Massimo A. Picardello

Spherical Functions and Harmonic Analysis on Free Groups

Let F,, r > 1, be a free group with r generators. In this paper, we study a principal series and a complementary series of irreducible unitary representations of F,, which are defined through the action of F, on its Poisson boundary, relative to a simple random walk. We show that the regular representation of F, can be written as a direct integral of the representations of the principal series and that the resulting harmonic analysis on the free group bears a close resemblance with the harmonic analysis of SL(2, IR).

Martin boundaries of random walks: ends of trees and groups

Consider a transient random walk Xn on an infinite tree T whose nonzero transition probabilities are bounded below. Suppose that Xn is uniformly irreducible and has bounded step-length. (Alternatively, Xn can be regarded as a random walk on a graph whose metric is equivalent to the metric of T.) The Martin boundary of Xn is shown to coincide with the space fl of all ends of T (or, equivalently, of the graph). This yields a boundary representation theorem on 0 for all positive eigenfunctions of the transition operator, and a nontangential Fatou theorem which describes their boundary behavior. These results apply, in particular, to finitely supported random walks on groups whose Cayley graphs admit a uniformly spanning tree. A class of groups of this type is constructed.

Weakly amenable groups and amalgamated products

Denote by B2(G) the Herz-Schur multiplier algebra of a locally compact group G and by B2x(G) the closure of the Fourier algebra in the topology of pointwise convergence boundedly in the norm of B2(G). G is said to be weakly amenable if B2X(G) = B2(G). We show that every amal- gamated product of a countable collection of locally compact amenable groups over a compact open subgroup is weakly amenable. This improves and extends previous results that hold for amalgams of compact groups. Let G be a locally compact group, A(G) its Fourier algebra and B(G) its Fourier-Stieltjes algebra. Denote by Bx(G) the closure of A(G) in the topol- ogy of uniform convergence on compact sets, boundedly in norm. Then G is amenable if and only if B^(G) = B(G). Moreover, if G is amenable, then B(G)=JtA{G) (the algebra of multipliers of A(G)). Recent papers on amenability have devoted a significant amount of attention to the Herz-Schur multiplier algebra B2(G) (He). It was noted in (BF) that B2(G) coincides with the algebra Jf0A(G) of completely bounded multipliers of A(G), introduced in (dCH), where many of its interesting properties were investigated. Amenability can be characterized in terms of B2(G) as follows. G is amenable if and only if B(G) = B2(G) ((Lo); for discrete groups, (Bo)). Ac- tually, for the purpose of studying amenability, the multiplier algebra B2(G) is better suited than J?A , because it has nicer functorial properties. For instance, the Herz-Schur multiplier constant AG (i.e., the infimum of the norms of all approximate identities in B2(G)) is a von Neumann algebra invariant ((Ha2); see (CoH) for a deep study of this invariant on simple Lie groups of real rank 1), whereas a similar statement does not hold for J(A. Above all, B2(G) is nicer in taking products. In fact, B2(GX)®B2(G2) c B2(GX x G2) (dCH), but the same statement does not hold for JfA . This paper makes use of B2(G) to study a property related to amalgamation for another type of products: free products with amalgamation. This property, studied in §2, can be naturally phrased in

The plancherel measure for symmetric graphs

SummaryLet G be the free product of r copies of the cyclic groupZk.We obtain the Plancherel formula for the commutative O*-algebra of radial convolution operators on l2(G). The Plancherel measure is expressed in terms of the c-function appearing in the expansion of spherical functions on G as linear combinations of exponentials.

Representation Theory and Complex Analysis

A collection of advanced articles in Complex Analysis, Lie Groups, Unitary Representations and Quantum Computing, wirtten by the scientific leaders in these areas.

Twist points of planar domains

We establish a potential theoretic approach to the study of twist points in the boundary of simply connected planar domains.

Random walks on amalgams

Let Γ be a locally compact amalgam of compact groups. We use the action of Γ on a suitable tree to study all random walks on Γ which can be described as nearest neighbour random walks on the tree. In particular, we derive the asymptotic behaviour ofn-step transition probabilities.

A CONVERSE TO THE MEAN VALUE PROPERTY ON HOMOGENEOUS TREES

The homogeneous tree T of degree q + I (q ~ 2) may be consid- ered as a discrete analogue of the open unit disc D, On D, every harmonic function satisfies the mean value property (MVP) at every point. Conversely, positive functions on D having the MVP with respect to a ball with specified radius at each point of D are harmonic under certain assumptions concerning the radius function: results of this type are due to ), R, Baxter, W. Veech and others, Here we consider harmonic functions on T with respect to a natural choice of a discrete Laplacian: the analogous MVP is true in this setting. We present a Lipschitz-type condition on the radius function (which now has in- teger values and refers to the discrete metric of T) under which harmonicity holds for positive functions whose value at each point is the mean of its values over the ball of the radius assigned to this point. The method is based upon our previous results concerning the geometrical realization of Martin boundaries of certain transition operators as the space of ends of the underlying graph,

The Lusin area function and local admissible convergence of harmonic functions on homogeneous trees

We prove admissible convergence to the boundary of functions that are harmonic on a subset of a homogeneous tree by means of a discrete Green formula and an analogue of the Lusin area function.

Ends of Infinite Graphs, Potential Theory and Electrical Networks

We give an introductory survey on concepts and results concerning harmonic functions on infinite graphs with the goal of describing the interplay between graph structure and potential theory. A particular emphasis is on the connection between the Martin boundary for harmonic functions and the space of ends of the underlying graph. A variety of results is described.