Harmonic Functions on Groups and Fourier Algebras

Abstract

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Jordan Structures in Harmonic Functions and Fourier Algebras on Homogeneous Spaces

“…In the special case of = G and a complex measure σ , the space H σ (G, C) of complex σ -harmonic functions on G has been studied in [6], and the space H σ (G, M n ), for matrix-valued σ , was considered in [5]. Our results subsume some of those in [5,6] and are applicable to symmetric spaces.…”

“…Our results subsume some of those in [5,6] and are applicable to symmetric spaces. The intention of studying the matrix equation f = f * σ is to view it as a system of convolution equations.…”

“…The construction of such a projection is similar to that in [5,6] for groups. We outline the main steps below.…”

“…Let H be a closed subgroup of a locally compact group G. As in [14], one can define the Fourier algebra A(H\G) and the Fourier-Stieltjes algebra B(H\G) associated to the homogeneous space H\G. This extends the notion of the Fourier algebra A(G) and Fourier-Stieltjes algebra B(G) of a group G. The harmonic functionals on A(G) have been studied in [6]. We define harmonic functionals on A(H\G) and extend the results in [6] to A(H\G).…”

“…The Liouville theorem is closely related to amenability of the underlying groups as shown in Sect. 4 (see also [6,Proposition 2.1.3]). Given a compact subgroup H of a second countable group G, we show that G is amenable if, and only if, I σ (H\G) has a bounded approximate identity for every norm-one σ ∈ B(H\G), where A(H\G) * σ is the annihilator of I σ (H\G).…”