Dbrane Phase Transitions and Monodromy in K-theory

Majumder and Sen have given an explicit construction of a first order phase transition in a non-supersymmetric system of Dbranes that occurs when the B field is varied. We show that the description of this transition in terms of K-theory involves a bundle of K groups of non-commutative algebras over the Kahler cone with nontrivial monodromy. Thus the study of monodromy in K groups associated with quantized algebras can be used to predict the phase structure of systems of (non-supersymmetric) Dbranes.Comment: 8 pages, RevTeX, 1 figur

Traces on cores of C*-algebras associated with self-similar maps

We completely classify the extreme tracial states onthe cores of the C*-algebras associated with self-similar maps on compact metric spaces. We present a complete list of them. The extreme tracial states are the union of the discrete type tracial states given by measures supported on the finite orbits of the branch points and a continuous type tracial state given by the Hutchinson measure on the original self-similar set.Comment: LaTeX 2e, 25 page

Bivariant algebraic K-Theory

We show how methods from K-theory of operator algebras can be applied in a completely algebraic setting to define a bivariant, matrix-stable, homotopy-invariant, excisive K-theory of algebras over a fixed unital ground ring H, kk_*(A,B), which is universal in the sense that it maps uniquely to any other such theory. It turns out kk is related to C. Weibel's homotopy algebraic K-theory, KH. We prove that, if H is commutative and A is central as an H-bimodule, then kk_*(H,A)=KH_*(A). We show further that some calculations from operator algebra KK-theory, such as the exact sequence of Pimsner-Voiculescu, carry over to algebraic kk.Comment: 40 pages, no figures. Comparison with Kassel's K-group added (see 6.7). Final version to appear in Crelle's Journal, including galley proof correction

Operator *-correspondences in analysis and geometry

An operator *-algebra is a non-selfadjoint operator algebra with completely isometric involution. We show that any operator *-algebra admits a faithful representation on a Hilbert space in such a way that the involution coincides with the operator adjoint up to conjugation by a symmetry. We introduce operator *-correspondences as a general class of inner product modules over operator *-algebras and prove a similar representation theorem for them. From this we derive the existence of linking operator *-algebras for operator *-correspondences. We illustrate the relevance of this class of inner product modules by providing numerous examples arising from noncommutative geometry.Comment: 31 pages. This work originated from the MFO workshop "Operator spaces and noncommutative geometry in interaction

Exel's crossed product for non-unital C*-algebras

We consider a family of dynamical systems (A,alpha,L) in which alpha is an endomorphism of a C*-algebra A and L is a transfer operator for \alpha. We extend Exel's construction of a crossed product to cover non-unital algebras A, and show that the C*-algebra of a locally finite graph can be realised as one of these crossed products. When A is commutative, we find criteria for the simplicity of the crossed product, and analyse the ideal structure of the crossed product.Comment: 22 page

Exploration of finite dimensional Kac algebras and lattices of intermediate subfactors of irreducible inclusions

We study the four infinite families KA(n), KB(n), KD(n), KQ(n) of finite dimensional Hopf (in fact Kac) algebras constructed respectively by A. Masuoka and L. Vainerman: isomorphisms, automorphism groups, self-duality, lattices of coideal subalgebras. We reduce the study to KD(n) by proving that the others are isomorphic to KD(n), its dual, or an index 2 subalgebra of KD(2n). We derive many examples of lattices of intermediate subfactors of the inclusions of depth 2 associated to those Kac algebras, as well as the corresponding principal graphs, which is the original motivation. Along the way, we extend some general results on the Galois correspondence for depth 2 inclusions, and develop some tools and algorithms for the study of twisted group algebras and their lattices of coideal subalgebras. This research was driven by heavy computer exploration, whose tools and methodology we further describe.Comment: v1: 84 pages, 13 figures, submitted. v2: 94 pages, 15 figures, added connections with Masuoka's families KA and KB, description of K3 in KD(n), lattices for KD(8) and KD(15). v3: 93 pages, 15 figures, proven lattice for KD(6), misc improvements, accepted for publication in Journal of Algebra and Its Application

Wieler solenoids, Cuntz-Pimsner algebras and K-theory

We study irreducible Smale spaces with totally disconnected stable sets and their associated $K$-theoretic invariants. Such Smale spaces arise as Wieler solenoids, and we restrict to those arising from open surjections. The paper follows three converging tracks: one dynamical, one operator algebraic and one $K$-theoretic. Using Wieler's Theorem, we characterize the unstable set of a finite set of periodic points as a locally trivial fibre bundle with discrete fibres over a compact space. This characterization gives us the tools to analyze an explicit groupoid Morita equivalence between the groupoids of Deaconu-Renault and Putnam-Spielberg, extending results of Thomsen. The Deaconu-Renault groupoid and the explicit Morita equivalence leads to a Cuntz-Pimsner model for the stable Ruelle algebra. The $K$-theoretic invariants of Cuntz-Pimsner algebras are then studied using the Cuntz-Pimsner extension, for which we construct an unbounded representative. To elucidate the power of these constructions we characterize the KMS weights on the stable Ruelle algebra of a Wieler solenoid. We conclude with several examples of Wieler solenoids, their associated algebras and spectral triples

The Algebras of Large N Matrix Mechanics

Extending early work, we formulate the large N matrix mechanics of general bosonic, fermionic and supersymmetric matrix models, including Matrix theory: The Hamiltonian framework of large N matrix mechanics provides a natural setting in which to study the algebras of the large N limit, including (reduced) Lie algebras, (reduced) supersymmetry algebras and free algebras. We find in particular a broad array of new free algebras which we call symmetric Cuntz algebras, interacting symmetric Cuntz algebras, symmetric Bose/Fermi/Cuntz algebras and symmetric Cuntz superalgebras, and we discuss the role of these algebras in solving the large N theory. Most important, the interacting Cuntz algebras are associated to a set of new (hidden) local quantities which are generically conserved only at large N. A number of other new large N phenomena are also observed, including the intrinsic nonlocality of the (reduced) trace class operators of the theory and a closely related large N field identification phenomenon which is associated to another set (this time nonlocal) of new conserved quantities at large N.Comment: 70 pages, expanded historical remark

Levinson's theorem and higher degree traces for Aharonov-Bohm operators

We study Levinson type theorems for the family of Aharonov-Bohm models from different perspectives. The first one is purely analytical involving the explicit calculation of the wave-operators and allowing to determine precisely the various contributions to the left hand side of Levinson's theorem, namely those due to the scattering operator, the terms at 0-energy and at infinite energy. The second one is based on non-commutative topology revealing the topological nature of Levinson's theorem. We then include the parameters of the family into the topological description obtaining a new type of Levinson's theorem, a higher degree Levinson's theorem. In this context, the Chern number of a bundle defined by a family of projections on bound states is explicitly computed and related to the result of a 3-trace applied on the scattering part of the model.Comment: 33 page

Weak Riemannian manifolds from finite index subfactors

Let $N\subset M$ be a finite Jones' index inclusion of II$_1$ factors, and denote by $U_N\subset U_M$ their unitary groups. In this paper we study the homogeneous space $U_M/U_N$, which is a (infinite dimensional) differentiable manifold, diffeomorphic to the orbit $${\cal O}(p) =\{u p u^*: u\in U_M\}$$ of the Jones projection $p$ of the inclusion. We endow ${\cal O}(p)$ with a Riemannian metric, by means of the trace on each tangent space. These are pre-Hilbert spaces (the tangent spaces are not complete), therefore ${\cal O}(p)$ is a weak Riemannian manifold. We show that ${\cal O}(p)$ enjoys certain properties similar to classic Hilbert-Riemann manifolds. Among them, metric completeness of the geodesic distance, uniqueness of geodesics of the Levi-Civita connection as minimal curves, and partial results on the existence of minimal geodesics. For instance, around each point $p_1$ of ${\cal O}(p)$, there is a ball $\{q\in {\cal O}(p):\|q-p_1\|<r\}$ (of uniform radius $r$) of the usual norm of $M$, such that any point $p_2$ in the ball is joined to $p_1$ by a unique geodesic, which is shorter than any other piecewise smooth curve lying inside this ball. We also give an intrinsic (algebraic) characterization of the directions of degeneracy of the submanifold inclusion ${\cal O}(p)\subset {\cal P}(M_1)$, where the last set denotes the Grassmann manifold of the von Neumann algebra generated by $M$ and $p$.Comment: 19 page