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Landstad’s duality theorem for C*-dynamical systems characterizes algebras arising as crossed products by actions of a given group. We state a dual version, characterizing cocrossed products by coactions.

Complement activation generates the C3a and C5a anaphylatoxins, which act through specific G-protein coupled receptors (R). Anaphylatoxins correlate with flares in patients with systemic lupus erythematosus. C3aR and C5aR mRNA and protein expression were significantly increased in brains of the lupus mouse model, MRL/lpr mice compared to those of the non-diseased congenic MRL/+ strain. To examine the role of C3aR and C5aR signaling in brain, specific antagonists, N2-[(2,2-diphenylethoxy)acetyl]-L-arginine and acetyl-Phe-(Orn-Pro-D-cyclohexylalanine)-Trp-Arg, respectively, were administered to MRL/lpr mice. Littermate controls were given vehicle alone. Control MRL/lpr mice had increased infiltration of blood cells into brain and an increase in a number of relevant cytokines and cell-bound accessory molecules (γ-IFN, TNF-α, IL-1β, MIP-2, MCP-1, RANTES, TNFR1, and ICAM-1) which were reduced by inhibition of C3aR and C5aR. In addition, the number of apoptotic cells that were increased in MRL/lpr mice compared to MRL/+ mice, was significantly reduced by inhibition of C3aR and C5aR, with a corresponding increase in Iκβα and phosphorylated Akt, and decrease in Bcl-xL and phosphorylated BAD. Thus, pharmacological blockade of anaphylatoxin receptors reduces pathological features in the brain in experimental lupus and represents a potentially viable treatment for human lupus cerebritis.

Using the strong relation between coactions of a discrete group G on C*-algebras and Fell bundles over G, we prove a new version of Mansfield's imprimitivity theorem for coactions of discrete groups. Our imprimitivity theorem works for the universally defined full crossed products and arbitrary subgroups of G, as opposed to the usual theory which uses the spatially defined reduced crossed products and normal subgroups of G. Moreover, our theorem factors through the usual one by passing to appropriate quotients. As applications we show that a Fell bundle over a discrete group is amenable in the sense of Exel if and only if the double dual action is amenable in the sense that the maximal and reduced crossed products coincide. We also give a new characterization of induced coactions in terms of their dual actions., Comment: 18 pages

A unitary representation induced from a normal subgroup of a separable locally compact group is irreducible if and only if the inducing representation is irreducible and the restriction of the induced representation to the normal subgroup is multiplicity-free.

In the duality for locally compact groups, much use is made of a version of the Hopf algebra technique in the context of von Neumann algebras, culminating in the theory of Kac algebras [6], [14]. It seems natural to ask whether something like a Hopf algebraic structure can be defined on the pre-dual of a Kac algebra. This leads to the question of whether the multiplication on a von Neumann algebra M, viewed as a linear map m from M ⊙ M (the algebraic tensor product) to M, can be pre-transposed to give a co-multiplication on the pre-dual M*, i.e., a linear map m* from M* to the completion of M* ⊙ M* with respect to some cross-norm. A related question is whether the multiplication on a C*-algebra A can be transposed to give a co-multiplication on the dual A*. Of course, this can be regarded as a special case of the preceding question by taking M = A**, where the double dual A** is identified with the enveloping von Neumann algebra of A.

A unitary representation induced from a normal subgroup of a separable locally compact group is irreducible if and only if the inducing representation is irreducible and the restriction of the induced representation to the normal subgroup is multiplicity-free.

Huruya [4] has proven that, for C ∗ {C^*} -algebras A 1 {A_1} and A 2 {A_2} , \[ ( A 1 ⊗ A 2 ) ∗ ∗ = A 1 ∗ ∗ ⊗ ¯ A 2 ∗ ∗ {({A_1} \otimes {A_2})^{**}} = A_1^{**}\overline \otimes A_2^{**} \] for every A 2 {A_2} if and only if A 1 {A_1} is scattered. We strengthen this by proving that ( A 1 ⊗ A 2 ) ∗ ∗ = A 1 ∗ ∗ ⊗ ¯ A 2 ∗ ∗ {({A_1} \otimes {A_2})^{**}} = A_1^{**}\overline \otimes A_2^{**} if and only if A 1 {A_1} or A 2 {A_2} is scattered. We discuss ramifications to representation theory and related questions regarding normal representations of W ∗ {W^*} -tensor products.