Automorphisms of C1-algebra bundles

Let B be a unital C¹-algebra and A = Γ(E) be the C¹-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 AW¹-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)^, $\text{Z}$) is obtained. The map η is onto whenever A = C(X, B) and B is the quotient of a purely infinite AW¹-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).     

Applications of the Connes spectrum to C1-dynamical systems,II

We show that for a C¹-dynamical system (A, G, α) with G discrete (abelian) the Connes spectrum Γ(α) is equal to $\text{G}\text{?}$ if and only if every nonzero closed ideal in G × _αA has a nonzero intersection with A. Denote by G_J the closed subgroup of G that leaves fixed the primitive ideal J of A. We show for a general group G that if all isotropy groups G_J are discrete, then GX_αA is simple if and only if A is G-simple and $\text{Γ(}\text{α}\text{) =}\text{G}\text{?}$. This result is applicable not only when G is discrete but also when G?$\text{R}$ or G?$\text{T}$ provided that A is not primitive. Specializing to single automorphisms (i.e., G=$\text{Z}$) we show that if (the transposed of) α acts freely on a dense set of points in $\text{A}\text{?}$, then Λ(α)=$\text{T}$. The converse is only proved when A is of type I.     

On a duality for crossed products of C1-algebras

Let $\text{U}$ be a C¹-algebra, and G be a locally compact abelian group. Suppose α is a continuous action of G on $\text{U}$. Then there exists a continuous action $\text{}\text{?}$ga of the dual group $\text{G}\text{?}$ of G on the C¹-crossed product by α such that the C¹-crossed product is isomorphic to the tensor product and the C¹-algebra of all compact operators on L²(G).     

The fixed-point algebra of tensor-product actions of finite Abelian groups on UHF-algebras

Let G be a finite Abelian group acting by tensor-product automorphisms on a UHF-C¹-algebra $\text{D}$. Extending a result of A. Kishimoto it is shown that the number of extremal traces on the fixed-point algebra $\text{D}$^G equals the cardinality of the subgroup K of automorphisms in G which are weakly inner in the trace representation of $\text{D}$.     

Idéaux primitifs induits dans les produits croisés

Let an amenable second countable locally compact group G act continuously on a separable C¹-algebra A: the structure of the prime ideal space of the crossed product $\text{C}{}^1\text{(G, A)}$ is investigated without any restricting hypothesis on the action of G.     

K-theory for rickart C*-algebras

We give an explicit index map for any properly infinite closed ideal of a Rickart C ^*-algebra. This generalizes Olsen's work on von Neumann algebras. We use our results to compute the topological and the algebraic K ₁-groups of any quotient algebra of a Rickart C ^*-algebra.     

Derivations of non-separable C1-algebras

The question of which C¹-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 C¹-tensor product of a von Neumann algebra and an abelian C¹-algebra has only inner derivations. Other special types of C¹-algebras are shown to have only inner derivations as well such as the C¹-tensor product of L(H) (all bounded operators on separable Hilbert space) and any separable C¹-algebra having only inner derivations. Derivations from a smaller C¹-algebra into a larger one are also considered, and this concept is generalized to include derivations between C¹-algebras connected by a ¹-homomorphism. Finally, we consider the general problem of a sequence of linear functionals on a C¹-algebra which converges to zero (in norm) when restricted to any abelian C¹-subalgebra. Does such a sequence converge to zero in norm? The answer is “yes” for normal functionals on L(H), but unknown in general.     

Unbounded derivations commuting with compact group actions. II

Let ($\text{A}$, G, α) be a $\text{C}{}^1$ dynamical system and let δ be a closed ¹ derivation in $\text{A}$ which commutes with α and satisfies $\text{A}$^G ? ker(δ). If $\text{A}$ is a separable Type I $\text{C}{}^1$ algebra and G is a second countable compact group, then δ generates a strongly continuous one parameter group of ¹ automorphisms of $\text{A}$.     

On lifting and extending derivations of approximately finite-dimensional C1-algebras

It is shown that if A is the C¹-algebra inductive limit of a sequence of finite-dimensional C¹-algebras, then for each closed two-sided ideal J of A derivations can be lifted to A from $\text{A}\text{J}$, and for each projection e in A derivations can be extended to A from eAe. An application of the second result is given.     

From Endomorphisms to Automorphisms and Back: Dilations and Full Corners

When S is a discrete subsemigroup of a discrete group G suchthat G = S^–1S, it is possible to extend circle-valuedmultipliers from S to G, to dilate (projective) isometric representationsof S to (projective) unitary representations of G, and to dilate/extendactions of S by injective endomorphisms of a C*-algebra to actionsof G by automorphisms of a larger C*-algebra. These dilationsare unique provided they satisfy a minimality condition. The(twisted) semigroup crossed product corresponding to an actionof S is isomorphic to a full corner in the (twisted) crossedproduct by the dilated action of G. This shows that crossedproducts by semigroup actions are Morita equivalent to crossedproducts by group actions, making powerful tools available tostudy their ideal structure and representation theory. The dilationof the system giving the Bost–Connes Hecke C*-algebrafrom number theory is constructed explicitly as an application:it is the crossed product C₀(A_f)Q*₊, corresponding to the multiplicativeaction of the positive rationals on the additive group A_f offinite adeles.     

The primitive dual space of [FC]− groups

We show that if G is a σ compact locally compact group with relatively compact conjugacy classes, then the enveloping C¹-algebra $\text{C}{}^1\text{(G)}$ has a Hausdorff primitive ideal space. We also discuss some open problems and a partial converse result.     

K-theoretic amenability for SL2(Qp), and the action on the associated tree

It is proved that if a locally compact group G acts simplicially on a tree in such a way that the stabilizers of the vertices are amenable, then G is K-amenable. In particular, the canonical map from the full C¹-algebra onto the reduced C¹-algebra of G induces isomorphisms in K-theory. The main corollary of our result is that SL₂(Q_p) and some other groups over local fields are K-amenable.     

Crossed products by finite group actions with certain tracial Rokhlin property

We introduce a special tracial Rokhlin property for unital C~*-algebras. Let A be a unital tracial rank zero C~*-algebra(or tracial rank no more than one C~*-algebra). Suppose that α : G → Aut(A) is an action of a finite group G on A, which has this special tracial Rokhlin property, and suppose that A is a α-simple C~*-algebra. Then, the crossed product C~*-algebra C~*(G, A, α) has tracia rank zero(or has tracial rank no more than one). In fact,we get a more general results.     

Automatic relative boundedness of derivations in C1-algebras

We show that a derivation of a C¹-algebra $\text{A}$ is automatically relative bounded with respect to any closed ¹derivation of $\text{A}$ with smaller domain. We give also some related results on the automatic continuity of derivations in certain Banach algebras.     

Groups with T1 primitive ideal spaces

We give a simple necessary and sufficient condition for the group C¹-algebra of a connected locally compact group to have a T₁ primitive ideal space, i.e., to have the property that all primitive ideals are maximal. A companion result settles the same question almost entirely for almost connected groups. As a by-product of the method used, we show that every point in the primitive ideal space of the group C¹-algebra of a connected locally compact group is at least locally closed. Finally, we obtain an analog of these results for discrete finitely generated groups; in particular the primitive ideal space of the group C¹-algebra of a discrete finitely generated solvable group is T₁ if and only if the group is a finite extension of a nilpotent group.     

Certain quotient spaces are countably separated,II

Let $\text{A}$ be a Type I C¹-algebra, $\text{B}$ a C¹-subalgebra, and G a group of ¹-automorphisms of $\text{A}$ which leave $\text{B}$ invariant. If $\text{A}\text{?}$/G is countably separated, then $\text{B}\text{?}$/G is also countably separated. This result has several applications in group representations.     

On nuclear C1-algebras

A C¹-algebra is called nuclear if there is a unique way of forming its tensor product with any other C¹-algebra. Takesaki [17] showed that all C¹-algebras of type I and all inductive limits of such algebras are nuclear, but that the C¹-algebra $\text{C}{}_{\mathrm{r}}{}^1\text{(G)}$ generated by the left regular representation of G on l²(G) is nonnuclear, where G is the free group on two generators. In this paper an extension property for tensor products of C¹-algebras is introduced, and is characterized in terms of the existence of a certain family of weak expectations on the algebra. Nuclearity implies the extension property, and this is used to show that for a discrete group $\text{G, C}{}_{\mathrm{r}}{}^1\text{(G)}$ is nuclear if and only if G is amenable.An approximation property in the dual of a C¹-algebra is introduced, and shown to imply nuclearity. It is not clear whether the converse implication holds, but it is proved that the known nuclear C¹-algebras satisfy the approximation property.     

Isomorphisms between unital C-algebras

It is shown that every almost linear bijection of a unital C^∗-algebra A onto a unital C^∗-algebra B is a C^∗-algebra isomorphism when h(n²uy)=h(n²u)h(y) for all unitaries u∈A, all y∈A, and n=0,1,2,…, and that almost linear continuous bijection of a unital C^∗-algebra A of real rank zero onto a unital C^∗-algebra B is a C^∗-algebra isomorphism when h(n²uy)=h(n²u)h(y) for all , all y∈A, and n=0,1,2,…. Assume that X and Y are left normed modules over a unital C^∗-algebra A. It is shown that every surjective isometry , satisfying T(0)=0 and T(ux)=uT(x) for all x∈X and all unitaries u∈A, is an A-linear isomorphism. This is applied to investigate C^∗-algebra isomorphisms between unital C^∗-algebras.     

On the Galois cohomology of dedekind rings

Let R be a Dedekind domain, G a finite group of automorphisms of R, and A an ambiguous ideal of R i.e., σA = A for all σ ∈ G. The Tate groups Hⁿ(G, A) are considered as R^G-modules. A localization theorem is proved and the precise R^G-module structure determined in a particular case. In addition some remarks are made concerning cohomological triviality.