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Let A be a thick building of type A 2 , and let V be its set of vertices. We study a commutative algebra B4 of 'averaging' operators acting on the space of complex valued functions on T. This algebra may be identified with a space of 'biradial functions' on ~V, or with a convolution algebra of bi-K-invariant functions on G, if G is a sufficiently large group of 'type-rotating' automorphisms of A, and K is the subgroup of G fixing a given vertex. We describe the multiplicative functionals on d and the corresponding spherical functions. We consider the C*-algebra induced by ei on £ 2 (T), find its spectrum E, prove positive definiteness of a kernel k z for each z e E, find explicitly the spherical Plancherel formula for any group G of type rotating automorphisms, and discuss the irreducibility of the unitary representations appearing therein. For the class of buildings A y arising from the groups Fy introduced in [2], this involves proving that the weak closure of si is maximal abelian in the von Neumann algebra generated by the left regular representation of Ty.

Abstract. We present an operator-valued version of the conditionally free product of states and measures, which in the scalar case was studied by Bożejko, Leinert and Speicher. The related combinatorics and limit theorems are provided.1. Introduction. The concept of free probability has been developed since the pioneering work of Voiculescu [V]. In this theory a probability space is a unital complex * -algebra A, elements of which are viewed as random variables, endowed with a state φ which plays the role of the expectation. Having a family (A i , φ i ) i∈I of such probability spaces we can define another one, (A, φ), where A is the unital free product * i∈I A i and φ is a state defined by the property that φ(a 1 . . . a m ) = 0 wheneverIn this situation one says that the family {A i } i∈I of subalgebras is free in the probability space (A, φ). This notion leads naturally to that of free convolution of probability measures on the real line. Speicher [S2, S3] has provided a combinatorial description of this theory by using the lattice of noncrossing partitions.Later on Bożejko, Leinert and Speicher [BS, BLS] investigated probability spaces A endowed with a pair (φ, ψ) of states. Having a family (A i , φ i , ψ i ) i∈I of such spaces they were able to construct a probability space (A, φ, ψ) (A, φ, ψ)). This led them to the notion of free convolution of pairs of probability measures on R.Here we are going to extend this theory to the situation when φ is allowed to be an operator-valued state, i.e. when φ is of the form φ(a) = P 0 π(a)| H 0 , where H 0 is a fixed Hilbert space, π is a * -representation of A in a Hilbert space H ⊃ H 0 and P 0 is the orthonormal projection of H onto H 0 , so that

Abstract. We prove that if p ≥ 1 and −1 ≤ r ≤ p − 1 then the binomial sequence np+r n , n = 0, 1, . . ., is positive definite and is the moment sequence of a probability measure ν(p, r), whose support is contained in 0, p p (p − 1) 1−p . If p > 1 is a rational number and −1 < r ≤ p − 1 then ν(p, r) is absolutely continuous and its density function V p,r can be expressed in terms of the Meijer G-function. In particular cases V p,r is an elementary function. We show that for p > 1 the measures ν(p, −1) and ν(p, 0) are certain free convolution powers of the Bernoulli distribution. Finally we prove that the binomial sequence np+r n is positive definite if and only if either p ≥ 1, −1 ≤ r ≤ p − 1 or p ≤ 0, p − 1 ≤ r ≤ 0. The measures corresponding to the latter case are reflections of the former ones.