Expand all Collapse all | Results 1 - 11 of 11 |
1. CJM 2013 (vol 66 pp. 625)
Classifying the Minimal Varieties of Polynomial Growth Let $\mathcal{V}$ be a variety of associative algebras generated by
an algebra with $1$ over a field of characteristic zero. This
paper is devoted to the classification of the varieties
$\mathcal{V}$ which are minimal of polynomial growth (i.e., their
sequence of codimensions growth like $n^k$ but any proper subvariety
grows like $n^t$ with $t\lt k$). These varieties are the building
blocks of general varieties of polynomial growth.
It turns out that for $k\le 4$ there are only a finite number of
varieties of polynomial growth $n^k$, but for each $k \gt 4$, the
number of minimal varieties is at least $|F|$, the cardinality of
the base field and we give a recipe of how to construct them.
Keywords:T-ideal, polynomial identity, codimension, polynomial growth, Categories:16R10, 16P90 |
2. CJM 2012 (vol 65 pp. 961)
A Hilbert Scheme in Computer Vision Multiview geometry is the study of
two-dimensional images of three-dimensional scenes, a foundational subject in computer vision.
We determine a universal GrÃ¶bner basis for the multiview ideal of $n$ generic cameras.
As the cameras move, the multiview varieties vary in a family of dimension $11n-15$.
This family is the distinguished component of a multigraded Hilbert scheme
with a unique Borel-fixed point.
We present a combinatorial study
of ideals lying on that Hilbert scheme.
Keywords:multigraded Hilbert Scheme, computer vision, monomial ideal, Groebner basis, generic initial ideal Categories:14N, 14Q, 68 |
3. CJM 2011 (vol 63 pp. 381)
A Complete Classification of AI Algebras with the Ideal Property Let $A$ be an AI algebra; that is, $A$ is the $\mbox{C}^{*}$-algebra inductive limit
of a sequence
$$
A_{1}\stackrel{\phi_{1,2}}{\longrightarrow}A_{2}\stackrel{\phi_{2,3}}{\longrightarrow}A_{3}
\longrightarrow\cdots\longrightarrow A_{n}\longrightarrow\cdots,
$$
where
$A_{n}=\bigoplus_{i=1}^{k_n}M_{[n,i]}(C(X^{i}_n))$,
$X^{i}_n$ are $[0,1]$, $k_n$, and
$[n,i]$ are positive integers.
Suppose that $A$ has the
ideal property: each closed two-sided ideal of $A$ is generated by
the projections inside the ideal, as a closed two-sided ideal.
In this article, we give a complete classification of AI algebras with the ideal property.
Keywords:AI algebras, K-group, tracial state, ideal property, classification Categories:46L35, 19K14, 46L05, 46L08 |
4. CJM 2009 (vol 61 pp. 124)
Characterizing Complete Erd\H os Space The space now known as {\em complete Erd\H os
space\/} $\cerdos$ was introduced by Paul Erd\H os in 1940 as the
closed subspace of the Hilbert space $\ell^2$ consisting of all
vectors such that every coordinate is in the convergent sequence
$\{0\}\cup\{1/n:n\in\N\}$. In a solution to a problem posed by Lex G.
Oversteegen we present simple and useful topological
characterizations of $\cerdos$.
As an application we determine the class
of factors of $\cerdos$. In another application we determine
precisely which of the spaces that can be constructed in the Banach
spaces $\ell^p$ according to the `Erd\H os method' are homeomorphic
to $\cerdos$. A novel application states that if $I$ is a
Polishable $F_\sigma$-ideal on $\omega$, then $I$ with the Polish
topology is homeomorphic to either $\Z$, the Cantor set $2^\omega$,
$\Z\times2^\omega$, or $\cerdos$. This last result answers a
question that was asked
by Stevo Todor{\v{c}}evi{\'c}.
Keywords:Complete Erd\H os space, Lelek fan, almost zero-dimensional, nowhere zero-dimensional, Polishable ideals, submeasures on $\omega$, $\R$-trees, line-free groups in Banach spaces Categories:28C10, 46B20, 54F65 |
5. CJM 2007 (vol 59 pp. 109)
On Fiber Cones of $\m$-Primary Ideals Two formulas for the multiplicity of the fiber cone
$F(I)=\bigoplus_{n=0}^{\infty} I^n/\m I^n$ of an $\m$-primary ideal of
a $d$-dimensional Cohen--Macaulay local ring $(R,\m)$ are derived in
terms of the mixed multiplicity $e_{d-1}(\m | I)$, the multiplicity
$e(I)$, and superficial elements. As a consequence, the
Cohen--Macaulay property of $F(I)$ when $I$ has minimal mixed
multiplicity or almost minimal mixed multiplicity is characterized
in terms of the reduction number of $I$ and lengths of certain ideals.
We also characterize the Cohen--Macaulay and Gorenstein properties of
fiber cones of $\m$-primary ideals with a $d$-generated minimal
reduction $J$ satisfying $\ell(I^2/JI)=1$ or
$\ell(I\m/J\m)=1.$
Keywords:fiber cones, mixed multiplicities, joint reductions, Cohen--Macaulay fiber cones, Gorenstein fiber cones, ideals having minimal and almost minimal mixed multiplicities Categories:13H10, 13H15, 13A30, 13C15, 13A02 |
6. CJM 2006 (vol 58 pp. 859)
Nonstandard Ideals from Nonstandard Dual Pairs for $L^1(\omega)$ and $l^1(\omega)$ The Banach convolution algebras $l^1(\omega)$
and their continuous counterparts $L^1(\bR^+,\omega)$
are much
studied, because (when the submultiplicative weight function
$\omega$ is radical) they are pretty much the prototypic examples
of commutative radical Banach algebras. In cases of ``nice''
weights $\omega$, the only closed ideals they have are the obvious,
or ``standard'', ideals. But in the
general case, a brilliant but very difficult paper of Marc Thomas
shows that nonstandard ideals exist in $l^1(\omega)$. His
proof was successfully exported to the continuous case
$L^1(\bR^+,\omega)$ by Dales and McClure, but remained
difficult. In this paper we first present a small improvement: a
new and easier proof of the existence of nonstandard ideals in
$l^1(\omega)$ and $L^1(\bR^+,\omega)$. The new proof is based on
the idea of a ``nonstandard dual pair'' which we introduce.
We are then able to make a much larger improvement: we
find nonstandard ideals in $L^1(\bR^+,\omega)$ containing functions
whose supports extend all the way down to zero in $\bR^+$, thereby solving
what has become a notorious problem in the area.
Keywords:Banach algebra, radical, ideal, standard ideal, semigroup Categories:46J45, 46J20, 47A15 |
7. CJM 2005 (vol 57 pp. 1178)
Asymptotic Behavior of the Length of Local Cohomology Let $k$ be a field of characteristic 0, $R=k[x_1, \ldots, x_d]$ be a polynomial ring,
and $\mm$ its maximal homogeneous ideal. Let $I \subset R$ be a homogeneous ideal in
$R$. Let $\lambda(M)$ denote the length of an $R$-module $M$. In this paper, we show
that
$$
\lim_{n \to \infty} \frac{\l\bigl(H^0_{\mathfrak{m}}(R/I^n)\bigr)}{n^d}
=\lim_{n \to \infty} \frac{\l\bigl(\Ext^d_R\bigl(R/I^n,R(-d)\bigr)\bigr)}{n^d}
$$
always exists. This limit has been shown to be ${e(I)}/{d!}$ for $m$-primary ideals
$I$ in a local Cohen--Macaulay ring, where $e(I)$ denotes the multiplicity
of $I$. But we find that this limit may not be rational in general. We give an example
for which the limit is an irrational number thereby showing that the lengths of these
extention modules may not have polynomial growth.
Keywords:powers of ideals, local cohomology, Hilbert function, linear growth Categories:13D40, 14B15, 13D45 |
8. CJM 2005 (vol 57 pp. 897)
Representation of Banach Ideal Spaces and Factorization of Operators Representation theorems are proved for Banach ideal spaces with the Fatou property
which are built by the Calder{\'o}n--Lozanovski\u\i\ construction.
Factorization theorems for operators in spaces more general than the Lebesgue
$L^{p}$ spaces are investigated. It is natural to extend the Gagliardo
theorem on the Schur test and the Rubio de~Francia theorem on factorization of the
Muckenhoupt $A_{p}$ weights to reflexive Orlicz spaces. However, it turns out that for
the scales far from $L^{p}$-spaces this is impossible. For the concrete integral operators
it is shown that factorization theorems and the Schur test in some reflexive Orlicz spaces
are not valid. Representation theorems for the Calder{\'o}n--Lozanovski\u\i\ construction
are involved in the proofs.
Keywords:Banach ideal spaces, weighted spaces, weight functions,, CalderÃ³n--Lozanovski\u\i\ spaces, Orlicz spaces, representation of, spaces, uniqueness problem, positive linear operators, positive sublinear, operators, Schur test, factorization of operators, f Categories:46E30, 46B42, 46B70 |
9. CJM 2004 (vol 56 pp. 225)
Complex Uniform Convexity and Riesz Measure The norm on a Banach space gives rise to a subharmonic function on the
complex plane for which the distributional Laplacian gives a Riesz measure.
This measure is calculated explicitly here for Lebesgue $L^p$ spaces and the
von~Neumann-Schatten trace ideals. Banach spaces that are $q$-uniformly
$\PL$-convex in the sense of Davis, Garling and Tomczak-Jaegermann are
characterized in terms of the mass distribution of this measure. This gives
a new proof that the trace ideals $c^p$ are $2$-uniformly $\PL$-convex for
$1\leq p\leq 2$.
Keywords:subharmonic functions, Banach spaces, Schatten trace ideals Categories:46B20, 46L52 |
10. CJM 2004 (vol 56 pp. 3)
Locally Compact Pro-$C^*$-Algebras Let $X$ be a locally compact non-compact Hausdorff topological space. Consider
the algebras $C(X)$, $C_b(X)$, $C_0(X)$, and $C_{00}(X)$ of respectively arbitrary,
bounded, vanishing at infinity, and compactly supported continuous functions on $X$.
Of these, the second and third are $C^*$-algebras, the fourth is a normed algebra,
whereas the first is only a topological algebra (it is indeed a pro-$C^\ast$-algebra).
The interesting fact about these algebras is that if one of them is given, the
others can be obtained using functional analysis tools. For instance, given the
$C^\ast$-algebra $C_0(X)$, one can get the other three algebras by
$C_{00}(X)=K\bigl(C_0(X)\bigr)$, $C_b(X)=M\bigl(C_0(X)\bigr)$, $C(X)=\Gamma\bigl(
K(C_0(X))\bigr)$, where the right hand sides are the Pedersen ideal, the
multiplier algebra, and the unbounded multiplier algebra of the Pedersen ideal of
$C_0(X)$, respectively. In this article we consider the possibility of these
transitions for general $C^\ast$-algebras. The difficult part is to start with a
pro-$C^\ast$-algebra $A$ and to construct a $C^\ast$-algebra $A_0$ such that
$A=\Gamma\bigl(K(A_0)\bigr)$. The pro-$C^\ast$-algebras for which this is
possible are called {\it locally compact\/} and we have characterized them using
a concept similar to that of an approximate identity.
Keywords:pro-$C^\ast$-algebras, projective limit, multipliers of Pedersen's ideal Categories:46L05, 46M40 |
11. CJM 2000 (vol 52 pp. 1221)
Nest Representations of TAF Algebras A nest representation of a strongly maximal TAF algebra $A$ with
diagonal $D$ is a representation $\pi$ for which $\lat \pi(A)$ is
totally ordered. We prove that $\ker \pi$ is a meet irreducible ideal
if the spectrum of $A$ is totally ordered or if (after an appropriate
similarity) the von Neumann algebra $\pi(D)''$ contains an atom.
Keywords:nest representation, meet irreducible ideal, strongly maximal TAF algebra Categories:47L40, 47L35 |