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Register a new userSmoothing operators and $C^*$-algebras for infinite dimensional Lie groups
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A host algebra of a (possibly infinite dimensional) Lie group $G$ is a $C^*$-algebra whose representations are in one-to-one correspondence with certain continuous unitary representations $pi colon G to U(cH)$. In this paper we present a new approach to host algebras for infinite dimensional Lie groups which is based on smoothing operators, i.e., operators whose range is contained in the space $cH^infty$ of smooth vectors. Our first major result is a characterization of smoothing operators $A$ that in particular implies smoothness of the maps $pi^A colon G to B(cH), g mapsto pi(g)A$. The concept of a smoothing operator is particularly powerful for representations $(pi,cH)$ which are semibounded, i.e., there exists an element $x_0 ing$ for which all operators $iddpi(x)$, $x in g$, from the derived representation are uniformly bounded from above in some neighborhood of $x_0$. Our second main result asserts that this implies that $cH^infty$ coincides with the space of smooth vectors for the one-parameter group $pi_{x_0}(t) = pi(exp tx_0)$. We then show that natural types of smoothing operators can be used to obtain host algebras and that, for every metrizable Lie group, the class of semibounded representations can be covered completely by host algebras. In particular, it permits direct integral decompositions.
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This paper is about the reduced group C*-algebras of real reductive groups, and about Hilbert C*-modules over these C*-algebras. We shall do three things. First we shall apply theorems from the tempered representation theory of reductive groups to determine the structure of the reduced C*-algebra (the result has been known for some time, but it is difficult to assemble a full treatment from the existing literature). Second, we shall use the structure of the reduced C*-algebra to determine the structure of the Hilbert C*-bimodule that represents the functor of parabolic induction. Third, we shall prove that the parabolic induction bimodule admits a secondary inner product, using which we can define a functor of parabolic restriction in tempered representation theory. We shall prove in the sequel to this paper that parabolic restriction is adjoint, on both the left and the right, to parabolic induction.
Let $G$ be a real or $p$-adic reductive group. We consider the tempered dual of $G$, and its connected components. For real groups, Wassermann proved in 1987, by noncommutative-geometric methods, that each connected component has a simple geometric structure which encodes the reducibility of induced representations. For $p$-adic groups, each connected component of the tempered dual comes with a compact torus equipped with a finite group action, and we prove that a version of Wassermanns theorem holds true under a certain geometric assumption on the structure of stabilizers for that action. We then focus on the case where $G$ is a quasi-split symplectic, orthogonal or unitary group, and explicitly determine the connected components for which the geometric assumption is satisfied.
For every finite dimensional Lie supergroup $(G,mathfrak g)$, we define a $C^*$-algebra $mathcal A:=mathcal A(G,mathfrak g)$, and show that there exists a canonical bijective correspondence between unitary representations of $(G,mathfrak g)$ and nondegenerate $*$-representations of $mathcal A$. The proof of existence of such a correspondence relies on a subtle characterization of smoothing operators of unitary representations. For a broad class of Lie supergroups, which includes nilpotent as well as classical simple ones, we prove that the associated $C^*$-algebra is CCR. In particular, we obtain the uniqueness of direct integral decomposition for unitary representations of these Lie supergroups.
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.
For a Lie algebra ${mathcal L}$ with basis ${x_1,x_2,cdots,x_n}$, its associated characteristic polynomial $Q_{{mathcal L}}(z)$ is the determinant of the linear pencil $z_0I+z_1text{ad} x_1+cdots +z_ntext{ad} x_n.$ This paper shows that $Q_{mathcal L}$ is invariant under the automorphism group $text{Aut}({mathcal L}).$ The zero variety and factorization of $Q_{mathcal L}$ reflect the structure of ${mathcal L}$. In the case ${mathcal L}$ is solvable $Q_{mathcal L}$ is known to be a product of linear factors. This fact gives rise to the definition of spectral matrix and the Poincar{e} polynomial for solvable Lie algebras. Application is given to $1$-dimensional extensions of nilpotent Lie algebras.
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