### A Banach principle for semifinite von Neumann algebras.

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Let $\U0001d504$ be a ${C}^{*}$-algebra, $G$ a compact abelian group, $\tau $ an action of $G$ by $*$-automorphisms of $\U0001d504,{\U0001d504}^{\tau}$ the fixed point algebra of $\tau $ and ${\U0001d504}_{F}$ the dense sub-algebra of $G$-finite elements in $\U0001d504$. Further let $H$ be a linear operator from ${\U0001d504}_{F}$ into $\U0001d504$ which commutes with $\tau $ and vanishes on ${\U0001d504}^{\tau}$. We prove that $H$ is a complete dissipation if and only if $H$ is closable and its closure generates a ${C}_{0}$-semigroup of completely positive contractions. These complete dissipations are classified in terms of certain twisted negative definite...

In this paper we extend to arbitrary number fields a construction of Bost-Connes of a ${C}^{*}$-dynamical system with spontaneous symmetry breaking and partition function the Riemann zeta function.

Let M be a finite von Neumann algebra acting on the standard Hilbert space L²(M). We look at the space of those bounded operators on L²(M) that are compact as operators from M into L²(M). The case where M is the free group factor is particularly interesting.

Let 𝓐 be a unital separable simple nuclear C*-algebra such that ℳ (𝓐 ⊗ 𝓚) has real rank zero. Suppose that ℂ is a separable simple liftable and purely large unital C*-subalgebra of ℳ (𝓐 ⊗ 𝓚)/ (𝓐 ⊗ 𝓚). Then the relative double commutant of ℂ in ℳ (𝓐 ⊗ 𝓚)/(𝓐 ⊗ 𝓚) is equal to ℂ.

The paper studies applications of ${C}^{*}$-algebras in geometric topology. Namely, a covariant functor from the category of mapping tori to a category of $AF$-algebras is constructed; the functor takes continuous maps between such manifolds to stable homomorphisms between the corresponding $AF$-algebras. We use this functor to develop an obstruction theory for the torus bundles of dimension $2$, $3$ and $4$. In conclusion, we consider two numerical examples illustrating our main results.