Narrow Results

We reconsider the (non-relativistic) quantum theory of indistinguishable particles on the basis of Rieffel’s notion of $C^{\ast }$-algebraic (“strict”) deformation quantization. Using this formalism, we relate the operator approach of Messiah and Greenberg (1964) to the configuration space approach pioneered by Souriau (1967), Laidlaw and DeWitt-Morette (1971), Leinaas and Myrheim (1977), and others. In dimension $d>2$, the former yields bosons, fermions, and paraparticles, whereas the latter seems to leave room for bosons and fermions only, apparently contradicting the operator approach as far as the admissibility of parastatistics is concerned. To resolve this, we first prove that in $d>2$ the topologically non-trivial configuration spaces of the second approach are quantized by the algebras of observables of the first. Secondly, we show that the irreducible representations of the latter may be realized by vector bundle constructions, among which the line bundles recover the results of the second approach. Mathematically speaking, representations on higher-dimensional bundles (which define parastatistics) cannot be excluded, which render the configuration space approach incomplete. Physically, however, we show that the corresponding particle states may always be realized in terms of bosons and/or fermions with an unobserved internal degree of freedom (although based on non-relativistic quantum mechanics, this conclusion is analogous to the rigorous results of the Doplicher–Haag–Roberts analysis in algebraic quantum field theory, as well as to the heuristic arguments which led Gell-Mann and others to qcd (i.e. Quantum Chromodynamics)).

Increasing tensor powers of the $k\times k$ matrices $M_k(ℂ)$ are known to give rise to a continuous bundle of $C^{\ast }$-algebras over $I=\{0\}\cup 1/\mathbb{N}\subset [0,1]$ with fibers $A_{1/N}=M_k{(ℂ)}^{\otimes N}$ and $A_0=C(X_k)$, where $X_k=S(M_k(ℂ))$, the state space of $M_k(ℂ)$, which is canonically a compact Poisson manifold (with stratified boundary). Our first result is the existence of a strict deformation quantization of $X_k$ à la Rieffel, defined by perfectly natural quantization maps $Q_{1/N}:{Ã}_0\to A_{1/N}$ (where ${Ã}_0$ is an equally natural dense Poisson subalgebra of $A_0$).

We apply this quantization formalism to the Curie–Weiss model (an exemplary quantum spin with long-range forces) in the parameter domain where its ${\mathbb{Z}}_2$ symmetry is spontaneously broken in the thermodynamic limit $N\to \infty$. If this limit is taken with respect to the macroscopic observables of the model (as opposed to the quasi-local observables), it yields a classical theory with phase space $X_2\cong B^3$ (i.e. the unit three-ball in ${ℝ}^3$). Our quantization map then enables us to take the classical limit of the sequence of (unique) algebraic vector states induced by the ground state eigenvectors ${\mathrm{\Psi }}_N^{(0)}$ of this model as $N\to \infty$, in which the sequence converges to a probability measure $\mu$ on the associated classical phase space $X_2$. This measure is a symmetric convex sum of two Dirac measures related by the underlying ${\mathbb{Z}}_2$-symmetry of the model, and as such the classical limit exhibits spontaneous symmetry breaking, too. Our proof of convergence is heavily based on Perelomov-style coherent spin states and at some stage it relies on (quite strong) numerical evidence. Hence the proof is not completely analytic, but somewhat hybrid.

After introducing the infinite Fermi $C^{\ast }$-tensor product of a single ${\mathbb{Z}}_2$-graded $C^{\ast }$-algebra as an inductive limit, we systematically study the structure of the so-called symmetric states, that is those which are invariant under the group consisting of all finite permutations of a countable set. Among the obtained results, we mention the extension of De Finetti theorem which asserts that a symmetric state is a “mixture” of product states, each of which is a product of a single even state. This result induces a canonical morphism of the simplexes made of the symmetric even states on the usual infinite $C^{\ast }$-tensor product and the symmetric states on the infinite Fermi $C^{\ast }$-tensor product. We then extend the so-called Klein transformation to the infinite Fermi $C^{\ast }$-tensor product, available when the parity automorphism is inner. In such a situation, we investigate further properties of product states, the last being the extremal symmetric states on such an infinite Fermi $C^{\ast }$-tensor product $C^{\ast }$-algebra. This paper is complemented with a finite dimensional illustrative example for which the Klein transformation is not implementable, and then the Fermi tensor product might not generate a usual tensor product. Therefore, in general, the study of the symmetric states on the Fermi algebra cannot be easily reduced to that of the corresponding symmetric states on the usual infinite tensor product, even if both share many common properties.

We prove that a tracially continuous W${}^{\ast }$-bundle ${ℳ}$ over a compact Hausdorff space $X$ with all fibers isomorphic to the hyperfinite II₁ factor ${ℛ}$ that is locally trivial already has to be globally trivial. The proof uses the contractibility of the automorphism group $\mathrm{\text{Aut}}({ℛ})$ shown by Popa and Takesaki. There is no restriction on the covering dimension of $X$.

We show that if $𝜃$ is an irrational number in $(0,1)$, ${𝜃}_1$ and ${𝜃}_2$ are in $[0,1),$$A=\left(\right.\begin{array}{cc}\hfill a\hfill & \hfill b\hfill \\ \hfill c\hfill & \hfill d\hfill \end{array}\left)\right.$ is a matrix of infinite order in SL${}_2(\mathbb{Z})$, either tr$(A)=3$ or tr$(A)=2$ and the greatest common divisor of the entries in $I_2-A^{-1}$ is one, then for any $𝜀>0,$ there exists $\delta >0$ satisfying the following: For any unital simple separable $C^{\ast }$-algebra ${𝒜}$ with tracial rank at most one, any three unitaries $u,v,w$ in ${𝒜}$, if $u,v,w$ satisfy certain trace conditions and

We show that the dimension of the Cuntz semigroup of a $C^{\ast }$-algebra is determined by the dimensions of the Cuntz semigroups of its separable sub-$C^{\ast }$-algebras. This allows us to remove separability assumptions from previous results on the dimension of Cuntz semigroups.

To obtain these results, we introduce a notion of approximation for abstract Cuntz semigroups that is compatible with the approximation of a $C^{\ast }$-algebra by sub-$C^{\ast }$-algebras. We show that many properties for Cuntz semigroups are preserved by approximation and satisfy a Löwenheim–Skolem condition.

We introduce a type of zero-dimensional dynamical system (a pair consisting of a totally disconnected compact metrizable space along with a homeomorphism of that space), which we call “fiberwise essentially minimal”, that is a class that includes essentially minimal systems and systems in which every orbit is minimal. We prove that the associated crossed product $C^{\ast }$-algebra of such a system is an A$𝕋$-algebra. Under the additional assumption that the system has no periodic points, we prove that the associated crossed product $C^{\ast }$-algebra has real rank zero, which tells us that such $C^{\ast }$-algebras are classifiable by $K$-theory. The associated crossed product $C^{\ast }$-algebras to these nontrivial examples are of particular interest because they are non-simple (unlike in the minimal case).

We investigate when discrete, amenable groups have $C^{\ast }$-algebras of real rank zero. While it is known that this happens when the group is locally finite, the converse is an open problem. We show that if $C^{\ast }(G)$ has real rank zero, then all normal subgroups of $G$ that are elementary amenable and have finite Hirsch length must be locally finite.

We prove rigidity results for large classes of corona algebras, assuming the Proper Forcing Axiom. In particular, we prove that a conjecture of Coskey and Farah holds for all separable ${\mathrm{\text{C}}}^{\ast }$-algebras with the metric approximation property and an increasing approximate identity of projections.

Perfect C^∗-algebras were introduced by Akeman and Shultz in [Perfect C*-algebras, Mem. Amer. Math. Soc. 55(326) (1985)] and they form a certain subclass of C*-algebras determined by their pure states, and for which the general Stone–Weierstrass conjecture is true. In this paper, we introduce the notion of perfect JC-algebras, and we use the strong relationship between a JC-algebra A and its universal enveloping C^∗-algebra $C^{\ast }(A)$, to establish that if $C^{\ast }(A)$ is perfect and A is of complex type, then A is perfect. It is also shown that every scattered JC-algebra of complex type is perfect, and the same conclusion holds for every JC-algebra of complex type whose primitive spectrum is Hausdorff.

We show that for any countable discrete maximally almost periodic group $G$ and any UHF algebra $A$, there exists a strongly outer product type action $\alpha$ of $G$ on $A$. When $G$ is also elementary amenable, Matui–Sato have shown that such actions have their tracial Rokhlin property. Consequently, the class of crossed products $A{⋊}_{\alpha }G$ satisfy Elliott’s classification conjecture. We also show the existence of the “Rokhlin” property for countable discrete almost abelian group actions on the universal UHF algebra.

An ATAI (or ATAF, respectively) algebra, introduced in [C. Jiang, A classification of non simple C*-algebras of tracial rank one: Inductive limit of finite direct sums of simple TAI C*-algebras, J. Topol. Anal.3 (2011) 385–404] (or in [X. C. Fang, The classification of certain non-simple C*-algebras of tracial rank zero, J. Funct. Anal.256 (2009) 3861–3891], respectively) is an inductive limit ${lim}_{n\to \infty }(A_n={\oplus }_{i=1}{A}_n^i,{\varphi }_{nm})$, where each $A_n^i$ is a simple separable nuclear TAI (or TAF) C*-algebra with UCT property. In [C. Jiang, A classification of non simple C*-algebras of tracial rank one: Inductive limit of finite direct sums of simple TAI C*-algebras, J. Topol. Anal.3 (2011) 385–404], the second author classified all ATAI algebras by an invariant consisting orderd total $K$-theory and tracial state spaces of cut down algebras under an extra restriction that all element in $K_1(A)$ are torsion. In this paper, we remove this restriction, and obtained the classification for all ATAI algebras with the Hausdorffized algebraic $K_1$-group as an addition to the invariant used in [C. Jiang, A classification of non simple C*-algebras of tracial rank one: Inductive limit of finite direct sums of simple TAI C*-algebras, J. Topol. Anal.3 (2011) 385–404]. The theorem is proved by reducing the class to the classification theorem of ${𝒜}{\mathscr{H}}{𝒟}$ algebras with ideal property which is done in [G. Gong, C. Jiang and L. Li, A classification of inductive limit C*-algebras with ideal property, preprint (2016), arXiv:1607.07681]. Our theorem generalizes the main theorem of [X. C. Fang, The classification of certain non-simple C*-algebras of tracial rank zero, J. Funct. Anal.256 (2009) 3861–3891], [C. Jiang, A classification of non simple C*-algebras of tracial rank one: Inductive limit of finite direct sums of simple TAI C*-algebras, J. Topol. Anal.3 (2011) 385–404] (see Corollary 4.3).