No full-text available
To read the full-text of this research,
you can request a copy directly from the author.
On derivations of $AW^*$-algebras
Abstract
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.
... As a result-as the rest of the methods pass to quotients-the full result holds for any quotient of a W*-algebra (or AW*-algebra). (As was shown for automorphisms in [6].) THEOREM 1. ...
... It follows that PROOF. The proof follows the proof for automorphisms given in [5] (see also [6])even more closely than does the proof given in [4], for completely positive maps in the properly infinite case. ...
... (In [5] and [6] this implication was established for automorphisms, and in [4] it was established in the case that q is equivalent to p. In neither these cases nor the present case does the proof use any special properties of the C*-algebra A.) The proof of the theorem is slightly simpler in the case that A is a quotient of a finite direct sum B of finite matrix algebras over abelian AW*-algebras. ...
It is shown that a sequence of completely positive linear maps on a W*-algebra that converges pointwise in norm to the identity converges uniformly.
... In this section we apply results of [6,81 to show that the tensor product L@ of a von Neumann algebra A and an abelian C*-algebra B has only inner derivations. This will be a corollary of a result about compact subsets of derivations (considering derivations as a subset of the Banach space L?(A) of bounded linear operators on rZ). ...
... The topology F of pointwise (norm) convergence on & is a Hausdorff topology on X (since derivations are weak* continuous) and is weaker than the point-norm topology; thus 3 is equivalent to it by compactness. Since 9 is obviously metrizable (since X is point-norm compact, hence bounded by the uniform boundedness theorem), every sequence in X has a point-norm convergent subsequence which is therefore norm convergent by [6,81. Thus X is compact (in norm). ...
... Also we needed that a point norm convergent sequence of derivations was norm convergent. Since both these conditions are satisfied by *-automorphisms of -4 [6] the following Corollary holds. ...
The question of which C∗-algebras have only inner derivations has been considered by a number of authors for 25 years. The separable case is completely solved, so this paper deals only with the non-separable case. In particular, we show that the C∗-tensor product of a von Neumann algebra and an abelian C∗-algebra has only inner derivations. Other special types of C∗-algebras are shown to have only inner derivations as well such as the C∗-tensor product of L(H) (all bounded operators on separable Hilbert space) and any separable C∗-algebra having only inner derivations. Derivations from a smaller C∗-algebra into a larger one are also considered, and this concept is generalized to include derivations between C∗-algebras connected by a ∗-homomorphism. Finally, we consider the general problem of a sequence of linear functionals on a C∗-algebra which converges to zero (in norm) when restricted to any abelian C∗-subalgebra. Does such a sequence converge to zero in norm? The answer is “yes” for normal functionals on L(H), but unknown in general.
... 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.
... . 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.
... Then llcy -ex||+ ||ex -ax|| + ||.xa Hence \\ad c \\ < 4y. By [4,Corollary 4.8] there exists an element b' e Z(B) such that \\ad c \\ = 2\\c -b'\\. Therefore \\a -b'\\ < ||a -c\\ + \\c -b'\\ < y + l/2\\ad c \\ < 3y. ...
The distance between two operator algebras acting on a Hilbert space H is defined to be the Hausdorff distance between their unit balls. We investigate the structural similarities between two close AW*-algebras A and B acting on a Hilbert space H. In particular, we prove that if A is of type I and separable, then A and B are *-isomorphic.
... 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.
... ProoJ: Let v be a unitary in Z(B) and chose c E C such that $(c) = V. By [ 8,Lemma 4.11 there is a z E Z(C) such that so that g(z) = v. Let r E Z(C) be the polar part of z so that $(r) = v. ...
Let B be a unital C∗-algebra and A = Γ(E) be the C∗-algebra of sections of a bundle over the separable compact space X with fibre B and structure group Inn B. If B is the quotient of an AW∗-algebra, then an exact sequence: 0 → Inn A → PInn A →ηH2(X, G), where PInn A is the group of pointwise inner automorphisms of A and G=H°(Z(B)^, ) is obtained. The map η is onto whenever A = C(X, B) and B is the quotient of a purely infinite AW∗-algebra. An essential part of the analysis is the result that Ad: U(B) → Inn B is a fibre bundle if and only if the space of inner derivations of B is norm closed. These results extend and clarify previous joint work with I. Raeburn (Indiana Univ. Math. J.29 (1980), 799).
... It is known that von Neumann algebras have only inner derivations. One of the main open questions in this area is whether quotients of von Neumann algebras ever admit outer derivations (see [11] and [7]). ...
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.
... By [El,Theorem 3.4], a sequence of derivations converges in the pointnorm topology only if it converges in the norm topology, so ad x n → 0. We also have that ad x n = 2 dist(x n , Z(M)) (proved independently in [Ga] and [Z]). Therefore (x n ) can also be represented by a sequence from Z(M). ...
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 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.
Every primitive C*-algebra is prime. An old theorem of Dixmier shows that the converse is true if the algebra is separable.
Recently Weaver has shown that without separability this is false. AW* -factors are easily seen to be prime. Are all their
ideals primitive? A number of partial results have appeared recently. It turns out that the answer is always positive and
that this follows immediately from a theorem of FB Wright proved nearly 50 years ago. His proof was hard and complicated.
He works in a more general setting but when his result is specialised to the class of operator algebras investigated here,
we are able to give a new, easy proof that the ideals of an AW* -factor form a well-ordered set; from this, it follows swiftly
that every proper ideal of an AW*-factor is primitive.
Every Glimm ideal of anAW
*-algebra is a (minimal) primitive ideal.
Using the theory of spectral subspaces associated with a group of isometries of a Banach space it is proved that each derivation of an AW*-algebra is inner. This constructive method of proof yields a generator b for the case of a skew- adjoint derivation which is seen to be the unique positive generator such that ‖bp‖= δ |Ap| for each central projection p in the AW*-algebra A. © 1974 Pacific Journal of Mathematics Manufactured and first issued in Japan.
It is proved that every derivation on an AW*-algebra of type II1 with central trace is inner. The proof employs a result on the algebraic decomposition of such algebras which is of interest even in the W* case.
A condition is given which is both necessary and sufficient for every derivation of a separable C*-algebra to be inner (or, if there is no unit, to be determined by a multiplier).
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(
$\mathfrak{A}$
)−.
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.
It is shown that in an AW∗-algebra a sequence of automorphisms which converges simply converges in norm. More generally, this is shown to hold in the quotient of an AW∗-algebra by a closed two-sided ideal. A similar result holds for derivations. Kallman had previously demonstrated this in a W∗-algebra (he assumed countable decomposability or finiteness), but his methods are not applicable to quotients of a W∗-algebra.
For each derivation δ of a C*algebra A with δ(x*)=-δ(x)* there exists a minimal positive element h in the enveloping von Neumann algebra A such that δ(x) = hx-xh. It is shown that the generator h belongs to the class of lower semi-continuous elements in A. From this it follows that if the function π→∥π.δ∥ is continuous on the spectrum of A then h multiplies A. This immediately implies that each derivation of a simple C*-algebra is given by a multiplier of the algebra. Another application shows that each derivation of a countably generated monotone sequentially closed C*- algebra is inner. © 1974 Pacific Journal of Mathematics Manufactured and first issued in Japan.
If pi: A-->B is a surjective morphism between separable C(*)-algebras, then for each derivation delta of B there is a derivation [unk]delta of A such that pi([unk]delta(a)) for each a in A.
- J G Glimm
- C Stone-Weierstrass Theorem For
J. G. GLIMM, A Stone-Weierstrass theorem for C*-algebras, Ann. of Math. 72 (1960),
216-244.
- W J Bouchers
- Uber Ableitungen Von
W. J. BOUCHERS, Uber Ableitungen von C*-Algebren, Nachr. Akad. Wiss. Gottingen
Math.-Phys. Kl. II, Nr. 2 (1973).
- S Sakai
- Tόhoku Kaplansky
- Math
S. SAKAI, On a conjecture of Kaplansky, Tόhoku Math. J. 12 (1960), 31-33.
The norm of an inner derivation of an A T7*-algebra, preprint
- H Halpern
H. HALPERN, The norm of an inner derivation of an A T7*-algebra, preprint.
DEEL, Derivations of APF*-algebras
The norm of an inner derivation of an A T*-algebra
- H Halpern