We consider the Markov operator Pφ on a discrete quantum group given by convolution with a q-tracial state φ. In the study of harmonic elements x, Pφ(x) = x, we define the Martin boundary Aφ. It is a separable C ∗-algebra carrying canonical actions of the quantum group and its dual. We establish a representation theorem to the effect that positive harmonic elements correspond to positive linear...

We define an analog of Voiculescu’s free entropy for n-tuples of unitaries u1, . . . , un in a tracial von Neumann algebra M which normalize a unital subalgebra L[0, 1] = B ⊂ M. Using this quantity, we define the free dimension δ0(u1, . . . , un G B). This number depends on u1, . . . , un only up to orbit equivalence over B. In particular, if R is a measurable equivalence relation on [0, 1] gen...

We present a formula for the norm of an elementary operator on a C∗-algebra that seems to be new. The formula involves (matrix) numerical ranges and a kind of geometrical mean for positive matrices, the tracial geometric mean, which seems not to have been studied previously and has interesting properties. In addition, we characterise compactness of elementary operators. We consider an elementar...

By a quasi-representation of a group G we mean an approximately multiplicative map of G to the unitary group of a unital C∗-algebra. A quasi-representation induces a partially defined map at the level K-theory. In the early 90s Exel and Loring associated two invariants to almostcommuting pairs of unitary matrices u and v: one a K-theoretic invariant, which may be regarded as the image of the Bo...