Expand all Collapse all | Results 1 - 6 of 6 |
1. CMB 2012 (vol 56 pp. 534)
A Cohomological Property of $\pi$-invariant Elements Let $A$ be a Banach algebra and $\pi \colon A \longrightarrow \mathscr L(H)$
be a continuous representation of $A$ on a separable Hilbert space $H$
with $\dim H =\frak m$. Let $\pi_{ij}$ be the coordinate functions of
$\pi$ with respect to an orthonormal basis and suppose that for each
$1\le j \le \frak m$, $C_j=\sum_{i=1}^{\frak m}
\|\pi_{ij}\|_{A^*}\lt \infty$ and $\sup_j C_j\lt \infty$. Under these
conditions, we call an element $\overline\Phi \in l^\infty (\frak m , A^{**})$
left $\pi$-invariant if $a\cdot \overline\Phi ={}^t\pi (a) \overline\Phi$ for all
$a\in A$. In this paper we prove a link between the existence
of left $\pi$-invariant elements and the vanishing of certain
Hochschild cohomology groups of $A$. Our results extend an earlier
result by Lau on $F$-algebras and recent results of Kaniuth-Lau-Pym
and the second named author in the special case that $\pi \colon A
\longrightarrow \mathbf C$ is a non-zero character on $A$.
Keywords:Banach algebras, $\pi$-invariance, derivations, representations Categories:46H15, 46H25, 13N15 |
2. CMB 2011 (vol 56 pp. 13)
Ordering the Representations of $S_n$ Using the Interchange Process Inspired by Aldous' conjecture for
the spectral gap of the interchange process and its recent
resolution by Caputo, Liggett, and Richthammer, we define
an associated order $\prec$ on the irreducible representations of $S_n$. Aldous'
conjecture is equivalent to certain representations being comparable
in this order, and hence determining the ``Aldous order'' completely is a
generalized question. We show a few additional entries for this order.
Keywords:Aldous' conjecture, interchange process, symmetric group, representations Categories:82C22, 60B15, 43A65, 20B30, 60J27, 60K35 |
3. CMB 2011 (vol 55 pp. 26)
A Mahler Measure of a $K3$ Surface Expressed as a Dirichlet $L$-Series We present another example of a $3$-variable polynomial defining a $K3$-hypersurface
and having a logarithmic Mahler measure expressed in terms of a Dirichlet
$L$-series.
Keywords:modular Mahler measure, Eisenstein-Kronecker series, $L$-series of $K3$-surfaces, $l$-adic representations, LivnÃ© criterion, Rankin-Cohen brackets Categories:11, 14D, 14J |
4. CMB 2009 (vol 52 pp. 39)
A Representation Theorem for Archimedean Quadratic Modules on $*$-Rings We present a new approach to noncommutative real algebraic geometry
based on the representation theory of $C^\ast$-algebras.
An important result in commutative real algebraic geometry is
Jacobi's representation theorem for archimedean quadratic modules
on commutative rings.
We show that this theorem is a consequence of the
Gelfand--Naimark representation theorem for commutative $C^\ast$-algebras.
A noncommutative version of Gelfand--Naimark theory was studied by
I. Fujimoto. We use his results to generalize
Jacobi's theorem to associative rings with involution.
Keywords:Ordered rings with involution, $C^\ast$-algebras and their representations, noncommutative convexity theory, real algebraic geometry Categories:16W80, 46L05, 46L89, 14P99 |
5. CMB 2007 (vol 50 pp. 85)
Classification of Finite Group-Frames and Super-Frames Given a finite group $G$, we examine the classification of all
frame representations of $G$ and the classification of all
$G$-frames, \emph{i.e.,} frames induced by group representations of $G$.
We show that the exact number of equivalence classes of $G$-frames
and the exact number of frame representations can be explicitly
calculated. We also discuss how to calculate the largest number
$L$ such that there exists an $L$-tuple of strongly disjoint
$G$-frames.
Keywords:frames, group-frames, frame representations, disjoint frames Categories:42C15, 46C05, 47B10 |
6. CMB 2002 (vol 45 pp. 337)
Surjectivity of $\mod\ell$ Representations Attached to Elliptic Curves and Congruence Primes For a modular elliptic curve $E/\mathbb{Q}$, we show a number of
links between the primes $\ell$ for which the mod $\ell$
representation of $E/\mathbb{Q}$ has projective dihedral image and
congruence primes for the newform associated to $E/\mathbb{Q}$.
Keywords:torsion points of elliptic curves, Galois representations, congruence primes, Serre tori, grossencharacters, non-split Cartan Categories:11G05, 11F80 |