In this paper we consider the algebraic crossed product${\mathcal{A}}:=C_{K}(X)\rtimes _{T}\mathbb{Z}$induced by a homeomorphism$T$on the Cantor set$X$, where$K$is an arbitrary field with involution and$C_{K}(X)$denotes the$K$-algebra of locally constant$K$-valued functions on$X$. We investigate the possible Sylvester matrix rank functions that one can construct on${\mathcal{A}}$by means of full ergodic$T$-invariant probability measures$\unicode[STIX]{x1D707}$on$X$. To do so, we present a general construction of an approximating sequence of$\ast$-subalgebras${\mathcal{A}}_{n}$which are embeddable into a (possibly infinite) product of matrix algebras over$K$. This enables us to obtain a specific embedding of the whole$\ast$-algebra${\mathcal{A}}$into${\mathcal{M}}_{K}$, the well-known von Neumann continuous factor over$K$, thus obtaining a Sylvester matrix rank function on${\mathcal{A}}$by restricting the unique one defined on${\mathcal{M}}_{K}$. This process gives a way to obtain a Sylvester matrix rank function on${\mathcal{A}}$, unique with respect to a certain compatibility property concerning the measure$\unicode[STIX]{x1D707}$, namely that the rank of a characteristic function of a clopen subset$U\subseteq X$must equal the measure of$U$.

For a division ring D, denote by 𝓜D the D-ring obtained as the completion of the direct limit with respect to themetric induced by its unique rank function. We prove that, for any ultramatricial D-ring 𝓑 and any non-discrete extremal pseudo-rank function N on 𝓑, there is an isomorphism of D-rings , where stands for the completion of 𝓑 with respect to the pseudo-metric induced by N. This generalizes a result of von Neumann. We also show a corresponding uniqueness result for *-algebras over fields F with positive definite involution, where the algebra МF is endowed with its natural involution coming from the *-transpose involution on each of the factors .

We study beyond-mean-field properties of interacting spin-1 Bose gases with synthetic Rashba-Dresselhaus spin-orbit coupling at low energies. We derive a many-body Hamiltonian following a tight-binding approximation in quasi-momentum space, where the effective spin dependence of the collisions that emerges from spin-orbit coupling leads to dominant correlated tunneling processes that couple the different bound states. We discuss the properties of the spectrum of the derived Hamiltonian and its experimental signatures. In a certain region of the parameter space, the system becomes integrable, and its dynamics becomes analogous to that of a spin-1 condensate with spin-dependent collisions. Remarkably, we find that such dynamics can be observed in existing experimental setups through quench experiments that are robust against magnetic fluctuations. :1909.13840v1 [cond-mat.quant-gas] arXiv

In spinor Bose-Einstein condensates, spin-changing collisions are a remarkable proxy to coherently realize macroscopic many-body quantum states. These processes have been, e.g., exploited to generate entanglement, to study dynamical quantum phase transitions, and proposed for realizing nematic phases in atomic condensates. In the same systems dressed by Raman beams, the coupling between spin and momentum induces a spin dependence in the scattering processes taking place in the gas. Here we show that, at weak couplings, such modulation of the collisions leads to an effective Hamiltonian which is equivalent to the one of an artificial spinor gas with spin-changing collisions that are tunable with the Raman intensity. By exploiting this dressed-basis description, we propose a robust protocol to coherently drive the spin-orbit-coupled condensate into the ferromagnetic stripe phase via crossing a quantum phase transition of the effective low-energy model in an excited state.

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