Let Γ be an Ã2 subgroup of PGL3(), where is a local field with residue field of order q. The module of coinvariants C(,ℤ)Γ is shown to be finite, where is the projective plane over . If the group Γ is of Tits type and if q ≢ 1 (mod 3) then the exact value of the order of the class [1]K0 in the K-theory of the (full) crossed product C*-algebra C(Ω) ⋊ Γ is determined, where Ω is the Furstenberg boundary of PGL3(). For groups of Tits type, this verifies a conjecture of G. Robertson and T. Steger.

In what follows the term C*-algebra will mean a complex C*-algebra with identity. We denote the identity element by 1. We shall also use the notation and terminology of Dixmier (3) without comment.

Let $\Gamma$ be a torsion free lattice in $G=\text{PGL}\left( 3,\mathbb{F} \right)$ where $\mathbb{F}$ is a nonarchimedean local field. Then $\Gamma$ acts freely on the affine Bruhat-Tits building $B$ of $G$ and there is an induced action on the boundary $\Omega$ of $B$. The crossed product ${{C}^{*}}$ -algebra $\mathcal{A}\left( \Gamma \right)=C\left( \Omega \right)\rtimes \Gamma$ depends only on $\Gamma$ and is classified by its $K$-theory. This article shows how to compute the $K$-theory of $\mathcal{A}\left( \Gamma \right)$ and of the larger class of rank two Cuntz-Krieger algebras.

A class of negative definite kernels is defined in terms of measure spaces. Using this concept, property (T) for a countable group $\Gamma$ is characterized in terms of measure-preserving actions of $\Gamma$, as follows. If a set $S$ is translated a finite amount by any fixed element of $\Gamma$, then there is a uniform bound on how far $S$ is translated.

Relative Property (T) for a subgroup H of a group G and its connection with positive definite functions are studied. A relation with the Haagerup approximation property is established. We show that if H is a non-normal subgroup of a group G which has Property (T) and G/H is amenable as a graph then H has finite index in G.

Injective matricial operator spaces have been classified up to Banach space isomorphism in [20]. The result is that every such space is isomorphic to l∞, l2, B(l2), or a direct sum of such spaces. A more natural project, given the matricial nature of the definitions involved, would be the classification of such spaces up to completely bounded isomorphism. This was done for injective von Neumann algebras in [6] and for injective operator systems (i.e. unital injective operator spaces) in [19]. It turns out that the spaces l∞ and B(l2) are in a natural way uniquely characterized up to completely bounded isomorphism. However, as shown in [20], a problem arises in the case of l2. For there are two injective operator spaces which are each isometrically isomorphic to l2 but not completely boundedly isomorphic to each other. We shall resolve this problem by showing that these are the only two possibilities, in the sense that any injective operator space which is isometric to l2 is completely isometric to one of them. (See Corollary 3 below.) The Hilbert spaces in von Neumann algebras investigated in [17], [13] turn out to be injective matricial operator spaces and are therefore completely isometric to one of our two examples. Another Hilbert space in B(l2) which has been much studied in operator theory, complex analysis and physics is the Cartan factor of type IV [10]. This is the complex linear span of a spin system and generates the Fermion C*-algebra ([3], §5·2). We show that a Cartan factor of type IV is not even completely boundedly isomorphic to an injective matricial operator space. One curious property of all the aforementioned Hilbert spaces is that every bounded operator on them is actually completely bounded, a fact that is crucial in our proofs.

It has long been known that there is a close connection between stochastic independence of continuous functions on an interval and space-filling curves [9]. In fact any two nonconstant continuous functions on [0, 1] which are independent relative to Lebesgue measure are the coordinate functions of a space filling curve. (The results of Steinhaus [9] have apparently been overlooked in more recent work in this area [3, 5, 6].)

In [2], R. Loewy and H. Schneider studied positive linear operators on circular cones. They characterised the extremal positive operators on these cones and noticed that such operators preserve the set of extreme rays of the cone in this case. They then conjectured that this property of extremal positive operators is true in general.

Let A be a noncommutative C*-algebra other than M2(I). We show that there exists a completely positive map φ of norm one on A and an element a ɛ A such that φ(a) = a, φ(a*a) = a*a, but φ(aa*) ≠ aa*.

Motivated by results of G. K. Pedersen, showing how a simple C*-algebra must contain an abundance of projections whenever it contains a single nontrivial projection, we provide generalisations and new proofs using more algebraic methods.

In recent years there has been considerable progress in the study of certain linear maps of C*-algebras which preserve the natural partial ordering. The most tractable such maps, the completely positive ones, have proved to be of great importance in the structure theory of C*-algebras(4). However general positive (order-preserving) linear maps are (at present) very intractable. For example, there is no algebraic formula which enables one to construct a general positive map, even on the algebra of 3 3 complex matrices. It is therefore of interest to study conditions stronger than positivity, but weaker than complete positivity.

A spin factor is a JW-factor of type I2. It is shown that certain automorphisms of finite dimensional spin factors extend to extremal positive linear maps on complex matrix algebras which are not decomposable, and hence, do not preserve extreme rays of the positive cone.

A unital C*-algebra A is said to have unitary 1-stable range (8) if for all pairs a, b of elements of A satisfying aA + bA = A there exists a unitary u in A such that a + bu is invertible. This concept is somewhat stronger than the usual stable range condition of algebraic K-theory ((3), chapter V). Handelman(8) shows among other things that finite AW*-algebras have unitary 1-stable range and uses this fact to study the algebraic K1 of a finite AW*-algebra. We prove below that a unital C*-algebra has unitary 1-stable range if and only if its group of invertible elements is dense. In addition we give some consequences of this fact and consider the related question of (unitary) polar decomposition in C*-algebras.

We investigate here the question of uniqueness of best approximation to operators in von Neumann algebras by elements of certain linear subspaces. Recall that a linear subspace V of a Banach space X is called a Chebyshev subspace if each vector in X has a unique best approximation by vectors in V. Our first main result characterizes the one-dimensional Chebyshev subspaces of a von Neumann algebra. This may be regarded as a generalization of a result of Stampfli [(4), theorem 2, corollary] which states that the scalar multiples of the identity operator form a Chebyshev subspace. Alternatively it may be regarded as a generalization of the commutative situation in which a continuous complex-valued function f on a compact Hausdorff space X spans a Chebyshev subspace of C(X) if and only if f does not vanish on X [(3), p. 215]. Our second main result is that a finite dimensional * subalgebra, of dimension > 1, of an infinite dimensional von Neumann algebra cannot be a Chebyshev subspace. This imposes limits to further generalization of Stampfli's result.

In what follows, B(H) will denote the C*-algebra of all bounded linear operators on a Hilbert space H. Suppose we are given a C*-subalgebra A of B(H), which we shall suppose contains the identity operator 1. We are concerned with the existence of states f of B(H) which satisfy the following trace-like relation relative to A:

Our first result shows the existence of states f satisfying (*), when A is the C*-algebra C*(x) generated by a normaloid operator × and the identity. This allows us to give simple proofs of some well-known results in operator theory. Recall that an operator × is normaloid if its operator norm equals its spectral radius.