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Spectrum of commutative Banach algebras and isomorphism of C*-algebras related to locally compact groups

Zhiguo Hu (1998)

Studia Mathematica

Let A be a semisimple commutative regular tauberian Banach algebra with spectrum Σ A . In this paper, we study the norm spectra of elements of s p a n ¯ Σ A and present some applications. In particular, we characterize the discreteness of Σ A in terms of norm spectra. The algebra A is said to have property (S) if, for all φ ∈ ¯ Σ A 0 , φ has a nonempty norm spectrum. For a locally compact group G, let ℳ 2 d ( Ĝ ) denote the C*-algebra generated by left translation operators on L 2 ( G ) and G d denote the discrete group G. We prove that the Fourier...

Spherical functions and uniformly bounded representations of free groups

Tadeusz Pytlik (1991)

Studia Mathematica

We give a construction of an analytic series of uniformly bounded representations of a free group G, through the action of G on its Poisson boundary. These representations are irreducible and give as their coefficients all the spherical functions on G which tend to zero at infinity. The principal and the complementary series of unitary representations are included. We also prove that this construction and the other known constructions lead to equivalent representations.

Spherical functions on ordered symmetric spaces

Jacques Faraut, Joachim Hilgert, Gestur Ólafsson (1994)

Annales de l'institut Fourier

We define on an ordered semi simple symmetric space ℳ = G / H a family of spherical functions by an integral formula similar to the Harish-Chandra integral formula for spherical functions on a Riemannian symmetric space of non compact type. Associated with these spherical functions we define a spherical Laplace transform. This transform carries the composition product of invariant causal kernels onto the ordinary product. We invert this transform when G is a complex group, H a real form of G , and when ℳ ...

Spherical Stein manifolds and the Weyl involution

Dmitri Akhiezer (2009)

Annales de l’institut Fourier

We consider an action of a connected compact Lie group on a Stein manifold by holomorphic transformations. We prove that the manifold is spherical if and only if there exists an antiholomorphic involution preserving each orbit. Moreover, for a spherical Stein manifold, we construct an antiholomorphic involution, which is equivariant with respect to the Weyl involution of the acting group, and show that this involution stabilizes each orbit. The construction uses some properties of spherical subgroups...

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