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The inner derivations and the primitive ideal space of a C * -algebra
... those of the form d a (x) = ax − xa with a ∈ M (A)) are important examples of elementary operators. In [35,Corollary 4.6] Somerset shows that if A is unital, {d a : a ∈ A} is norm closed if and only if Orc(A) < ∞, where Orc(A) is a constant defined in terms of a certain graph structure on Prim(A) (the primitive spectrum of A). If Orc(A) = ∞, the structure of outer derivations that are norm limits of inner derivations remains undescribed. ...
... If Orc(A) = ∞, the structure of outer derivations that are norm limits of inner derivations remains undescribed. In addition, if A is unital and separable, then by [19,Theorem 5.3] and [35,Corollary 4.6] Orc(A) < ∞ if and only if the set {M u,u * : u ∈ A, u unitary} of inner automorphisms is norm closed. ...
... Since φ t (1) = 1, c(t) = b(t) −1 for each t and so c = b −1 ∈ M (A) = Γ b (E).The proof for the InnAut(A) is similar.Remark 6.8. The results that InnAut(A) is norm closed if the C * -algebra A is prime or homogeneous (in Corollaries 3.5 and 6.7) can also be deduced from[35,19,3].To explain the deductions, we first identify InnAut(A) with InnAut(M (A)). If A is prime, then M (A) is also prime (by [2, Lemma 1.1.7]). ...
For a C*-algebra A we consider the problem of when the set $TM_0(A)$ of all two-sided multiplications $x \mapsto axb$ ($a,b \in A$) on A is norm closed, as a subset of B(A). We first show that $TM_0(A)$ is norm closed for all prime C*-algebras A. On the other hand, if $A\cong \Gamma_0(E )$ is an n-homogeneous C*-algebra, where E is the canonical $\mathbb{M}_n $-bundle over the primitive spectrum X of A, we show that $TM_0(A)$ fails to be norm closed if and only if there exists a $\sigma$-compact open subset U of X and a phantom complex line subbundle L of E over U (i.e. L is not globally trivial, but is trivial on all compact subsets of U). This phenomenon occurs whenever $n \geq 2$ and X is a CW-complex (or a topological manifold) of dimension $3 \leq d<\infty$.
... 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. ...
... We begin by recalling some terminology from [35]. Let X be a topological space. ...
... It was shown in [35,Theorem 4.4] that, if A is a unital C * -algebra, K s (A) = 1 2 Orc(A). It follows that if A is any C * -algebra then K s (M(A)) = 1 2 Orc(M(A)) and so 1 2 Orc(M(A)) ≤ K(M(A)). ...
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 $N\geq3$, $$K(M(C^*(G_N)))=
\frac{1}{2}\left\lceil \frac{N}{2}\right\rceil.$$
... Indeed there are examples of C * -algebras in which elements can be found arbitrarily far from the centre but inducing inner derivations of arbitrarily small norm [13]. To investigate such behaviour in more detail the numerical invariants K(A) and K s (A) have been introduced, defined as follows [1]: The properties of the invariant K s (A) were established fairly fully in [18], where it was shown that for an arbitrary unital, noncommutative C * -algebra A, K s (A) = 1 2 Orc(A) where Orc(A) is an integer, or ∞, determined by the extent to which Prim(A), the primitive ideal space of A equipped with Jacobson's hull-kernel topology, departs from being a Hausdorff space. Examples in [18] show that the invariant K s (A) takes all possible values in the set { 1 2 , 1, 3 2 , . . . ...
... To investigate such behaviour in more detail the numerical invariants K(A) and K s (A) have been introduced, defined as follows [1]: The properties of the invariant K s (A) were established fairly fully in [18], where it was shown that for an arbitrary unital, noncommutative C * -algebra A, K s (A) = 1 2 Orc(A) where Orc(A) is an integer, or ∞, determined by the extent to which Prim(A), the primitive ideal space of A equipped with Jacobson's hull-kernel topology, departs from being a Hausdorff space. Examples in [18] show that the invariant K s (A) takes all possible values in the set { 1 2 , 1, 3 2 , . . . , ∞}. ...
... implies that for P , Q ∈ Prim(A), P and Q can be separated by a continuous complex-valued function on Prim(A) if and only if P and Q intersect the centre of A in distinct maximal ideals, and hence if and only if they contain distinct Glimm ideals. Evidently if P and Q can be separated by a continuous function then P Q, but the converse is not true in general, unless Orc(A) = 1 [18]. Exploring the failure of this converse is one of the themes of the present paper. ...
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.
... To describe this, we need a slight variant of the definition of ∼. Recall from [33] that for P, Q ∈ Prim(A) we write P ∼ Q if P and Q cannot be separated by disjoint open subsets of Prim(A) (for a fuller discussion, see Section 5 below). For the multiplier algebra ...
... We begin by explaining the notation Orc(A) [33]. Recall that for a C * -algebra A and for P, Q ∈ Prim(A) we write P ∼ Q if P and We also need the following topological lemma characterizing separation by open sets. ...
... Let A be the C * -algebra defined as follows (see [33,Example 2.8]). Let B be the C * -algebra consisting of all continuous functions from the interval [0, 1] into the 2 × 2 complex matrices. ...
Let A be a C-0(X)-algebra with continuous map phi from Prim(A), the primitive ideal space of A, to a locally compact Hausdorff space X. Then the multiplier algebra M(A) is a C(beta X)-algebra with continuous map (phi) over bar : Prim(M(A)) -> beta X extending phi. For x epsilon Im(phi), let J(x) boolean AND{P epsilon Prim(A) : phi (P)=x} and H-x = boolean AND {Q epsilon Prim(M(A)) : (phi) over bar (Q) = x}. Then J(x) subset of H-x subset of J(x), the strict closure of J(x) in M(A). Thus, H-x is strictly closed if and only if H-x = (J) over tilde (x), and the ' spectral synthesis' question asks when this happens. In this paper, it is shown that, for cr-unital A, H-x is strictly closed for all x epsilon Im(phi) if and only if J(x) is locally modular for all x epsilon Im(phi) and phi is a closed map relative to its image. Various related results are obtained.
... Define a graph structure on Prim(A) by saying that P and Q are adjacent if P ∼ Q, and let Orc(A) be the supremum of the diameters of the connected components of Prim(A) in this graph structure (with the convention that a singleton has diameter 1). The work in [17] shows that for a unital C * -algebra A the following are equivalent: ...
... One of the main results of [17] is that Orc(A) = 1 if and only if D a = 2d(a, Z(A)) for all self-adjoint a ∈ A. ...
Let A be a C -algebra with an identity and let Z be the canon- ical map from A Z A, the central Haagerup tensor product of A, to CB(A), the algebra of completely bounded operators on A. It is shown that if ev- ery Glimm ideal of A is primal then Z is an isometry. This covers unital quasi-standard C -algebras and quotients of AW -algebras.
... Johnson [5] and Kyle [7] showed that the equality holds and sometimes does not. For quotients of W * -algebras, Sommerset [10] showed that (1) holds while recently, Bonyo and Agure [3] showed that when J is a proper two-sided ideal then (1) is true. In order to examine the behaviour of norms of these norm-attainable inner derivations, we shall determine the relationship between two constants due to Archbold [2] i.e. ...
... Sommerset [10] established the relationship between C s (Ω) and the order of connecting Ω and OrcΩ. His main result showed that when Ω is a weakly central C * -algebra, then OrcΩ ≤ 2 so that C s (Ω) ≤ 1, hence the relation C s (Ω) = 1 2 Orc(Ω). ...
We investigate the relationship between inner derivations imple-mented by a norm-attainable element of a C*-algebra to those of ideals and primitive ideals. Moreover, we give related results on the relation-ship between the constants C(Ω) and C s (Ω) of C * -algebras to those of ideals and primitive ideals.
... Bresar, Zalar [9] showed that a Jordan * -derivation is the map δ a (x) = ax − x * a for fixed a ∈ U; hence, the derivation is inner. Douglas [15] continued the study of W s (Y ) which was considerably more amenable where Archbold [1] defined the smallest numbers to be [0, ∞] and introduced two constants W (Y ) and W t (Y ) such that d(y , Z(Y )) ≤ W (Y ) D(y , Y ) , for all y ∈ Y and d(y , Z(Y )) ≤ W s (Y ) D(y , Y ) , for all y = y * ∈ Y. The author in [26] showed that for the nth ...
In this note, we provide detailed characterization of operators in terms of norm-attainability and norm estimates in Banach algebras. In particular, we establish the necessary and sufficient conditions for norm-attainability of the derivations and also give their norm bounds in the norm-attainable classes.
... 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 is not prime, one considers the central Haagerup tensor product A⊗ Z,h A and the induced contraction [31]). The problem of when θ Z A is isometric has been recently completely solved by Archbold, Somerset and Timoney in [32,Theorem 4] and [8,Theorem 7] (see also [7]); θ Z A is isometric if and only if each Glimm ideal of A is primal. As an easy consequence of this result, we obtain: ...
Let A be a unital C*-algebra with the canonical (H) C*-bundle over the maximal ideal space of the centre of A, and let E(A) be the set of all elementary operators on A. We consider derivations on A which lie in the completely bounded norm closure of E(A), and show that such derivations are necessarily inner in the case when each fibre of is a prime C*-algebra. We also consider separable C*-algebras A for which is an (F) bundle. For these C*-algebras we show that the following conditions are equivalent: E(A) is closed in the operator norm; A as a Banach module over its centre is topologically finitely generated; fibres of have uniformly finite dimensions, and each restriction bundle of over a set where its fibres are of constant dimension is of finite type as a vector bundle.
... (i) The distance from a to Z(A) is of interest because it is related to the norm of the inner derivation induced by a; see [9, 10]. Since a and b induce the same inner derivations if and only if a&b # Z(A), Theorem 3 implies that for each inner derivation on a unital C*-algebra there exists an element of minimal norm implementing the derivation. ...
It is shown that ifAis a unitalC*-algebra thenZ(A), the centre ofA, is a proximinal subspace. In other words, for eacha∈Athere existsz∈Z(A) such that ‖a−z‖ is equal to the distance fromatoZ(A).
... 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)
... More recent results in this direction, and further references, can be found in [16]; the whole area is related to Hilbert's 13th Problem. We now show the connection between the terminology of Proposition 2.3 and that used in [12]. The following definitions are usually made in the context of the primitive ideal space of A, rather than the spectrum of A. For 2-subhomogeneous C*-algebras, however, these spaces are homeomorphic, and it seems simpler for us to work with the spectrum. ...
A characterization is given of those unital, 2-subhomogeneous, Fell C*-algebras which have only inner derivations. This proves Sproston and Strauss's conjecture from 1992. Various examples are given of phenomena which cannot occur for separable C*-algebras. In particular, an example is given of a C*-algebra with only inner derivations which has a quotient algebra admitting outer derivations. This answers a question of Akemann, Elliott, Pedersen and Tomiyama from 1976.
... 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.
The Mautner groups are the 5-dimensional solvable Lie groups that have non-type-I factor representations. We show that their corresponding group $C^*$-algebras are quasi-standard and we describe the topology of their spaces of minimal primal ideals and Glimm ideals.
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.
We consider derivations in the image of the canonical contraction 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 central. We also provide an example of a C*-algebra which has an outer derivation implemented by an elementary operator.
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.
We use recent work on spectral synthesis in multiplier algebras to give an intrinsic characterization of the separable C*-algebras A for which Orc (M(A))=1, i.e., for which the relation of inseparability on the topological space of primitive ideals of the multiplier algebra M(A) is an equivalence relation. This characterization has applications to the calculation of norms of inner derivations and other elementary operators on A and M(A). For example, we give necessary and sufficient conditions on the ideal structure of a separable C*-algebra A for the norm of every inner derivation to be twice the distance of the implementing element to the centre of M(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.
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.
The ideal space Id(A) of a Banach algebra A is studied as a bitopological space Id(A), τu, τn, where τu is the weakest topology for which all the norm functions I → ‖ a + I‖ (with a ∈ A and I ∈ Id(A)) are upper semi-continuous, and τn is the de Groot dual of τu. When A is separable, τn∨τu is either a compact, metrizable topology, or it is neither Hausdorff nor first countable. TAF-algebras are shown to exhibit
the first type of behaviour. Applications to Banach bundles (which motivate the study), and to PI-Banach algebras, are given.
1991 Mathematics Subject Classification: 46H10, 46J20.
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