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The Boidol group is the smallest non-*-regular exponential Lie group. It is of dimension 4 and its Lie algebra is an extension of the Heisenberg Lie algebra by the reals with the roots 1 and -1. We describe the C*-algebra of the Boidol group as an algebra of operator fields defined over the spectrum of the group. It is the only connected solvable Lie group of dimension less than or equal to 4 whose group C*-algebra had not yet been determined.

Let $G=K\ltimes A$ be the semi-direct product group of a compact group $K$ acting on an abelian locally compact group $A$. We describe the $C^*$-algebra $C^*(G)$ of $G$ in terms of an algebra of operator fields defined over the spectrum of $G $, generalizing previous results obtained for some special classes of such groups.

We investigate Beurling-Fourier algebras, a weighted version of Fourier algebras, on various Lie groups focusing on their spectral analysis. We will introduce a refined general definition of weights on the dual of locally compact groups and their associated Beurling-Fourier algebras. Constructions of nontrivial weights will be presented focusing on the cases of representative examples of Lie groups, namely $SU(n)$, the Heisenberg group $\mathbb{H}$, the reduced Heisenberg group $\mathbb{H}_r$, the Euclidean motion group $E(2)$ and its simply connected cover $\widetilde{E}(2)$. We will determine the spectrum of Beurling-Fourier algebras on each of the aforementioned groups emphasizing its connection to the complexification of underlying Lie groups. We also demonstrate "polynomially growing" weights does not change the spectrum and show the associated regularity of the resulting Beurling-Fourier algebras., Comment: A few minor updates, to appear in Adv. Math.; 88 pages

We show that the tensor product $A\otimes B$ over $\mathbb{C}$ of two $C^* $-algebras satisfying the \textit{NCDL} conditions has again the same property. We use this result to describe the $C^* $-algebra of the Heisenberg motion groups $G_n = \mathbb{T}^n \ltimes \mathbb{H}_n$ as algebra of operator fields defined over the spectrum of $G_n $.

Let $G_n=U(n)\ltimes {\mathbb H}_n $ be the semi-direct product of the unitary group acting by automorphisms on the Heisenberg group ${\mathbb H}_n$. According to Lipsman, the unitary dual $\widehat {G_n} $ of $G_n $ is in one to one correspondence with the space of admissible coadjoint orbits $\mathfrak g_n^\ddagger /G_n $ of $G_n $. In this paper, we determine the topology of the space $\mathfrak g_n^\ddagger /G_n $ and we show that the correspondence with $\widehat {G_n} $ is a homeomorphism., Comment: 49 pages

Let $G= \exp(\g)$ be a connected, simply connected nilpotent Lie group. We show that for every $G$-invariant smooth sub-manifold $M$ of $g^*$, there exists an open relatively compact subset $\mathcal{M}$ of $M$ such that for any smooth adapted field of operators $(F(l))_{l\in M}$ supported in $G\cdot \mathcal{M}$ there exists a Schwartz function $f$ on $G$ such that $\pi_l(f)= \op_{F(l)}$ for all $l\in M$. This retract theorem can then be used to show that for every Lie group $\G$ of automorphisms of $G$ containing the inner automorphisms of $G$ with locally closed $\G$-orbits in $\g^*$, the proper $\G$-prime two-sided closed ideals of $L^1(G)$ are the kernels of $\G$-orbits in $\hat{G}$.

We show that for a connected Lie group $G$, its Fourier algebra $A(G)$ is weakly amenable only if $G$ is abelian. Our main new idea is to show that weak amenability of $A(G)$ implies that the anti-diagonal, $\check{\Delta}_G=\{(g,g^{-1}):g\in G\}$, is a set of local synthesis for $A(G\times G)$. We then show that this cannot happen if $G$ is non-abelian. We conclude for a locally compact group $G$, that $A(G)$ can be weakly amenable only if it contains no closed connected non-abelian Lie subgroups. In particular, for a Lie group $G$, $A(G)$ is weakly amenable if and only if its connected component of the identity $G_e$ is abelian., Comment: The final version to appear in Adv. Math.;Proposition 3.4 and related comments from the previous version have been deleted; 25 pages

We determine the space of primary ideals in the group algebra $L^1(G)$ of a connected nilpotent Lie group by identifying for every $\pi\in\hat G $ the family ${\mathcal I}^\pi $ of primary ideals with hull $\{\pi\}$ with the family of invariant polynomials of a certain finite dimensional subspace ${\mathcal P}_Q^\pi $ of the space of polynomials ${\mathcal P}(G) $ on $G $., Comment: 25 pages

For any nilpotent Lie group $G$ we provide a description of the image of its $C^*$-algebra through its operator-valued Fourier transform. Specifically, we show that $C^*(G)$ admits a finite composition series such that that the spectra of the corresponding quotients are Hausdorff sets in the relative topology, defined in terms of the fine stratification of the space of coadjoint orbits of $G$, and the canonical fields of elementary $C^*$-algebras defined by the successive subquotients are trivial. We give a description of the image of the Fourier transform as a $C^*$-algebra of piecewise continuous operator fields on the spectrum, determined by the boundary behavior of the restrictions of operator fields to the spectra of the subquotients in the composition series. For uncountable families of 3-step nilpotent Lie groups and also for a sequence of nilpotent Lie groups of arbitrarily high nilpotency step, we prove that every continuous trace subquotient of their $C^*$-algebras has its Dixmier-Douady invariant equal to zero., Comment: 27 pages

Using the operator valued Fourier transform, the C*-algebras of connected real two-step nilpotent Lie groups are characterized as algebras of operator fields defined over their spectra. In particular, it is shown by explicit computations, that the Fourier transform of such C*-algebras fulfills the norm controlled dual limit property., Comment: 37 pages, submitted to "Revista Matem\'atica Complutense"

Motivated by the description of the C*-algebra of the affine automorphism group $N_{6,28}$ of the Siegel upper half-plane of degree 2 as an algebra of operator fields defined over the unitary dual $\widehat{N_{6,28}}$ of the group, we introduce a family of C*-algebras, which we call almost $C_0(\mathcal{K})$, and we show that the C*-algebra of the group $N_{6,28}$ belongs to this class., Comment: 49 pages

Motivated by the description of the C*-algebras of 5 dimensional nilpotent Lie groups as algebras of operator fields defined over their spectra, we introduce the family of C* -algebras with norm controlled dual limits and we show that the C* -algebras of the 5 dimensional nilpotents Lie groups belong to this class.

Let g=g_1+g_2, [g,g] =g_2, be a nilpotent Lie algebra of step 2, V_1,..., V_m a basis of g_1 and L=\sum_{j,k} a_{jk} V_j V_k be a left-invariant differential operator on G=exp (g), where the coefficients a_{jk} form a real, symmetric mxm-matrix. It is shown that if a solution w(t,x) to the Schr\"odinger equation \partial_t w(t,g)=i Lw(t,g), w(0,g)=f(g), satisfies a suitable Gaussian type estimate at time t= 0 and at some time t=T\ne 0, then w=0 . The proof is based on Hardy's uncertainty principle and explicit computations within Howe's oscillator semigroup. Our results extend work by Ben Said and Thangavelu in which the authors study the Schr\"odinger equation associated to the sub-Laplacian on the Heisenberg group., Comment: 15 pages

For a compact group $G$ we define the Beurling-Fourier algebra $A_\omega(G)$ on $G$ for weights $\omega$ defined on the dual $\what G$ and taking positive values. The classical Fourier algebra corresponds to the case $\omega$ is the constant weight 1. We study the Gelfand spectrum of the algebra realizing it as a subset of the complexification $G_{\mathbb C}$ defined by McKennon and Cartwright and McMullen. In many cases, such as for polynomial weights, the spectrum is simply $G$. We discuss the questions when the algebra $A_\omega(G)$ is symmetric and regular. We also obtain various results concerning spectral synthesis for $A_\omega(G)$., Comment: 37 pages

We describe the C*-algebras of the Heisenberg group H_n, n\geq 1, and the thread-like Lie group G_N, N\geq 3, in terms of C*-algebras of operator fields., Comment: 31 pages

We extend the results by Froelich and Spronk and Turowska on the connection between operator synthesis and spectral synthesis for A(G) to second countable locally compact groups G. This gives us another proof that one-point subset of G is a set of spectral synthesis and that any closed subgroup is a set of local spectral synthesis. Furthermore we show that ``non-triangular'' sets are strong operator Ditkin sets and we establish a connection between operator Ditkin sets and Ditkin sets. These results are applied to prove that any closed subgroup of $G$ is a local Ditkin set., Comment: 21 pages

We study the problem of determining all connected Lie groups $G$ which have the following property (hlp): every sub-Laplacian $L$ on $G$ is of holomorphic $L^p$-type for $1\leq p<\infty, p\ne 2.$ First we show that semi-simple non-compact Lie groups with finite center have this property. We then apply an $L^p$-transference principle, essentially due to Anker, to show that every connected Lie group $G$ whose semi-simple quotient by its radical is non-compact has property (hlp). For the convenience of the reader, we give a self-contained proof of this transference principle, which generalizes the well-known Coifman-Weiss principle. One is thus reduced to studying compact extensions of solvable Lie groups. We extend previous work of Hebisch, Ludwig and M\"uller to compact extensions of certain classes of exponential solvable Lie groups., Comment: 39 pages