Orientation in operator algebras

A concept of orientation is relevant for the passage from Jordan structure to associative structure in operator algebras. The research reported in this paper bridges the approach of Connes for von Neumann algebras and ourselves for C*-algebras in a general theory of orientation that is of geometric nature and is related to dynamics.

The operator algebra approach to quantum groups

A relatively simple definition of a locally compact quantum group in the C*-algebra setting will be explained as it was recently obtained by the authors. At the same time, we put this definition in the historical and mathematical context of locally compact groups, compact quantum groups, Kac algebras, multiplicative unitaries, and duality theory.

Prime factors

We use Voiculescu’s free probability theory to prove the existence of prime factors, hence answering a longstanding problem in the theory of von Neumann algebras.

On representations of finite type

Representations of the (infinite) canonical anticommutation relations and the associated operator algebra, the fermion algebra, are studied. A “coupling constant” (in (0,1]) is defined for primary states of “finite type” of that algebra. Primary, faithful states of finite type with arbitrary coupling are constructed and classified. Their physical significance for quantum thermodynamical systems at high temperatures is discussed. The scope of this study is broadened to include a large class of operator algebras sharing some of the structural properties of the fermion algebra.

New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan’s tau function

In this paper, we give two infinite families of explicit exact formulas that generalize Jacobi’s (1829) 4 and 8 squares identities to 4n2 or 4n(n + 1) squares, respectively, without using cusp forms. Our 24 squares identity leads to a different formula for Ramanujan’s tau function τ(n), when n is odd. These results arise in the setting of Jacobi elliptic functions, Jacobi continued fractions, Hankel or Turánian determinants, Fourier series, Lambert series, inclusion/exclusion, Laplace expansion formula for determinants, and Schur functions. We have also obtained many additional infinite families of identities in this same setting that are analogous to the η-function identities in appendix I of Macdonald’s work [Macdonald, I. G. (1972) Invent. Math. 15, 91–143]. A special case of our methods yields a proof of the two conjectured [Kac, V. G. and Wakimoto, M. (1994) in Progress in Mathematics, eds. Brylinski, J.-L., Brylinski, R., Guillemin, V. & Kac, V. (Birkhäuser Boston, Boston, MA), Vol. 123, pp. 415–456] identities involving representing a positive integer by sums of 4n2 or 4n(n + 1) triangular numbers, respectively. Our 16 and 24 squares identities were originally obtained via multiple basic hypergeometric series, Gustafson’s Cℓ nonterminating 6φ5 summation theorem, and Andrews’ basic hypergeometric series proof of Jacobi’s 4 and 8 squares identities. We have (elsewhere) applied symmetry and Schur function techniques to this original approach to prove the existence of similar infinite families of sums of squares identities for n2 or n(n + 1) squares, respectively. Our sums of more than 8 squares identities are not the same as the formulas of Mathews (1895), Glaisher (1907), Ramanujan (1916), Mordell (1917, 1919), Hardy (1918, 1920), Kac and Wakimoto, and many others.

Quantum groups with invariant integrals

Quantum groups have been studied intensively for the last two decades from various points of view. The underlying mathematical structure is that of an algebra with a coproduct. Compact quantum groups admit Haar measures. However, if we want to have a Haar measure also in the noncompact case, we are forced to work with algebras without identity, and the notion of a coproduct has to be adapted. These considerations lead to the theory of multiplier Hopf algebras, which provides the mathematical tool for studying noncompact quantum groups with Haar measures. I will concentrate on the *-algebra case and assume positivity of the invariant integral. Doing so, I create an algebraic framework that serves as a model for the operator algebra approach to quantum groups. Indeed, the theory of locally compact quantum groups can be seen as the topological version of the theory of quantum groups as they are developed here in a purely algebraic context.

Inositol 1,4,5-trisphosphate receptor subtype 3 in pancreatic islet cell secretory granules revisited.

It has been reported that the inositol 1,4,5-trisphosphate receptor subtype 3 is expressed in islet cells and is localized to both insulin and somatostatin granules [Blondel, O., Moody, M. M., Depaoli, A. M., Sharp, A. H., Ross, C. A., Swift, H. & Bell, G. I. (1994) Proc. Natl. Acad. Sci. USA 91, 7777-7781]. This subcellular localization was based on electron microscope immunocytochemistry using antibodies (affinity-purified polyclonal antiserum AB3) directed to a 15-residue peptide of rat inositol trisphosphate receptor subtype 3. We now show that these antibodies cross-react with rat, but not human, insulin. Accordingly, the anti-inositol trisphosphate receptor subtype 3 (AB3) antibodies label electron dense cores of mature (insulin-rich) granules of rat pancreatic beta cells, and rat granule labeling was blocked by preabsorption of the AB3 antibodies with rat insulin. The immunostaining of immature, Golgi-associated proinsulin-rich granules with AB3 antibodies was very weak, indicating that cross-reactivity is limited to the hormone and not its precursor. Also, the AB3 antibodies labeled pure rat insulin crystals grown in vitro but failed to stain crystals grown from pure human insulin. By immunoprecipitation, the antibodies similarly displayed a higher affinity for rat than for human insulin. We could not confirm the labeling of somatostatin granules using AB3 antibodies.