Simple infinite-dimensional quotients of the group C★-algebras ofcertain discrete 6-dimensional nilpotent groups

Discrete Cocompact Subgroups of the Four-Dimensional Nilpotent Connected Lie Group and Their Group C*-Algebras

By a theorem of Dixmier, primitive quotients of enveloping algebras of finite-dimensional complex nilpotent Lie algebras are isomorphic to Weyl algebras. In view of this result, it is natural to consider simple quotients of positive parts of quantized enveloping algebras (and more generally of uniparameter Quantum Nilpotent Algebras) as quantum analogues of Weyl algebras. In this note, we study the Lie algebra of derivations of the simple quotients of U q + ( s o 5 ) U_q^+(\mathfrak {so}_5) of Gelfand-Kirillov dimension 2. For a specific family of such simple quotients, we prove that all derivations are inner (as in the case of Weyl algebras) whereas all other such algebras are quantum Generalized Weyl Algebras over a commutative Laurent polynomial algebra in one variable and have a first Hochschild cohomology group of dimension 1.

The K-groups, the range of trace on K 0 , and isomorphism classifications are obtained for simple infinite dimensional quotient C * -algebras of the group C * -algebras of six lattice subgroups, corresponding to each of the six non-isomorphic 5-dimensional connected, simply connected, nilpotent Lie groups. Connes' non-commutative geometry involving cyclic cocycles and the Chern character play a key role in the proofs.

For connected nilpotent groups, 7 is the lowest dimension\nwhere there are infinitely many non-isomorphic groups, and also\nwhere some groups have no discrete cocompact subgroups. Here\none infinite family of 7-dimensional connected groups is studied,\ndiscrete cocompact subgroups H are found for some of them, and then\nthe faithful simple quotients A of $C^{*}(\\roman H)$ are identified. Such A are\nshown to be isomorphic to $C^{*}$-crossed products $C^{*}(\\roman H_\n3,\\Cal C(\\Bbb T^3))$ generated\nby some intriguing effective minimal distal flows $(\\roman H_3,\\Bbb T^\n3)$, where $\\roman H_3$ is the\ndiscrete 3-dimensional Heisenberg group.

Let G be a torsion free, finitely generated nilpotent group, and H be a subgroup of finite index of G. Mal'cev proved in [4] that G can be embedded in a connected, simply connected, nilpotent rational Lie group 65, such that G and H are discrete subgroups of 65 and the coset spaces 6/G and 6/H are compact. Nomizu showed in [5] that if L is the associated Lie algebra of 65, then Hn(L, Q+) -Hn(6/G, Q+) where Q+ is the additive group of rational numbers. Let Hn(G, Q+) denote the rational cohomology group of the abstract group G. It follows from Lie group theory that Hn(65/G, Q+)'t-Hn(G, Q+), hence Hn(L, Q+)--Hn(G, Q+), and for the same reason we have Hn(L, Q+) ,-Hn(H, Q+), whence Hn(G, Q+)>Hn(H, Q+). In this paper we shall, however, indicate a purely algebraic approach to this theorem in the following strengthened form:

Let K be a field of characteristic zero. For a torsion-free finitely generated nilpotent group G, we naturally associate four finite dimensional nilpotent Lie algebras over K, ℒ K (G), grad(ℓ)(ℒ K (G)), grad(g)(exp ℒ K (G)), and L K (G). Let 𝔗 c be a torsion-free variety of nilpotent groups of class at most c. For a positive integer n, with n ≥ 2, let F n (𝔗 c ) be the relatively free group of rank n in 𝔗 c . We prove that ℒ K (F n (𝔗 c )) is relatively free in some variety of nilpotent Lie algebras, and ℒ K (F n (𝔗 c )) ≅ L K (F n (𝔗 c )) ≅ grad(ℓ)(ℒ K (F n (𝔗 c ))) ≅ grad(g)(exp ℒ K (F n (𝔗 c ))) as Lie algebras in a natural way. Furthermore, F n (𝔗 c ) is a Magnus nilpotent group. Let G 1 and G 2 be torsion-free finitely generated nilpotent groups which are quasi-isometric. We prove that if G 1 and G 2 are relatively free of finite rank, then they are isomorphic. Let L be a relatively free nilpotent Lie algebra over ℚ of finite rank freely generated by a set X. Give on L the structure of a group R, say, by means of the Baker–Campbell–Hausdorff formula, and let H be the subgroup of R generated by the set X. We show that H is relatively free in some variety of nilpotent groups; freely generated by the set X, H is Magnus and L ≅ ℒℚ(H) ≅ L ℚ(H) as Lie algebras. For relatively free residually torsion-free nilpotent groups, we prove that ℒ K and L K are isomorphic as Lie algebras. We also give an example of a finitely generated Magnus nilpotent group G, not relatively free, such that ℒℚ(G) is not isomorphic to L ℚ(G) as Lie algebras.

Δ-weakly mixing subset in positive entropy actions of a nilpotent group

Let $G$ be a locally compact group and $\mu$ a probability measure on $G$. Given a unitary representation $\pi$ of $G$, let $P_\mu$ denote the $\mu$-average $\int_G\pi(g)\,\mu(dg)$. $\mu$ is called neat if for every unitary representation $\pi$ and every $a$ in the support of $\mu$, $\slim_{n\to\infty}\bigl(P_\mu^n -\pi(a)^n E_\mu\bigr) =0$, where $E_\mu$ is a canonically defined orthogonal projection. $G\/$ is called neat if every almost aperiodic probability measure on $G$ is neat. Previously known results show that every almost aperiodic spread out probability measure is neat, in particular, every discrete group is neat; furthermore, identity excluding groups, in particular, compact groups and nilpotent groups, are neat. In this work neatness of solvable Lie groups, connected algebraic groups, Euclidian motion groups, [SIN] groups, and extensions of abelian groups by discrete groups is established. Neatness of ergodic probability measures on any locally compact group is also proven. The key to these results is the result that when $\{X_n\}_{n=1}^\infty$ is the left random walk of law $\mu$ on $G$ and $\pi$ a unitary representation in a separable Hilbert space, then for every $k=0,1,\dots$\,, the sequence $\pi(X_n)^{-1}P_\mu^{n-k}$ converges almost surely in the strong operator topology.

We establish pointwise almost everywhere convergence for ergodic averages along polynomial sequences in nilpotent groups of step two of measure-preserving transformations on \(\sigma \)-finite measure spaces. We also establish corresponding maximal inequalities on \(L^p\) for \(1<p\le \infty \) and \(\rho \)-variational inequalities on \(L^2\) for \(2<\rho <\infty \). This gives an affirmative answer to the Furstenberg–Bergelson–Leibman conjecture in the linear case for all polynomial ergodic averages in discrete nilpotent groups of step two. Our proof is based on almost-orthogonality techniques that go far beyond Fourier transform tools, which are not available in the non-commutative, nilpotent setting. In particular, we develop what we call a nilpotent circle method that allows us to adapt some of the ideas of the classical circle method to the setting of nilpotent groups.

In this paper we investigate the nature of projections in theC*-algebras of nilpotent locally compact groups and of the associated compact open subsets in their dual spaces. We obtain explicit descriptions of all projections inL1(G) whenG is a connected nilpotent group and inC*(G) whenG is an infinite discrete nilpotent group the torsion subgroup of which is contained in the center.

Extensions of characters of a discrete, torsion-free, nilpotent group to characters of some of its supergroups in its Mal'tsev completion are examined. Some sufficient conditions, as well as some necessary conditions, for extensibility are provided. In particular, the situation for nilpotent groups of class 2 is discussed. © de Gruyter 2005.

The characters (extremal positive definite central functions) of discrete nilpotent groups are studied. The relationship between the set of characters of G and the primitive ideals of the group C*-algebra C*(G) is investigated. It is shown that for a large class of nilpotent groups these objects are in 1–1 correspondence. One proof of this exploits the fact that faithful characters of certain nilpotent groups vanish off the finite conjugacy class subgroup. An example is given where the latter property fails.

An infinitesimal algebraic group over a field k is a (connected) algebraic group scheme whose coordinate ring is a finite dimension~al k-algebra. If one begins with a reductive algebraic group G defined over Fp, then one has a close relationship between the representation theory of G, that of its infinitesimal subgroups G, (defined as the kernels of the Frobenius endomorphisms ~rr: G-+ G), and that of its finite (Chevalley) subgroups G(Fq) of Fq-rational points (q = pd). A basic theorem of Cline et al. I-9, Theorem 6.6] relates the cohomology of G in a rational representation to that of the finite groups G(Fq). Related work of Cline et al. [6] and of the authors 1-16] relates the cohomology of G to that of its infinitesimal subgroups. A determination of the cohomology of the first infinitesimal subgroup G 1 of G with trivial coefficients has enabled the authors in [14] to make certain explicit cohomology calculations for the finite groups G(Fq) with certain nontrivial coefficients. In this paper, we continue our study of infinitesimal subgroups of simple algebraic groups by proving various qualitative properties of their cohomology and by extending our computational knowledge. In particular, we provide a computation which applies in a range of degrees to Chevalley groups over the prime field. We also investigate the cohomology of infinitesimal subgroups of nilpotent groups, enabling us for example to determine the cohomology of their Lie algebras in certain cases. We apply this Lie algebra cohomology computation to determine the integral cohomology of various torsion free, nilpotent groups modulo "small prime" torsion, thereby extending a theorem of Dwyer [12]. In more detail, the paper begins in Sect. 1 with a spectral sequence converging to the cohomology of the first infinitesimal subgroup G~ which was previously studied in [14]. From this spectral sequence, it easily follows that H*(G1, M) is a Noetherian H*(G1, k)-module for each finite dimensional rational Gl-module M. We then apply the work of Andersen and Jantzen [1] on the Gl-cohomology of induced modules to obtain the analogous result for the second infinitesimal subgroup Gz. Our analysis also permits a determination of the "generic

Letbe a length function on a group G, and let Mdenote the operator of pointwise multiplication byon l2(G). Following Connes, M𝕃can be used as a “Dirac” operator for the reduced group C*-algebra(G). It deûnes a Lipschitz seminorm on(G), which defines a metric on the state space of(G). We show that for any length function satisfying a strong form of polynomial growth on a discrete group, the topology from this metric coincides with the weak-* topology (a key property for the definition of a “compact quantum metric space”). In particular, this holds for all word-length functions on ûnitely generated nilpotent-by-finite groups.

equations (corresponding to U V .V U) that determine a faithful representation of H5,i which generates a simple C*-algebra Jt with a unique tracial state (correspond- ing to the irrational rotation algebra A03).These algebras are infinite dimensional simple quotients of C* (H5,i).Each section concludes by identifying the other infinite dimensional simple quotients of C* (H5,i) namely, those arising from a non-faithful representation of H5,, and we present them as matrix algebras over an irrational rotation algebra in most cases.Analogues of the simple quotients arising from the non-faithful representations (as described above) exist for C* (H4), but not for C* (H3); the irrational rotation algebras A03 exhaust the infinite dimensional simple quotients of C* (H3). (This situation for H3 also holds for the group Hs,; see Theorem 1.2.)Here are some further comments about the structure of the paper.In the Prelimi- naries, notation is established for C*-crossed products; also we give a brief summary of the results we need about the irrational rotation algebras A03 and, more especially,