Let p be a prime integer, 1≤s≤r be integers and F be a field of characteristic different from p. We find upper and lower bounds for the essential p-dimension ed_p($ Al{{g}_{{{{p}^r},{{p}^s}}}} $) of the class $ Al{{g}_{{{{p}^r},{{p}^s}}}} $ of central simple algebras of degree p^r and exponent dividing p^s. In particular, we show that ed(Alg_8,2)=ed₂(Alg_8,2)=8 and ed_p($ Al{{g}_{{{{p}^2},p}}} $)=p²+p for p odd.

We construct a new subfactor planar algebra, and as a corollary a new subfactor, with the ‘extended Haagerup’ principal graph pair. This completes the classification of irreducible amenable subfactors with index in the range ($ {4},{3} + \sqrt {{3}} $), which was initiated by Haagerup in 1993. We prove that the subfactor planar algebra with these principal graphs is unique. We give a skein-theoretic description, and a description as a subalgebra generated by a certain element in the graph planar algebra of its principal graph. In the skein-theoretic description there is an explicit algorithm for evaluating closed diagrams. This evaluation algorithm is unusual because intermediate steps may increase the number of generators in a diagram. This is the published version of arXiv:0909.4099 [math.OA].

We use techniques from both real and complex algebraic geometry to study K-theoretic and related invariants of the algebra C(X) of continuous complex-valued functions on a compact Hausdorff topological space X. For example, we prove a parameterized version of a theorem by Joseph Gubeladze; we show that if M is a countable, abelian, cancellative, torsion-free, semi-normal monoid, and X is contractible, then every finitely generated projective module over C(X)[M] is free. The particular case $ M = \mathbb{N}_0^n $ gives a parameterized version of the celebrated theorem proved independently by Daniel Quillen and Andrei Suslin that finitely generated projective modules over a polynomial ring over a field are free. The conjecture of Jonathan Rosenberg which predicts the homotopy invariance of the negative algebraic K-theory of C(X) follows from the particular case $ M = {\mathbb{Z}^n} $. We also give algebraic conditions for a functor from commutative algebras to abelian groups to be homotopy invariant on C^*-algebras, and for a homology theory of commutative algebras to vanish on C^*-algebras. These criteria have numerous applications. For example, the vanishing criterion applied to nil K-theory implies that commutative C^*-algebras are K-regular. As another application, we show that the familiar formulas of Hochschild–Kostant–Rosenberg and Loday–Quillen for the algebraic Hochschild and cyclic homology of the coordinate ring of a smooth algebraic variety remain valid for the algebraic Hochschild and cyclic homology of C(X). Applications to the conjectures of Beĭlinson-Soulé and Farrell–Jones are also given.

For any unital separable simple infinite-dimensional nuclear C^∗-algebra with finitely many extremal traces, we prove that $ \mathcal{Z} $-absorption, strict comparison and property (SI) are equivalent. We also show that any unital separable simple nuclear C^∗-algebra with tracial rank zero is approximately divisible, and hence is $ \mathcal{Z} $-absorbing.