We present the definition of crossed products of Hilbert C*-bimodules by Hilbert bundles with commuting finite group actions and finite dimensional fibers. This is a general construction containing the bundle construction and crossed products of Hilbert C*-bimodule by finite groups. We also study the structure of endomorphism algebras of the tensor products of bimodules. We also define the multiple crossed products using three bimodules containing more than 2 bundles and show the associativity law. Moreover, we present some examples of crossed product bimodules easily computed by our method.

Let G be a second countable locally compact group possessing a normal subgroup N with G/N abelian. We prove that if G/N is discrete then G has T1 primitive ideal space if and only if the G-quasiorbits in Prim N are closed. This condition on G-quasiorbits arose in Pukanzky's work on connected and simply connected solvable Lie groups where it is equivalent to the condition of Auslander and Moore that G be type R on N (-nilradical). Using an abstract version of Pukanzky's arguments due to Green and Pedersen we establish that if G is a connected and simply connected Lie group then Prim G is T1 whenever G-quasiorbits in [G, G] are closed.

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.

Let A be a C*-algebra and K the C*-algebra of all compact operators on a countably infinite dimensional Hilbert space. In this note, we shall show that there is an isomorphism of a semigroup of equivalence classes of certain partial automorphisms of A ⊗ K onto a semigroup of equivalence classes of certain countably generated A-A-Hilbert bimodules.

Let (N,ℝ,θ) be a centrally ergodic W* dynamical system. When N is not a factor, we show that for each nonzero real number t, the crossed product induced by the time t automorphism θt is not a factor if and only if there exist a rational number r and an eigenvalue s of the restriction of θ to the center of N, such that rst=2π. In the C* setting, minimality seems to be the notion corresponding to central ergodicity. We show that if (A,ℝ,α) is a minimal unital C* dynamical system and A is not simple, then, for each nonzero real number t, the crossed product induced by the time t automorphism αt is not simple if there exist a rational number r and an eigenvalue s of the restriction of α to the center of A, such that rst=2π. The converse is true if, in addition, A is commutative or prime.

Suppose that σ:𝔐→𝔐 is an ultraweakly continuous surjective *-linear mapping and d:𝔐→𝔐 is an ultraweakly continuous *-σ-derivation such that d(I) is a central element of 𝔐. We provide a Kadison–Sakai-type theorem by proving that ℌ can be decomposed into and d can be factored as the form , where δ:𝔐→𝔐 is an inner *-σ𝔎-derivation, Z is a central element, 2τ=2σ𝔏 is a *-homomorphism, and σ𝔎 and σ𝔏 stand for compressions of σ to 𝔎and 𝔏 , respectively.

Given two unital continuous C*-bundles, A and B, over the same compact Hausdorff base space X, we study the continuity properties of their different amalgamated free products over C(X).

We analyse Hecke pairs (G,H) and the associated Hecke algebra when G is a semi-direct product N ⋊ Q and H = M ⋊ R for subgroups M ⊂ N and R ⊂ Q with M normal in N. Our main result shows that, when (G,H) coincides with its Schlichting completion and R is normal in Q, the closure of in C*(G) is Morita–Rieffel equivalent to a crossed product I⋊βQ/R, where I is a certain ideal in the fixed-point algebra C*(N)R. Several concrete examples are given illustrating and applying our techniques, including some involving subgroups of GL(2,K) acting on K2, where K = ℚ or K = ℤ[p−1]. In particular we look at the ax + b group of a quadratic extension of K.

Vincent Lafforgue's bivariant K-theory for Banach algebras is invariant in the second variable under a rather general notion of Morita equivalence. In particular, the ordinary topological K-theory for Banach algebras is invariant under Morita equivalences.

We provea weak form of the mean ergodic theorem for actions of amenable locally compact quantum groups in the von Neumann algebra setting.

It is shown that the complexification of a positive linear map on a real C*-algebra need not be positive whereas the complexification of a completely positive linear map is completely positive. It is further shown that a real C*-algebra is amenable if and only if its complexification is amenable and that a completely positive linear map is amenable if and only if its complexification is. Finally, a real version of the Choi–Effros lifting theorem is established.

Let A be a C*-algebra, and let X be a Banach A-bimodule. Johnson [B. E. Johnson, ‘Local derivations on C*-algebras are derivations’, Trans. Amer. Math. Soc. 353 (2000), 313–325] showed that local derivations from A into X are derivations. We extend this concept of locality to the higher cohomology of a C*-algebra and show that, for every , bounded local n-cocycles from A(n) into X are n-cocycles.

Let be a higher rank Exel–Laca algebra generated by an alphabet . If contains d commuting isometries corresponding to rank d and the transition matrices do not have finite rows, then is trivial and is isomorphic to K0 of the abelian subalgebra of generated by the source projections of .