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Asymptotic behavior of the invariant measure for a diffusion related to an NA group

Ewa Damek, Andrzej Hulanicki (2006)

Colloquium Mathematicae

On a Lie group NA that is a split extension of a nilpotent Lie group N by a one-parameter group of automorphisms A, the heat semigroup μ t generated by a second order subelliptic left-invariant operator ∑ j = 0 m Y j + Y is considered. Under natural conditions there is a μ ̌ t -invariant measure m on N, i.e. μ ̌ t * m = m . Precise asymptotics of m at infinity is given for a large class of operators with Y₀,...,Yₘ generating the Lie algebra of S.

Asymptotic spherical analysis on the Heisenberg group

Jacques Faraut (2010)

Colloquium Mathematicae

The asymptotics of spherical functions for large dimensions are related to spherical functions for Olshanski spherical pairs. In this paper we consider inductive limits of Gelfand pairs associated to the Heisenberg group. The group K = U(n) × U(p) acts multiplicity free on 𝓟(V), the space of polynomials on V = M(n,p;ℂ), the space of n × p complex matrices. The group K acts also on the Heisenberg group H = V × ℝ. By a result of Carcano, the pair (G,K) with G = K ⋉ H is a Gelfand pair. The main results...

Automatic continuity of operators commuting with translations

J. Alaminos, J. Extremera, A. R. Villena (2006)

Studia Mathematica

Let τ X and τ Y be representations of a topological group G on Banach spaces X and Y, respectively. We investigate the continuity of the linear operators Φ: X → Y with the property that Φ ∘ τ X ( t ) = τ Y ( t ) ∘ Φ for each t ∈ G in terms of the invariant vectors in Y and the automatic continuity of the invariant linear functionals on X.

Automorphisms and derivations of a Fréchet algebra of locally integrable functions

F. Ghahramani, J. McClure (1992)

Studia Mathematica

We find representations for the automorphisms, derivations and multipliers of the Fréchet algebra L ¹ l o c of locally integrable functions on the half-line ℝ + . We show, among other things, that every automorphism θ of L ¹ l o c is of the form θ = φ a e λ X e D , where D is a derivation, X is the operator of multiplication by coordinate, λ is a complex number, a > 0, and φ a is the dilation operator ( φ a f ) ( x ) = a f ( a x ) ( f ∈ L ¹ l o c , x ∈ ℝ + ). It is also shown that the automorphism group is a topological group with the topology of uniform convergence on bounded...

Averages of unitary representations and weak mixing of random walks

Michael Lin, Rainer Wittmann (1995)

Studia Mathematica

Let S be a locally compact (σ-compact) group or semigroup, and let T(t) be a continuous representation of S by contractions in a Banach space X. For a regular probability μ on S, we study the convergence of the powers of the μ-average Ux = ʃ T(t)xdμ(t). Our main results for random walks on a group G are: (i) The following are equivalent for an adapted regular probability on G: μ is strictly aperiodic; U n converges weakly for every continuous unitary representation of G; U is weakly mixing for any...

Banach algebras associated with Laplacians on solvable Lie groups and injectivity of the Harish-Chandra transform

Detlev Poguntke (2010)

Colloquium Mathematicae

For any connected Lie group G and any Laplacian Λ = X²₁ + ⋯ + X²ₙ ∈ 𝔘𝔤 (X₁,...,Xₙ being a basis of 𝔤) one can define the commutant 𝔅 = 𝔅(Λ) of Λ in the convolution algebra ℒ¹(G) as well as the commutant ℭ(Λ) in the group C*-algebra C*(G). Both are involutive Banach algebras. We study these algebras in the case of a "distinguished Laplacian" on the "Iwasawa part AN" of a semisimple Lie group. One obtains a fairly good description of these algebras by objects derived from the semisimple group....

Banach algebras with unique uniform norm II

S. J. Bhatt, H. V. Dedania (2001)

Studia Mathematica

Semisimple commutative Banach algebras 𝓐 admitting exactly one uniform norm (not necessarily complete) are investigated. 𝓐 has this Unique Uniform Norm Property iff the completion U(𝓐) of 𝓐 in the spectral radius r(·) has UUNP and, for any non-zero spectral synthesis ideal ℐ of U(𝓐), ℐ ∩ 𝓐 is non-zero. 𝓐 is regular iff U(𝓐) is regular and, for any spectral synthesis ideal ℐ of 𝓐, 𝓐/ℐ has UUNP iff U(𝓐) is regular and for any spectral synthesis ideal ℐ of U(𝓐), ℐ = k(h(𝓐 ∩ ℐ)) (hulls...

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