ,ψ)$-derivation $d$ from $\calak$ into $B(\calhk)$ is continuous if and only if the $*$-…
In this article we shall focus on the derivations on module extension of Banach algebras and determine the general structure of them. Then we obtain some results concerning the automatic continuity of these mappings.
A linear map $ϕ:{\mathcal A}\to {\mathcal B} $ between (Banach) algebras is called 3-homomorphism if $ϕ(abc)=ϕ(a)ϕ(b)ϕ(c)$ for each $a, b, c \in {\mathcal A}$. We investigate 3-homomorphisms on Banach algebras with bounded approximate identities and establish in two ways that every involution preserving homomorphism between $C^*$-algebras is norm decreasing.
On automatic continuity of operators from ordered to topological vector spaces
Sobolev Institute of Mathematics（索博列夫数学研究所）；Southern Mathematical Institute of the Vladikavkaz Scientific Center of the Russian Academy of Sciences（俄罗斯科学院弗拉季高加索科学中心南方数学研究所）
We prove that a biseparating map between spaces of vector-valued continuous functions is usually automatically continuous. However, we also discuss special cases when it is not true.
We give characterizations of unital uniform topological algebras and saturated locally multiplicatively convex algebras by means of multiplicative linear functionals. Some automatic continuity theorems in advertibly complete uniform topological algebras are extended to a larger class of algebras. Consequences and applications are given.
Let $\mathcal{A}$ and $\mathcal{U}$ be Banach algebras and $θ$ be a nonzero character on $\mathcal{A}$. Then the \textit{Lau product Banach algebra} $\mathcal{A}\times_θ\mathcal{U}$ associated with the Banach algebras $\mathcal{A}$ and $\mathcal{U}$ is the $l^1$-direct sum $\mathcal{A}\oplus\mathcal{U}$ equipped with the algebra multiplication $(a,u)(a',u')=(ab,θ(a)u'+θ(a')u+uu')\, (a,a'\in\mathcal{A},u,u'\in\mathcal{U})$ and $l^1$-norm. In this paper we shall investigate the derivations and multipliers from this Banach algebras and study the automatic continuity of these mappings. We also stu…
In a first result we prove that every continuous local triple derivation on a JB$^*$-triple is a triple derivation. We also give an automatic continuity result, that is, we show that local triple derivations on a JB$^*$-triple are continuous even if not assumed a priori to be so. In particular every local triple derivation on a C$^*$-algebra is a triple derivation. We also explore the connections between (bounded local) triple derivations and generalised (Jordan) derivations on a C$^*$-algebra.
Continuity equation on metric spaces via measure-valued derivations and BV-Wasserstein curves
Institute of Mathematics TU Berlin（柏林工业大学数学研究所）；Faculty of Mathematics Bielefeld University（比勒费尔德大学数学系）；Department of Mathematics and Statistics University of Jyväskylä（于韦斯屈莱大学数学与统计系）
Let A and B be commutative locally convex algebras with unit. A is assumed to be a uniform topological algebra. Let h be an injective homomorphism from A to B. Under additional assumptions, we characterize the continuity of the homomorphism h^(-1) / Im(h) by the fact that the radical (or strong radical) of the closure of Im(h) has only zero as a common point with Im(h). This gives an answer to a conjecture concerning some automatic continuity theorems on uniform topological algebras.
We prove a general criterion for a von Neumann algebra $M$ in order to be in standard form. It is formulated in terms of an everywhere defined, invertible, antilinear, a priori not necessarily bounded operator, intertwining $M$ with its commutant $M'$ and acting as the $*$-operation on the centre. We also prove a generalized version of the BT-Theorem which enables us to see that such an intertwiner must be necessarily bounded. It is shown that this extension of the BT-Theorem leads to the automatic boundedness of quite general operators which intertwine the identity map of a von Neumann algebr…
It is proved that every linear biseparating map between spaces of vector-valued differentiable functions is a weighted composition map. As a consequence, such a map is always continuous.
Let $ω$ be a continuous weight on $\mathbb R^+$ and let $L^1(ω)$ be the corresponding convolution algebra. By results of Grønbæk and Bade & Dales the continuous derivations from $L^1(ω)$ to its dual space $L^{\infty}(1/ω)$ are exactly the maps of the form $$(D_ϕf)(t)=\int_0^{\infty}f(s)\,\frac{s}{t+s}\,ϕ(t+s)\,ds\qquad\text{($t\in\mathbb R^+$ and $f\in L^1(ω)$)}$$ for some $ϕ\in L^{\infty}(1/ω)$. Also, every $D_ϕ$ has a unique extension to a continuous derivation $\bar{D}_ϕ:M(ω)\to L^{\infty}(1/ω)$ from the corresponding measure algebra. We show that a certain condition on $ϕ$ implies that $\b…
We prove that in a certain class E of nonseparable Banach spaces the norm topology of the dual ball is definable in terms of its weak* topology. Thus, any weak* homeomorphism between duals balls of spaces in E is automatically norm-continuous.
Let A and U be Banach algebras such that U is also a Banach A- bimodule with compatible algebra operations, module actions and norm. By defining an approprite action, we turn l1-direct product A item U into a Banach algebra such that A is closed subalgebra and U is a closed ideal of it. This algebra, is in fact semidirect product of A and U which we denote it by A litem U and every semidirect products of Banach algebras can be represented as this form. In this paper we consider the Banach algebra A litem U as mentioned and study the derivations on it. In fact we consider the automatic continui…
Let $Ω$ be a locally compact Hausdorff space. We show that any local $\mathbb{C}$-linear map (where "local" is a weaker notion than $C_0(Ω)$-linearity) between Banach $C_0(Ω)$-modules are "nearly $C_0(Ω)$-linear" and "nearly bounded". As an application, a local $\mathbb{C}$-linear map $θ$ between Hilbert $C_0(Ω)$-modules is automatically $C_0(Ω)$-linear. If, in addition, $Ω$ contains no isolated point, then any $C_0(Ω)$-linear map between Hilbert $C_0(Ω)$-modules is automatically bounded. Another application is that if a sequence of maps $\{θ_n\}$ between two Banach spaces "preserve $c_0$-sequ…
We investigate the automatic regularity of bounded derivations from a Banach lattice algebra of regular operators A into a Banach A-module with a Banach lattice structure compatible with the module operations.
We study the question how quickly products of a fixed conjugacy class in the projective unitary group of a II${}_1$-factor von Neumann algebra cover the entire group. Our result is that the number of factors that are needed is essentially as small as permitted by the $1$-norm - in analogy to a result of Liebeck-Shalev for non-abelian finite simple groups. As an application of the techniques, we prove that every homomorphism from the projective unitary group of a II${}_1$-factor to a polish SIN group is continuous. Moreover, we show that the projective unitary group of a II${}_1$-factor carries…
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…
Let ${\mathcal M}$ be a von Neumann algebra with no central summands of type I$_1$. It is shown that every nonlinear $ξ-$Lie derivation ($ξ\neq 1$) on $\mathcal M$ is an additive derivation.
It is well known that weakly continuous semigroups defined over $\mathbb{R}_{+}$ are automatically strongly continuous. We extend this result to more generally defined semigroups, including multiparameter semigroups.
In 1952, Michael posed a question about the functional continuity of commutative Frechet algebras in his memoir, known as Michael problem in the literature. We settle this in the affirmative along with its various equivalent forms, even for the non-commutative case. Indeed, we continue our recent works, and develop two approaches to directly attack these problems. The first approach is to show that the test case for this problem, the Frechet algebra of all entire functions on the Banach space of all bounded complex sequences, is, in fact, a Frechet algebra of all complex formal power series in…
We study continuity and equicontinuity of semigroups on norming dual pairs with respect to topologies defined in terms of the duality. In particular, we address the question whether continuity of a semigroup already implies (local/quasi) equicontinuity. We apply our results to transition semigroups and show that, under suitable hypothesis on $E$, every transition semigroup on $C_b(E)$ which is continuous with respect to the strict topology $β_0$ is automatically quasi-equicontinuous with respect to that topology. We also give several characterizations of $β_0$-continuous semigroups on $C_b(E)$…
Starting from Sinclair's 1976 work {\it Automatic Continuity of Linear Operators}, Cambridge University Press, (1976), on automatic continuity of linear operators on Banach spaces, we prove that sequences of intertwining continuous linear maps are eventually constant with respect to the separating space of a fixed linear map. Our proof uses a gliding hump argument. We also consider aspects of continuity of linear functions between locally convex spaces and prove that such that a linear function $T$ from the locally convex space $X$ to the locally convex space $Y$ is continuous whenever the sep…
We give a survey of Husain, Ng and Liang's results concerning the automatic continuity of algebra homomorphisms. We also give the improvements obtained by Joseph, Akkar, Oudadess, Zelazko and ourselves.
