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The Norm of a Derivation in a W ∗ -Algebra
Abstract
The norm of an inner derivation δa of a (non-necessary separable) W*-algebra M is shown to satisfy $\|\delta_a\| = 2 \inf \{\|a - z\|; z \in Z, \quad\text{the center of} M \}$ , and some related resulted results are obtained.
... In addition, we use the technique of J -dual spaces in order to obtain the estimation δ a I→J 2 a J :I for an arbitrary derivation δ = δ a : I → J , a ∈ J : I. This result extends a well-known estimation δ a M→M 2 a B(H) , a ∈ B(H) obtained by L.Zsido [22] for a derivation δ a acting in an arbitrary von Neumann algebra M ⊂ B(H). ...
... It is of interest to compare the estimates obtained in Theorem 4.18 with the result of L.Zsido [22], who established that for any derivation δ = δ a , a ∈ B(H) acting in a von Neumann algebra M ∈ B(H) the inequality δ a M→M 2 a B(H) holds. The inequalities established in Theorem 4.18 for derivations acting in arbitrary symmetric quasi-Banach ideals of compact operators are very similar to those of Zsido. ...
... The inequalities established in Theorem 4.18 for derivations acting in arbitrary symmetric quasi-Banach ideals of compact operators are very similar to those of Zsido. Note, that our proof of these inequalities is based on the technique of J -dual spaces and differs quite markedly from the techniques used in [22]. ...
Let $\mathcal{I,J}$ be symmetric quasi-Banach ideals of compact operators on
an infinite-dimensional complex Hilbert space $H$, let $\mathcal{J:I}$ be a
space of multipliers from $\mathcal{I}$ to $\mathcal{J}$. Obviously, ideals
$\mathcal{I}$ and $\mathcal{J}$ are quasi-Banach algebras and it is clear that
ideal $\mathcal{J}$ is a bimodule for $\mathcal{I}$. We study the set of all
derivations from $\mathcal{I}$ into $\mathcal{J}$. We show that any such
derivation is automatically continuous and there exists an operator
$a\in\mathcal{J:I}$ such that $\delta(\cdot)=[a,\cdot]$, moreover
$\|a\|_{\mathcal{B}(H)}\leq\|\delta\|_\mathcal{I\to J}\leq
2C\|a\|_\mathcal{J:I}$, where $C$ is the modulus of concavity of the quasi-norm
$\|\cdot\|_\mathcal{J}$. In the special case, when $\mathcal{I=J=K}(H)$ is a
symmetric Banach ideal of compact operators on $H$ our result yields the
classical fact that any derivation $\delta$ on $\mathcal{K}(H)$ may be written
as $\delta(\cdot)=[a,\cdot]$, where $a$ is some bounded operator on $H$ and
$\|a\|_{\mathcal{B}(H)}\leq\|\delta\|_\mathcal{I\to I}\leq
2\|a\|_{\mathcal{B}(H)}$.
... The question whether the inequality in (3.1) is in fact an equality has been the goal of study of many researchers (compare [20,36,15,38,12,4,33,34], and [5], among many others). In [20, Example 6.2] the authors exhibit a unital C * -algebra U containing a sequence of unitary elements (u n ) ⊂ U such that ad un = [u n , . ...
... It is proved in [20, Theorem 3.1] that K s (A) ≤ 1 2 when A is a von Neumann algebra, and in the general setting, K(A) < ∞ if and only if the space of all inner derivations on A is closed in the Banach space of all derivations on A. Clearly K(A) = 0 when A is commutative. It is known that K(A) ≤ 1 whenever A is unital and primitive [36], or just prime [34, Corollary 2.9], or a von Neumann algebra [38], or an AW * -algebra [12]. For an arbitrary unital, non-commutative C * -algebra A either K(A) = 1 2 , or K(A) = 1 √ 3 , or K(A) ≥ 1, depending on the topological properties of the primitive and primal ideals of A (see [34]). ...
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 results representing derivations on $C_0(L,B(H))$ and on $C_0(L,K(H))$ as inner derivations determined by multipliers.
... [37,Theorem 3.6]; but there is a ∈ Z(A) realizing this distance, see e.g. [18] and [43]). ...
We study some “almost preserver” problems on von Neumann algebra modules. More precisely, we study (1) maps which “almost preserve” the right or left annihilator; (2) the “almost band preservers” – that is, maps which “almost preserve corners”, and (3) “almost centralizers,” which almost preserve module actions. Under certain conditions, we show that maps of these types are automatically continuous, and can be approximated by maps which precisely preserve these relations; often, the operators from the latter class are multiplication operators.
... Therefore we may choose c ∈ B with desired properties. We apply now Theorem 3, [8], Corollary, [20] and Lemma 8, [10] to obtain ...
Here we present an alternative proof using Bures distance that the generator L of a norm continuous completely positive semigroup acting on a C∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^*$$\end{document}-algebra B⊂B(H)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {B}}\subset \mathcal B(H)$$\end{document} has the form L(b)=Ψ(b)+k∗b+bk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ L(b) = \Psi (b) + k^*b+bk$$\end{document}, b∈B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b\in {\mathcal {B}}$$\end{document} for some completely positive map Ψ:B→B(H)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Psi :{\mathcal {B}}\rightarrow {\mathcal {B}}(H)$$\end{document} and k∈B(H)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k\in {\mathcal {B}}(H)$$\end{document}.
... . If A is a von Neumann algebra (or, more generally, an AW * -algebra) or a unital primitive C * -algebra (in particular, a unital simple C * -algebra) then K(A) = 1 2 ([24, 26, 40, 79, 89]). These and other such cases are covered by Somerset's characterisation for unital A: K(A) = 1 2 if and only if the ideal P ∩ Q ∩ R is primal whenever P , Q and R are primitive ideals of A such that P ∩ Z(A) = Q ∩ Z(A) = R ∩ Z(A) ([77]). ...
It is shown that a unital C*-algebra A has the Dixmier property if and only if it is weakly central and satisfies certain tracial conditions. This generalises the Haagerup-Zsido theorem for simple C*-algebras. We also study a uniform version of the Dixmier property, as satisfied for example by von Neumann algebras and the reduced C*-algebras of Powers groups, but not by all C*-algebras with the Dixmier property, and we obtain necessary and sufficient conditions for a simple unital C*-algebra with unique tracial state to have this uniform property. We give further examples of C*-algebras with the uniform Dixmier property, namely all C*-algebras with the Dixmier property and finite radius of comparison-by-traces. Finally, we determine the distance between two Dixmier sets, in an arbitrary unital C*-algebra, by a formula involving tracial data and algebraic numerical ranges.
... For each a ∈ A we have the inner derivation induced by A, ∆ a : x → ax − xa. When A is a finite-dimensional C * -algebra, for each a ∈ A we have ||∆ a || = 2 dist(a, Z(A)), see [Zsi73]. In [Arc78], Archbold defined K(A) = inf{K : dist(a, Z(A)) ≤ K||∆ a || ∀a ∈ A}. ...
We study the class of pseudocompact C*-algebras, which are the logical limits of finite-dimensional C*-algebras. The pseudocompact C*-algebras are unital, stably finite, real rank zero, stable rank one, and tracial. We show that the pseudocompact C*-algebras have trivial K_1 groups and the Dixmier property. The class is stable under direct sums, tensoring by finite-dimensional C*-algebras, taking corners, and taking centers. We give an explicit axiomatization of the commutative pseudocompact C*-algebras. We also study the subclass of pseudomatricial C*-algebras, which have unique tracial states, strict comparison of projections, and trivial centers. We give some information about the K_0 groups of the pseudomatricial C*-algebras.
... If the elements a are restricted to be self-adjoint then the corresponding constant is denoted by K s (A). If A = B(H) (or, more generally, a non-commutative von Neumann algebra on a Hilbert space H = C) then K(A) = 1 2 [37,38]. For unital noncommutative C * -algebras, K s (A) = 1 2 Orc(A) [35], where the connecting order Orc(A) ∈ N ∪ {∞} is determined by a graph structure in the primitive ideal space Prim(A) (see Section 2), and for the constant K(A) it has been shown that the only possible positive values less than or equal to 1 2 + 1 √ 3 are: 10,11]. ...
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 $N\geq3$, $$K(M(C^*(G_N)))=
\frac{1}{2}\left\lceil \frac{N}{2}\right\rceil.$$
... One way is to calculate the distance d(a, Z(A)) from a to Z(A), the centre of A; the other way is to compute the operator norm of ad(a), the inner derivation of A induced by a. For von Neumann algebras and some closely related algebras one has the interesting result that these two methods yield essentially the same answer: for all elements a in the algebra, ad(a) = 2d(a, Z(A)) [5,7,8,[10][11][12]16,20,21]. The unital C * -algebras for which this equality holds have been characterized in [19] as those C * -algebras A for which the ideal P ∩ Q ∩ R is primal whenever P , Q, and R are primitive ideals of A such that P ∩ Z(A) = Q ∩ Z(A) = R ∩ Z(A). ...
It is well known that if A is a von Neumann algebra then the norm of any inner derivation ad(a) is equal to twice the distance from the element a to the centre Z(A) of the algebra. More generally, this property holds in a unital C∗-algebra if and only if the ideal P∩Q∩R is primal whenever P, Q, and R are primitive ideals of A such that P∩Z(A)=Q∩Z(A)=R∩Z(A). In this paper we give a characterization, in terms of ideal structure, of those unital C∗-algebras A for which the norm of any inner derivation ad(a) at least dominates the distance from a to the centre Z(A). In doing so, we show that if A does not have this property then it necessarily contains an element a, with ‖ad(a)‖=1, whose distance from Z(A) is greater than or equal to 3+8214. We also show how this number is related to the numbers 415 and 12+13 which have previously arisen in the study of norms of inner derivations. The techniques used in this work include spectral theory, the theory of primitive and primal ideals, and constrained geometrical optimisation.
... Norms of inner derivations of C*-algebras have long been a matter of interest (see [2,6,18,19,21,25]). For as A, \etD(a,A) denote the inner derivation of A defined by a and let d(a,Z(A)) denote the distance of a from Z(A), the centre of A. Archbold [2] [20] succeeded in showing that if A is non-commutative and unital, theneitheriC(vl) = 1/2, or K(A) = 1/V3, or K(A) ^ 1, and for each of the three possibilities he obtained necessary and sufficient conditions in terms of the ideal structure of A. Applying these criteria as well as some of our results mentioned above we relate K(A ® a 5) to K(A) and K(B) (Theorem 3-3). ...
An ideal I in a C*-algebra A is called primal if whenever n ≥ 2 and J1,…, Jn are ideals in A with zero product then Jk I for at least one k. The topologized space of minimal primal ideals of A, Min-Primal (A), has been extensively studied by Archbold[3]. Very much in the spirit of Fell's work [14] it was shown in [3, theorem 5·3] (see also [5, theorem 3·4]) that if A is quasi-standard, then A is *-isomorphic to a maximal full algebra of cross-sections of Min-Primal (A). Moreover, if A is separable the fibre algebras are primitive throughout a dense subset. On the other hand, the complete regularization of the primitive ideal space of A gives rise to the space of so-called Glimm ideals of A, Glimm (A). It turned out that A is quasi-standard exactly when Min-Primal (A) and Glimm (A) coincide as sets and topologically [5, theorem 3·3].(Received September 01 1994)(Revised December 13 1994)
In the last few decades, the concept of Birkhoff–James orthogonality has been used in several applications. In this survey article, the results known on the necessary and sufficient conditions for Birkhoff–James orthogonality in certain Banach spaces are mentioned. Their applications in studying the geometry of normed spaces are given. The connections between this concept of orthogonality, and the Gateaux derivative and the subdifferential set of the norm function are provided. Several interesting distance formulas can be obtained using the characterizations of Birkhoff–James orthogonality, which are also mentioned. In the end, some new results are obtained.
We consider Sturm-Liouville problems with a boundary condition linearly dependent on the eigenparameter. We concentrate the study on the cases where non-real or non-simple (multiple) eigenvalues are possible. We prove that the system of root (i.e. eigen and associated) functions of the corresponding operator, with an arbitrary function removed, form a minimal system in L2(0, 1), except some cases where this system is neither complete nor minimal. The method used is based on the determination of the explicit form of the biorthogonal system. These minimality results can be extended to basis properties in L2(0, 1).
The theory of operator algebras, that is, C*-algebras and von Neumann algebras on complex Hilbert spaces is of increasing importance to many branches of mathematics, for example, integration theory, operator theory, algebraic topology, and particularly mathematical physics and quantum mechanics. Because C*-algebras provide a natural framework for the foundations of quantum mechanics and quantum field theory, it is an important problem to characterize the class of C*-algebras by certain properties, for instance, motivated by physical experiments. Two characterizations of operator algebras in different categories exist: (1) A. Connes' characterization of von Neumann algebras in terms of self-dual homogeneous Hilbert cones and (2) the work of Alfsen and Shultz characterizing the state spaces of C*-algebras using the geometry of compact convex sets and their affine function spaces.
The completely bounded norm of an inner derivation of a C * -algebra is determined in terms of the central Haagerup tensor norm. As a consequence, it is equal to twice the distance of an implementing element to the center of the local multiplier algebra.
A well-known theorem of E. Posner [10] states that if the composition d 1 d 2 of derivations d 1 d 2 of a prime ring A of characteristic not 2 is a derivation, then either d 1 = 0 or d 2 = 0. A number of authors have generalized this theorem in several ways (see e.g. [1], [2], and [5], where further references can be found). Under stronger assumptions when A is the algebra of all bounded linear operators on a Banach space (resp. Hilbert space), Posner's theorem was reproved in [3] (resp. [12]). Recently, M. Mathieu [8] extended Posner's theorem to arbitrary C * -algebras.
Let A be a strongly maximal TAF-algebra. It is shown that 1 2Orc(A)≤K(A)≤4 3Orc(A), where K(A) and Orc(A) are constants determined by the norms of inner derivations of A, and by the hull-kernel topology on the space of meet-irreducible ideals of A, respectively. It follows that the set of inner derivations of A is closed in the Banach space of all bounded derivations of A if and only if Orc(A)<∞. These results are analogous to those for C * -algebras.
It is shown that if A is the C∗-algebra inductive limit of a sequence of finite-dimensional C∗-algebras, then for each closed two-sided ideal J of A derivations can be lifted to A from , and for each projection e in A derivations can be extended to A from eAe. An application of the second result is given.
Let A be a C *-algebra with centre Z . If a ∈ A , the bounded linear mapping x → + ax – xa ( x ∈ A ) is called the inner derivation of A induced by a , and we denote it by D ( a, A ). A simple application of the triangle inequality shows that
where d ( a; Z ) denotes the distance from a to Z in the normed space A .
Relativizing an idea from multiplicity theory, we say that an element x of a von Neumann algebra M is n-divisible if (W*(x)' cap M) unitally contains a factor of type I_n. We decide the density of the n-divisible operators, for various n, M, and operator topologies. The most sensitive case is sigma-strong density in II_1 factors, which is closely related to the McDuff property. We make use of Voiculescu's noncommutative Weyl-von Neumann theorem to obtain several descriptions of the norm closure of the n-divisible operators in B(ell^2). Here are two consequences: (1) in contrast to the reducible operators, of which they form a subset, the divisible operators are nowhere dense; (2) if an operator is a norm limit of divisible operators, it is actually a norm limit of unitary conjugates of a single divisible operator. This is related to our ongoing work on unitary orbits by the following theorem, which is new even for B(ell^2): if an element of a von Neumann algebra belongs to the norm closure of the aleph_0-divisible operators, then the sigma-weak closure of its unitary orbit is convex.
The theorem that each derivation of aC*-algebra
$\mathfrak{A}$
extends to an inner derivation of the weak-operator closure ϕ(
$\mathfrak{A}$
)− of
$\mathfrak{A}$
in each faithful representation ϕ of
$\mathfrak{A}$
is proved in sketch and used to study the automorphism group of
$\mathfrak{A}$
in its norm topology. It is proved that the connected component of the identity ı in this group contains the open ball ℬ of radius 2 with centerl and that each automorphism in ℬ extends to an inner automorphism of ϕ(
$\mathfrak{A}$
)−.
The norm of a derivation
- J G Stampfli Of Mathematics
- Calea Academy Of The Socialist Republic Of Romania
- Grivitei
J. G. Stampfli, The norm of a derivation, Pacific J. Math. 33 (1970), 737-747. MR 42 #861. INSTITUTE OF MATHEMATICS, ACADEMY OF THE SOCIALIST REPUBLIC OF ROMANIA, CALEA GRIVITEI, 21, BUCHAREST 12, ROMANIA This content downloaded from 128.230.234.162 on Fri, 9 Aug 2013 07:56:12 AM All use subject to JSTOR Terms and Conditions