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Operator Algebras / Algèbres des opérateurs
(Michael Lamoureaux and Ian Putnam, Organizers)

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BERNDT BRENKEN, University of Calgary, Calgary, Alberta  T2N 1N4
Representations of graph C^*-algebras and endomorphisms of type I algebras with discrete center

Cuntz-Krieger algebras associated with infinite, possibly infinite valued matrices with any number of zero entries were introduced previously, and found to correspond exactly to C^*-algebras of directed graphs with any number of edges, sources, sinks, and isolated vertices. Here we show that the correspondence between representations and endomorphisms established earlier involving the original Cuntz-Krieger algebras extends to this setting, so to one between representations of Cuntz-Krieger algebras for infinite matrices and endomorphisms of a direct sum of type I factor von Neumann algebras. This further demonstrates that this is a natural approach to defining Cuntz-Krieger algebras in the more general setting.

KEN DAVIDSON, Department of Pure Mathematics, University of Waterloo, Ontario  N2L 3G1
Primitive limit algebras and C^*-envelopes

We study irreducible representations of regular limit subalgebras of AF-algebras. The main result is twofold: every closed prime ideal of a limit of direct sums of nest algebras (NSAF) is primitive, and every prime regular limit algebra is primitive. A key step is that the quotient of a NSAF algebra by any closed ideal has an AF C^*-envelope, and this algebra is exhibited as a quotient of a concretely represented AF algebra. When the ideal is prime, the C^*-envelope is primitive. The GNS construction is used to produce algebraically irreducible representations for quotients of NSAF algebras by closed prime ideals. Thus the closed prime ideals of NSAF algebras coincide with the primitive ideals. Moreover these representations extend to *-representations of the C^*-envelope of the quotient, so that a fortiori these algebras are also operator primitive. The same holds true for arbitrary limit algebras and the {0} ideal.

GEORGE ELLIOTT, University of Toronto, Toronto, Ontario
On the classification of non-simple C^*-algebras

A brief survey is given of progress in the classification of non-simple amenable C^*-algebras. (Many classification results, but by no means all, are focussed on the simple case.) Some recent work in progress is described.

JULIANA ERLIJMAN, University of Regina, Regina, Saskatchewan  S4S 0A2
N-sided braid type subfactors

We generalize the construction of two-sided pairs of braid subfactors of the hyperfinite II₁ factor to n-sided pairs. The n-sided inclusion contains a sequence of intermediate subfactors with the property that the inclusion of any two consecutive subfactors is conjugate to the two-sided pair.

DOUGLAS R. FARENICK, Department of Mathematics and Statistics, University of Regina Regina, Saskatchewan  S4S 0A2
An algebraic analogue of the Fukamiya-Kaplansky Lemma

The Fukamiya-Kaplansky Lemma is recast as: in any complex algebraic algebra with positive involution, elements of the form a*a have nonnegative spectrum. One application of this result is an algebraic characterisation of finite-dimensional C^*-algebras (real or complex) among all finite-dimensional involutive algebras.

DAVID KRIBS, University of Iowa, Iowa, USA
Completely positive maps in dilation and wavelet decompositions

Certain representations of the Cuntz C^*-algebra arise in wavelet analysis and through the minimal isometric dilations of row contractions acting on Hilbert space. Decomposition theories for dealing

with such representations have recently been developed; however, the computations required can become quite involved. An alternative method will be presented for obtaining this information strictly in terms of a completely positive map on finite dimensional space.

DAN KUCEROVSKY, Department of Mathematics and Statistics, University of New Brunswick, Fredericton, New Brunswick  E3B 5A3
Ideal-related extensions

We explain the extension picture of KK-theory. The deeper applications of the theory depend on the properties of so-called absorbing extensions. We give a characterization of absorbing extensions, and introduce an ``ideal-related'' version of the absorption property.

HUAXIN LIN, Department of Mathematics, University of Oregon, Eugene, Oregon  97403, USA
Classification of simple amanable C^*-algebras

We give a classification theorem for unital separable simple nuclear C^*-algebras with tracial topological rank zero which satisfy the Universal Coefficient Theorem. We prove that if A and B are two such C^*-algebras and

æ è K0(A), K0(A)+, [1A], K1(A) ö ø @ æ è K0(B), K0(B)+, [1B], K1(B) ö ø ,

then A @ B.

LAURENT MARCOUX, Department of Mathematics, University of Alberta, Edmonton, Alberta  T6G 2G1
Linear combinations of projections in certain C^*-algebras

In this talk we shall show that if a C^*-algebra A admits a certain 3 ×3 matrix decomposition, then every commutator in A can be expressed as a linear combination of at most 210 (and often fewer) projections in A.

In certain C^*-algebras, this is sufficient to allow us to express every element as a linear combination of a (fixed) finite number of projections.

JAMES MINGO, Queen's University
q-circular and z-circular elements of a C^*-algebra

(joint work with Alexandru Nica, University of Waterloo)

If -1 £ q £ 1 the q-commutation relations for a Hilbert space K are

a(x) a(h)* -q a(h)* a(x) = áx, hñ1    for x,h Î K

These interpolate between the canonical commutation (q = 1) and the anti-commutation (q = -1) relations.

Bo\.zejko and Speicher showed that the annihilation operators {a(x)}_x can be realized as bounded operators on q-Fock space. Elements of the form a(x) + a(x)^* are the q analogues of semi-circular elements and so for x^h the q-circular elements

a(x)+a(x)*+i æ è a(h) + a(h)* ö ø Ö2

are the analogues of the circular elements introduced by Voiculescu. q-circular elements can be characterized by combinatorial relations which we show can be extended to the case where q is a complex number z with |z| < 1. For these elements we show that there is an asymptotic model of random matrices.

ALEXANDRU NICA, University of Waterloo, Waterloo, Ontario  N2L 3G1
R-cyclic families of matrices in free probability

We introduce the concept of R-cyclic family of matrices with entries in a non-commutative probability space; the definition consists in asking that only the `cyclic' non-crossing cumulants of the entries of the matrices are allowed to be different from zero.

Let A₁, ¼, A_s be an R-cyclic family of d ×d matrices over a non-commutative probability space ( A, j). We prove a convolution-type formula for the explicit computation of the joint distribution of A₁, ¼, A_s (considered in M_d(A), with the natural state), in terms of the joint distribution (considered in the original space (A,j)) of the entries of the s matrices. We present several applications of this formula.

Moreover, let A₁,¼,A_s be as above, and let D Ì M_d (A) denote the algebra of scalar diagonal matrices. We prove that the R-cyclicity of the family A₁,¼,A_s is equivalent to a freeness requirement, when one works with amalgamation over the subalgebra D.

JOHN PHILLIPS, Victoria
Spectral Flow and the Dixmier trace

(joint work with A. Carey and F. Sukochev)

Suppose that (H,D) is an unbounded Fredholm Module for the C^*-algebra, A. That is, (1+D²)^-1 is compact and A is represented on H in such a way that the commutators [D,a] are bounded for a in a dense *-subalgebra, A, of A. Let {m_n} be the decreasing sequence of eigenvalues of the compact operator (1+D²)^-1/2 (which is a smooth replacement for |D|^-1 which may not exist). If

N å n = 0  mn = O(logN)

then we say that (H,D) is L^(1,¥)-summable. We observe that:

mN £ 1 N+1 N å n = 0  mn = O æ è logN N+1 ö ø .

Hence the sequence {m_n} (and therefore the Fredholm Module (H,D)) is p-summable for each p > 1.

Now for each unitary u Î A, the spectral flow along the linear path joining D to uDu^* makes sense and is known to be the Fredholm index of PuP on P(H) where P is the projection on the nonnegative eigenspace of D. By previous work of Carey and Phillips, we have the following analytic formula for this index for each p > 1:

sf(D,uDu*) = ind(PuP) = 1 Mp ó õ 1 0  Tr æ è u[D,u*](1+Dt2)-p/2 ö ø  dt,

where D_t = (1-t)D+tuDu^* and M_p = ò_-¥^¥(1+x²)^-p/2 dx.

By taking the limit of this formula (which is by no means obvious) we are able to deduce a formula of Connes' implicit in his book. That is,

sf(D,uDu*) = ind(PuP) = 1 2 Tw æ è u[D,u*](1+D2)-1/2 ö ø .

Where, T_w is ``the'' Dixmier trace on the ideal L^(1,¥). In the case that Connes usually considers ( i.e., |D|^-1 exists as a bounded operator) our formula is easily seen to imply:

sf(D,uDu*) = ind(PuP) = 1 2 Tw(u[D,u*]|D|-1),

which is the usual form of Connes' formula.

N. CHRISTOPHER PHILLIPS, Department of Mathematics, University of Oregon, Eugene, Oregon  97403-1222, USA
C^*-algebras of minimal diffeomorphisms

This work is joint with Lin Qing.

Let M be a compact smooth manifold, and let h: M ® M be a minimal diffeomorphism. Our main theorem is that the crossed product C^*-algebra C^*(Z,M,h) can be represented as a direct limit, with no dimension growth, of what we call recursive subhomogeneous C^*-algebras. Many (but not all) properties of simple direct limits, with no (or very slow) dimension growth, of homogeneous C^*-algebras, carry over to such direct limits. In particular, the crossed product always has stable rank one. Moreover, if h is uniquely ergodic, so that C^*(Z,M,h) has a unique normalized trace t, then C^*(Z, M, h) has real rank zero if and only if the range of t_* on K₀(C^*(Z,M,h)) is dense in R, and, in this case, C^* (Z,M,h) belongs to the class of simple separable nuclear C^*-algebras known to be determined up to isomorphism by the Elliott invariant.

With suitable choices of M and h, we can show that the crossed product C^*-algebra preserves much less information about the diffeomorphism h than is the case for minimal homeomorphisms of the Cantor set (as in the work of Giordano-Putnam-Skau). In particular, we can give two minimal diffeomorphisms of the 3-torus (S¹)³ which are not orbit equivalent but whose crossed product C^*-algebras are isomorphic. We can also give two minimal diffeomorphisms, one of S³×S¹ and one of S⁵ ×S¹, whose crossed product C^*-algebras are isomorphic; since the manifolds don't even have the same dimension, orbit equivalence is clearly impossible.

CHRISTIAN SKAU, Norwegian University of Science and Technology(NTNU), Norway
Toeplitz flows and Choquet simplices-using Bratteli diagrams and dimension groups to establish a connection

Toeplitz flows,defined via arithmetical progressions,form a distinguished class of minimal dynamical systems. It was shown by T. Downarowicz in 1991 (a preliminary result had been obtained by S. Williams in 1984) by a long and arduous argument that every Choquet simplex can be realized as the set of invariant probability measures of a Toeplitz flow. We will present a more conceptual proof, extending Downarowicz result, using Bratteli diagrams and the basic theory of dimension groups. We will simultanously obtain a characterization of the K-0 groups associated to Toeplitz flows.

KEITH TAYLOR, University of Saskatchewan, Saskatoon, Saskatchewan
A connection between projections in the L¹-algebra of a locally compact group and wavelets

We discuss a strong connection between projections (selfadjoint idempotents) in L¹(G) for the group G formed as a semidirect product of the reals acting on an n-dimensional vector space on the one hand and functions which generate continuous frames on the other hand.