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Inner derivations and primal ideals of C*-algebras, II
... We now try to give some insight into our methods. Theorem 2.3 of [3] gave a sufficient condition on a unital C * -algebra A for K(A) n. In the case n = 1, the condition is that whenever P , Q, R are primitive ideals of A which cannot be separated by continuous functions on Prim(A) there should exist a primitive ideal T of A such that P , Q, R are all adjacent to T (where two primitive ideals are 'adjacent' if they cannot be separated by disjoint open sets). ...
... In this section we collect some of the definitions and notation which are needed later. We give the definitions of primal and Glimm ideals, of the graph structure ∼ on Prim(A), and of Orc(A), but for a fuller preliminary discussion we must refer the reader to [3]. ...
... Proposition 1.3. [3] Let A be a unital C * -algebra and let a ∈ A. Suppose that ad(a) 1. Let P , Q ∈ Primal(A) with P ⊆ Q. ...
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.
... 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 , . ...
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.
... 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]. These results use the fine structure of the topology on Prim(A) together with spectral constructions and the constrained optimization of the bounding radii of planar sets. ...
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 $N\geq3$, $$K(M(C^*(G_N)))=
\frac{1}{2}\left\lceil \frac{N}{2}\right\rceil.$$
... Subsequent work has generalised this equality to various classes of C * -algebras but [22, 3.2, 3.3] implies a characterisation of those A where equality always holds (those where all Glimm ideals of M(A) are 3-primal). Moreover in case this condition is not true, then there is a ∈ M(A) with δ a ≤ √ 3 inf z∈Z(M (A)) a − z (and further related work is to be found in [4,21,22,23,6]). An example of [7] shows that the condition on Glimm ideals of M(A) is difficult to relate to the structure of the primitive ideal space of A, so that the results are perhaps most satisfactory in the unital case where M(A) = A. ...
We present a formula for the norm of an elementary operator on a C*-algebra that seems to be new. The formula involves (matrix) numerical ranges and a kind of geometrical mean for positive matrices, the tracial geometric mean, which seems not to have been studied previously and has interesting properties. In addition, we characterise compactness of elementary operators.
We extend the nonstandard hull construction of Luxemburg and we show that the proposed extension covers various other constructions such as the tracial and the weak nonstandard hull. We provide a novel proof that the tracial nonstandard hull of an internal von Neumann algebra is itself a von Neumann algebra. We prove that some extensively studied properties of C*-algebras are preserved and reflected by the nonstandard hull construction. Eventually, we show how to obtain hull-like representations of the two K-groups associated to the nonstandard hull of an internal C*-algebra.
If G is a locally compact group, then for each derivation D from L 1 (G) into L1 (G) there is a bounded measure μ ∈ M(G) with D(a) = a * μ - μ * a for a ∈ L1 (G) ("derivation problem" of B. E. Johnson).
Abstact
A necessary and sufficient condition is given for a separable C *-algebra to be *-isomorphic to a maximal full algebra of cross-sections over a base space such that the fibre algebras are primitive throughout a dense subset. The condition is that the relation of inseparability for pairs of points in the primitive ideal space should be an open equivalence relation.
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.
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(
$\mathfrak{A}$
)−.
Let A be a C*-algebra. For a ∈ A let D(a, A) denote the inner derivation induced by a, regarded as a bounded operator on A, and let d(a, Z(A)) denote the distance of a from Z(A), the centre of A. Let K(A) be the smallest number in [0, ∞] such that d(a, Z(A))≤ K(A)∥D(a, A)∥ for all a ∈ A. It is shown that if A is non-commutative and has an identity then either K(A) = , or K(A) = 1 / √3, or K(A) ≥ 1. Necessary and sufficient conditions for these three possibilities are given in terms of the primitive and primal ideals of A. If A is a quotient of an AW*-algebra then K(A) ≤ . Helly's Theorem is used to show that if A is a weakly central C*-algebra then K(A) ≤ 1.
A derivation on an algebra W is a linear function 3): ® ® satisfying 3) (a6) = 3) (a)b + a 3) (6) for all a, b in 0. If there exists an a in 21 such that 3)(b) = ab — ba for b in H, then 3) is called the inner derivation induced by a. If S is a von Neumann algebra, then by a theorem of Sakai [7], every derivation on H is inner. In this paper we compute the norm of a derivation on a von Neumann algebra. Specifically we prove that if SE is a von Neumann algebra acting on a separable Hilbert space K, T is in tt, and 3)^ is the derivation induced by Tt then 3)^,?i} = 2 infi \\T — Z, Z in centre?.
For a C∗-algebra A let M(A) denote the two-sided multipliers of A in its enveloping von Neumann algebra. A complete description of M(A) is given in the case where the spectrum of A is Hausdorff. The formula M(A ⊗αB) = M(A) ⊗αM(B) is discussed and examples are given where is non-simple even though A is simple and separable. As a generalization of Tietze's Extension Theorem it is shown that a multiplier of a quotient of A is the image of an element from M(A), if A is separable. Finally, deriving algebras and thin operators and their relations to multipliers are discussed.
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 .
Some elementary results on derivations of continuous fields of C*-algebras are used to prove that every derivation of an A lF*-algebra of type III (or of type I) is inner, and also that if a given quotient of an A T7*-algebra is known to have only inner derivations, its tensor product with a separable commutative C*-algebra with unit also has this property.