Expand all Collapse all | Results 1 - 25 of 142 |
1. CJM Online first
Spectral flow for nonunital spectral triples We prove two results about nonunital index theory left open in a
previous paper. The
first is that the spectral triple arising from an action of the reals on a $C^*$-algebra with invariant trace
satisfies the hypotheses of the nonunital local index formula. The second result concerns the meaning of spectral flow in the nonunital case. For the special case of paths
arising from the odd
index pairing for smooth spectral triples in the nonunital setting we are able to connect with earlier approaches to the analytic definition of spectral flow.
Keywords:spectral triple, spectral flow, local index theorem Category:46H30 |
2. CJM Online first
Rotation algebras and the Exel trace formula We found that if $u$ and $v$ are any two unitaries in
a unital $C^*$-algebra with $\|uv-vu\|\lt 2$ and $uvu^*v^*$ commutes with
$u$ and $v,$ then the $C^*$-subalgebra $A_{u,v}$ generated by $u$ and
$v$ is isomorphic to a quotient of some rotation algebra $A_\theta$
provided that $A_{u,v}$ has a unique tracial state.
We also found that the Exel trace formula holds in any unital
$C^*$-algebra.
Let $\theta\in (-1/2, 1/2)$ be a real number. We prove the
following:
For any $\epsilon\gt 0,$ there exists $\delta\gt 0$ satisfying the following:
if $u$ and $v$ are two unitaries in any unital simple $C^*$-algebra
$A$ with tracial rank zero such that
\[
\|uv-e^{2\pi i\theta}vu\|\lt \delta
\text{ and }
{1\over{2\pi i}}\tau(\log(uvu^*v^*))=\theta,
\]
for all tracial state $\tau$ of $A,$ then there exists a pair
of unitaries $\tilde{u}$ and $\tilde{v}$ in $A$
such that
\[
\tilde{u}\tilde{v}=e^{2\pi i\theta} \tilde{v}\tilde{u},\,\,
\|u-\tilde{u}\|\lt \epsilon
\text{ and }
\|v-\tilde{v}\|\lt \epsilon.
\]
Keywords:rotation algebras, Exel trace formula Category:46L05 |
3. CJM Online first
The C*-algebras of Compact Transformation Groups We investigate the representation theory of the
crossed-product $C^*$-algebra associated to a compact group $G$
acting on a locally compact space $X$ when the stability subgroups
vary discontinuously.
Our main result applies when $G$ has a principal stability subgroup or
$X$ is locally of finite $G$-orbit type. Then the upper multiplicity
of the representation of the crossed product induced from an
irreducible representation $V$ of a stability subgroup is obtained by
restricting $V$ to a certain closed subgroup of the stability subgroup
and taking the maximum of the multiplicities of the irreducible
summands occurring in the restriction of $V$. As a corollary we obtain
that when the trivial subgroup is a principal stability subgroup, the
crossed product is a Fell algebra if and only if every stability
subgroup is abelian. A second corollary is that the $C^*$-algebra of
the motion group $\mathbb{R}^n\rtimes \operatorname{SO}(n)$ is a Fell algebra. This uses
the classical branching theorem for the special orthogonal group
$\operatorname{SO}(n)$ with respect to $\operatorname{SO}(n-1)$. Since proper transformation
groups are locally induced from the actions of compact groups, we
describe how some of our results can be extended to transformation
groups that are locally proper.
Keywords:compact transformation group, proper action, spectrum of a C*-algebra, multiplicity of a representation, crossed-product C*-algebra, continuous-trace C*-algebra, Fell algebra Categories:46L05, 46L55 |
4. CJM Online first
Minimal Dynamical Systems on Connected Odd Dimensional Spaces Let $\beta\colon S^{2n+1}\to S^{2n+1}$ be a minimal homeomorphism ($n\ge 1$). We show that
the crossed product $C(S^{2n+1})\rtimes_\beta \mathbb{Z}$ has rational tracial rank at most one.
Let $\Omega$ be a connected compact metric space with finite covering dimension and
with $H^1(\Omega, \mathbb{Z})=\{0\}.$ Suppose that $K_i(C(\Omega))=\mathbb{Z}\oplus G_i,$ where $G_i$ is a finite abelian group, $i=0,1.$
Let $\beta\colon \Omega\to\Omega$ be a minimal homeomorphism. We also show that
$A=C(\Omega)\rtimes_\beta\mathbb{Z}$ has rational tracial rank at most one and is
classifiable.
In particular, this applies to the minimal dynamical systems on
odd dimensional real projective spaces.
This is done by studying minimal homeomorphisms on $X\times \Omega,$ where
$X$ is the Cantor set.
Keywords:minimal dynamical systems Categories:46L35, 46L05 |
5. CJM Online first
The Bochner-Schoenberg-Eberlein property and spectral synthesis for certain Banach algebra products Associated with two commutative Banach algebras $A$ and $B$ and
a character $\theta$ of $B$ is a certain Banach algebra product
$A\times_\theta B$, which is a splitting extension of $B$ by
$A$. We investigate two topics for the algebra $A\times_\theta
B$ in relation to the corresponding ones of $A$ and $B$. The
first one is the Bochner-Schoenberg-Eberlein property and the
algebra of Bochner-Schoenberg-Eberlein functions on the spectrum,
whereas the second one concerns the wide range of spectral synthesis
problems for $A\times_\theta B$.
Keywords:commutative Banach algebra, splitting extension, Gelfand spectrum, set of synthesis, weak spectral set, multiplier algebra, BSE-algebra, BSE-function Categories:46J10, 46J25, 43A30, 43A45 |
6. CJM 2013 (vol 65 pp. 1287)
$K$-theory of Furstenberg Transformation Group $C^*$-algebras The paper studies the $K$-theoretic invariants of the crossed product
$C^{*}$-algebras associated with an important family of homeomorphisms
of the tori $\mathbb{T}^{n}$ called Furstenberg transformations.
Using the Pimsner-Voiculescu theorem, we prove that given $n$, the
$K$-groups of those crossed products, whose corresponding $n\times n$
integer matrices are unipotent of maximal degree, always have the same
rank $a_{n}$. We show using the theory developed here that a claim
made in the literature about the torsion subgroups of these $K$-groups
is false. Using the representation theory of the simple Lie algebra
$\frak{sl}(2,\mathbb{C})$, we show that, remarkably, $a_{n}$ has a
combinatorial significance. For example, every $a_{2n+1}$ is just the
number of ways that $0$ can be represented as a sum of integers
between $-n$ and $n$ (with no repetitions). By adapting an argument
of van Lint (in which he answered a question of ErdÅs), a simple,
explicit formula for the asymptotic behavior of the sequence
$\{a_{n}\}$ is given. Finally, we describe the order structure of the
$K_{0}$-groups of an important class of Furstenberg crossed products,
obtaining their complete Elliott invariant using classification
results of H. Lin and N. C. Phillips.
Keywords:$K$-theory, transformation group $C^*$-algebra, Furstenberg transformation, Anzai transformation, minimal homeomorphism, positive cone, minimal homeomorphism Categories:19K14, 19K99, 46L35, 46L80, , 05A15, 05A16, 05A17, 15A36, 17B10, 17B20, 37B05, 54H20 |
7. CJM 2013 (vol 66 pp. 1143)
Maps Preserving Complementarity of Closed Subspaces of a Hilbert Space Let $\mathcal{H}$ and $\mathcal{K}$ be infinite-dimensional separable
Hilbert spaces and ${\rm Lat}\,\mathcal{H}$ the lattice of all closed subspaces oh $\mathcal{H}$.
We describe the general form of pairs of bijective maps $\phi , \psi :
{\rm Lat}\,\mathcal{H} \to {\rm Lat}\,\mathcal{K}$ having the property that for every pair
$U,V \in {\rm Lat}\,\mathcal{H}$ we have $\mathcal{H} = U \oplus V \iff \mathcal{K} = \phi (U) \oplus \psi (V)$. Then we reformulate this theorem as a description
of bijective image equality and kernel equality preserving maps acting on bounded linear idempotent operators. Several known
structural results for maps on idempotents are easy consequences.
Keywords:Hilbert space, lattice of closed subspaces, complemented subspaces, adjacent subspaces, idempotents Categories:46B20, 47B49 |
8. CJM 2013 (vol 65 pp. 1005)
Uniformly Continuous Functionals and M-Weakly Amenable Groups Let $G$ be a locally compact group. Let $A_{M}(G)$ ($A_{0}(G)$)denote
the closure of $A(G)$, the Fourier algebra of $G$ in the space of
bounded (completely bounded) multipliers of $A(G)$.
We call a locally compact group M-weakly amenable if
$A_M(G)$
has a
bounded approximate identity. We will show that when $G$ is M-weakly
amenable, the algebras $A_{M}(G)$ and $A_{0}(G)$ have
properties that are characteristic of the Fourier algebra of an
amenable group. Along the way we show that the sets of tolopolically
invariant means associated with these algebras have the same
cardinality as those of the Fourier algebra.
Keywords:Fourier algebra, multipliers, weakly amenable, uniformly continuous functionals Categories:43A07, 43A22, 46J10, 47L25 |
9. CJM 2013 (vol 66 pp. 596)
The Ordered $K$-theory of a Full Extension Let $\mathfrak{A}$ be a $C^{*}$-algebra with real rank zero which has
the stable weak cancellation property. Let $\mathfrak{I}$ be an ideal
of $\mathfrak{A}$ such that $\mathfrak{I}$ is stable and satisfies the
corona factorization property. We prove that
$
0 \to \mathfrak{I} \to \mathfrak{A} \to \mathfrak{A} / \mathfrak{I} \to 0
$
is a full extension if and only if the extension is stenotic and
$K$-lexicographic. {As an immediate application, we extend the
classification result for graph $C^*$-algebras obtained by Tomforde
and the first named author to the general non-unital case. In
combination with recent results by Katsura, Tomforde, West and the
first author, our result may also be used to give a purely
$K$-theoretical description of when an essential extension of two
simple and stable graph $C^*$-algebras is again a graph
$C^*$-algebra.}
Keywords:classification, extensions, graph algebras Categories:46L80, 46L35, 46L05 |
10. CJM 2013 (vol 66 pp. 373)
Uniform Convexity and Bishop-Phelps-BollobÃ¡s Property A new characterization of the uniform convexity of
Banach space is obtained in the sense of Bishop-Phelps-BollobÃ¡s
theorem. It is also proved that the couple of Banach spaces $(X,Y)$
has the bishop-phelps-bollobÃ¡s property for every banach space $y$
when $X$ is uniformly convex. As a corollary, we show that the
Bishop-Phelps-BollobÃ¡s theorem holds for bilinear forms on
$\ell_p\times \ell_q$ ($1\lt p, q\lt \infty$).
Keywords:Bishop-Phelps-BollobÃ¡s property, Bishop-Phelps-BollobÃ¡s theorem, norm attaining, uniformly convex Categories:46B20, 46B22 |
11. CJM 2013 (vol 66 pp. 721)
On Whitney-type Characterization of Approximate Differentiability on Metric Measure Spaces We study approximately differentiable functions on metric measure spaces admitting a Cheeger differentiable structure. The main result is a Whitney-type characterization of approximately differentiable functions in this setting.
As an application, we prove a Stepanov-type theorem and consider approximate differentiability of Sobolev, $BV$ and maximal functions.
Keywords:approximate differentiability, metric space, strong measurable differentiable structure, Whitney theorem Categories:26B05, 28A15, 28A75, 46E35 |
12. CJM 2013 (vol 65 pp. 783)
Generalised Triple Homomorphisms and Derivations We introduce generalised triple homomorphism between Jordan Banach
triple systems as a concept which extends the notion of generalised homomorphism between
Banach algebras given by K. Jarosz and B.E. Johnson in 1985 and 1987, respectively.
We prove that every generalised triple homomorphism between JB$^*$-triples
is automatically continuous. When particularised to C$^*$-algebras, we rediscover
one of the main theorems established by B.E. Johnson. We shall also consider generalised
triple derivations from a Jordan Banach triple $E$ into a Jordan Banach triple $E$-module,
proving that every generalised triple derivation from a JB$^*$-triple $E$ into itself or into $E^*$
is automatically continuous.
Keywords:generalised homomorphism, generalised triple homomorphism, generalised triple derivation, Banach algebra, Jordan Banach triple, C$^*$-algebra, JB$^*$-triple Categories:46L05, 46L70, 47B48, 17C65, 46K70, 46L40, 47B47, 47B49 |
13. CJM 2013 (vol 65 pp. 1073)
From Quantum Groups to Groups In this paper we use the recent developments in the
representation theory of locally compact quantum groups,
to assign, to each locally compact
quantum group $\mathbb{G}$, a locally compact group $\tilde {\mathbb{G}}$ which
is the quantum version of point-masses, and is an
invariant for the latter. We show that ``quantum point-masses"
can be identified with several other locally compact groups that can be
naturally assigned to the quantum group $\mathbb{G}$.
This assignment preserves compactness as well as
discreteness (hence also finiteness), and for large classes of quantum
groups, amenability. We calculate this invariant for some of the most
well-known examples of
non-classical quantum groups.
Also, we show that several structural properties of $\mathbb{G}$ are encoded
by $\tilde {\mathbb{G}}$: the latter, despite being a simpler object, can carry very
important information about $\mathbb{G}$.
Keywords:locally compact quantum group, locally compact group, von Neumann algebra Category:46L89 |
14. CJM 2013 (vol 66 pp. 641)
Heat Kernels and Green Functions on Metric Measure Spaces We prove that, in a setting of local Dirichlet forms on metric measure
spaces, a two-sided sub-Gaussian estimate of the heat kernel is equivalent
to the conjunction of the volume doubling propety, the elliptic Harnack
inequality and a certain estimate of the capacity between concentric balls.
The main technical tool is the equivalence between the capacity estimate and
the estimate of a mean exit time in a ball, that uses two-sided estimates of
a Green function in a ball.
Keywords:Dirichlet form, heat kernel, Green function, capacity Categories:35K08, 28A80, 31B05, 35J08, 46E35, 47D07 |
15. CJM 2012 (vol 65 pp. 1236)
Higher Connectedness Properties of Support Points and Functionals of Convex Sets We prove that the set of all support points of a nonempty closed convex bounded set $C$ in a real infinite-dimensional Banach space $X$ is $\mathrm{AR(}\sigma$-$\mathrm{compact)}$ and contractible. Under suitable conditions, similar results are proved also for the set of all support functionals of $C$ and for the domain, the graph and the range of the subdifferential map of a proper convex l.s.c. function on $X$.
Keywords:convex set, support point, support functional, absolute retract, Leray-Schauder continuation principle Categories:46A55, 46B99, 52A07 |
16. CJM 2012 (vol 65 pp. 863)
Cumulants of the $q$-semicircular Law, Tutte Polynomials, and Heaps The $q$-semicircular distribution is a probability law that
interpolates between the Gaussian law and the semicircular law. There
is a combinatorial interpretation of its moments in terms of matchings
where $q$ follows the number of crossings, whereas for the free
cumulants one has to restrict the enumeration to connected matchings.
The purpose of this article is to describe combinatorial properties of
the classical cumulants. We show that like the free cumulants, they
are obtained by an enumeration of connected matchings, the weight
being now an evaluation of the Tutte polynomial of a so-called
crossing graph. The case $q=0$ of these cumulants was studied by
Lassalle using symmetric functions and hypergeometric series. We show
that the underlying combinatorics is explained through the theory of
heaps, which is Viennot's geometric interpretation of the
Cartier-Foata monoid. This method also gives a general formula for
the cumulants in terms of free cumulants.
Keywords:moments, cumulants, matchings, Tutte polynomials, heaps Categories:05A18, 05C31, 46L54 |
17. CJM 2012 (vol 65 pp. 989)
Automatic Continuity of Homomorphisms in Non-associative Banach Algebras We introduce the concept of a rare element in a non-associative normed
algebra and show that the existence of such element is the only obstruction
to continuity of a surjective homomorphism from a non-associative Banach
algebra to a unital normed algebra with simple completion. Unital
associative algebras do not admit any rare element and hence automatic
continuity holds.
Keywords:automatic continuity, non-associative algebra, spectrum, rare operator, rare element Categories:46H40, 46H70 |
18. CJM 2012 (vol 66 pp. 102)
Continuity of convolution of test functions on Lie groups For a Lie group $G$, we show that the map
$C^\infty_c(G)\times C^\infty_c(G)\to C^\infty_c(G)$,
$(\gamma,\eta)\mapsto \gamma*\eta$
taking a pair of
test functions to their convolution is continuous if and only if $G$ is $\sigma$-compact.
More generally, consider $r,s,t
\in \mathbb{N}_0\cup\{\infty\}$ with $t\leq r+s$, locally convex spaces $E_1$, $E_2$
and a continuous bilinear map $b\colon E_1\times E_2\to F$
to a complete locally convex space $F$.
Let $\beta\colon C^r_c(G,E_1)\times C^s_c(G,E_2)\to C^t_c(G,F)$,
$(\gamma,\eta)\mapsto \gamma *_b\eta$ be the associated convolution map.
The main result is a characterization of those $(G,r,s,t,b)$
for which $\beta$ is continuous.
Convolution
of compactly supported continuous functions on a locally compact group
is also discussed, as well as convolution of compactly supported $L^1$-functions
and convolution of compactly supported Radon measures.
Keywords:Lie group, locally compact group, smooth function, compact support, test function, second countability, countable basis, sigma-compactness, convolution, continuity, seminorm, product estimates Categories:22E30, 46F05, 22D15, 42A85, 43A10, 43A15, 46A03, 46A13, 46E25 |
19. CJM 2012 (vol 65 pp. 1043)
Convolution of Trace Class Operators over Locally Compact Quantum Groups We study locally compact quantum groups $\mathbb{G}$ through the
convolution algebras $L_1(\mathbb{G})$ and $(T(L_2(\mathbb{G})),
\triangleright)$. We prove that the reduced quantum group
$C^*$-algebra $C_0(\mathbb{G})$ can be recovered from the convolution
$\triangleright$ by showing that the right $T(L_2(\mathbb{G}))$-module
$\langle K(L_2(\mathbb{G}) \triangleright T(L_2(\mathbb{G}))\rangle$ is
equal to $C_0(\mathbb{G})$. On the other hand, we show that the left
$T(L_2(\mathbb{G}))$-module $\langle T(L_2(\mathbb{G}))\triangleright
K(L_2(\mathbb{G})\rangle$ is isomorphic to the reduced crossed product
$C_0(\widehat{\mathbb{G}}) \,_r\!\ltimes C_0(\mathbb{G})$, and hence is
a much larger $C^*$-subalgebra of $B(L_2(\mathbb{G}))$.
We establish a natural isomorphism between the completely bounded
right multiplier algebras of $L_1(\mathbb{G})$ and
$(T(L_2(\mathbb{G})), \triangleright)$, and settle two invariance
problems associated with the representation theorem of
Junge-Neufang-Ruan (2009). We characterize regularity and discreteness
of the quantum group $\mathbb{G}$ in terms of continuity properties of
the convolution $\triangleright$ on $T(L_2(\mathbb{G}))$. We prove
that if $\mathbb{G}$ is semi-regular, then the space
$\langle T(L_2(\mathbb{G}))\triangleright B(L_2(\mathbb{G}))\rangle$ of right
$\mathbb{G}$-continuous operators on $L_2(\mathbb{G})$, which was
introduced by Bekka (1990) for $L_{\infty}(G)$, is a unital $C^*$-subalgebra
of $B(L_2(\mathbb{G}))$. In the representation framework formulated by
Neufang-Ruan-Spronk (2008) and Junge-Neufang-Ruan, we show that the
dual properties of compactness and discreteness can be characterized
simultaneously via automatic normality of quantum group bimodule maps
on $B(L_2(\mathbb{G}))$. We also characterize some commutation
relations of completely bounded multipliers of $(T(L_2(\mathbb{G})),
\triangleright)$ over $B(L_2(\mathbb{G}))$.
Keywords:locally compact quantum groups and associated Banach algebras Categories:22D15, 43A30, 46H05 |
20. CJM 2012 (vol 65 pp. 481)
Correction of Proofs in "Purely Infinite Simple $C^*$-algebras Arising from Free Product Constructions'' and a Subsequent Paper |
Correction of Proofs in "Purely Infinite Simple $C^*$-algebras Arising from Free Product Constructions'' and a Subsequent Paper The proofs of Theorem 2.2 of K. J. Dykema and M. RÃ¸rdam, Purely infinite simple
$C^*$-algebras arising from free product constructions}, Canad. J.
Math. 50 (1998), 323--341 and
of Theorem 3.1 of K. J. Dykema, Purely infinite simple
$C^*$-algebras arising from free product constructions, II, Math.
Scand. 90 (2002), 73--86 are corrected.
Keywords:C*-algebras, purely infinite Category:46L05 |
21. CJM 2012 (vol 65 pp. 559)
Extreme Version of Projectivity for Normed Modules Over Sequence Algebras We define and study the so-called extreme version of the notion of a
projective normed module. The relevant definition takes into account
the exact value of the norm of the module in question, in contrast
with the standard known definition that is formulated in terms of norm
topology.
After the discussion of the case where our normed algebra $A$ is just
$\mathbb{C}$, we concentrate on the case of the next degree of complication,
where $A$ is a sequence algebra, satisfying some natural conditions.
The main results give a full characterization of extremely projective
objects within the subcategory of the category of non-degenerate
normed $A$--modules, consisting of the so-called homogeneous modules.
We consider two cases, `non-complete' and `complete', and the
respective answers turn out to be essentially different.
In particular, all Banach non-degenerate homogeneous modules,
consisting of sequences, are extremely projective within the category
of Banach non-degenerate homogeneous modules. However, neither of
them, provided it is infinite-dimensional, is extremely projective
within the category of all normed non-degenerate homogeneous modules.
On the other hand, submodules of these modules, consisting of finite
sequences, are extremely projective within the latter category.
Keywords:extremely projective module, sequence algebra, homogeneous module Category:46H25 |
22. CJM 2012 (vol 65 pp. 331)
Lushness, Numerical Index 1 and the Daugavet Property in Rearrangement Invariant Spaces We show that for spaces with 1-unconditional bases
lushness, the alternative Daugavet property and numerical
index 1 are equivalent. In the class of rearrangement
invariant (r.i.) sequence spaces the only examples of spaces with
these properties are $c_0$, $\ell_1$ and $\ell_\infty$.
The only lush r.i. separable function space on $[0,1]$ is $L_1[0,1]$;
the same space is the only r.i. separable function space on $[0,1]$
with the Daugavet property over the reals.
Keywords:lush space, numerical index, Daugavet property, KÃ¶the space, rearrangement invariant space Categories:46B04, 46E30 |
23. CJM 2012 (vol 65 pp. 52)
C$^*$-algebras Nearly Contained in Type $\mathrm{I}$ Algebras In this paper we consider near inclusions $A\subseteq_\gamma B$ of C$^*$-algebras. We show that if $B$ is a separable type $\mathrm{I}$ C$^*$-algebra and $A$ satisfies Kadison's similarity problem, then $A$ is also type $\mathrm{I}$ and use this to obtain an embedding of $A$ into $B$.
Keywords:C$^*$-algebras, near inclusions, perturbations, type I C$^*$-algebras, similarity problem Category:46L05 |
24. CJM 2012 (vol 65 pp. 485)
Filters in C*-Algebras In this paper we analyze states on C*-algebras and
their relationship to filter-like structures of projections and
positive elements in the unit ball. After developing the basic theory
we use this to investigate the Kadison-Singer conjecture, proving its
equivalence to an apparently quite weak paving conjecture and the
existence of unique maximal centred extensions of projections coming
from ultrafilters on the natural numbers. We then prove that Reid's
positive answer to this for q-points in fact also holds for rapid
p-points, and that maximal centred filters are obtained in this case.
We then show that consistently such maximal centred filters do not
exist at all meaning that, for every pure state on the Calkin algebra,
there exists a pair of projections on which the state is 1, even
though the state is bounded strictly below 1 for projections below
this pair. Lastly we investigate towers, using cardinal invariant
equalities to construct towers on the natural numbers that do and do
not remain towers when canonically embedded into the Calkin algebra.
Finally we show that consistently all towers on the natural numbers
remain towers under this embedding.
Keywords:C*-algebras, states, Kadinson-Singer conjecture, ultrafilters, towers Categories:46L03, 03E35 |
25. CJM 2011 (vol 64 pp. 755)
Homotopy Classification of Projections in the Corona Algebra of a Non-simple $C^*$-algebra We study projections in the corona algebra of $C(X)\otimes K$, where K
is the $C^*$-algebra of compact operators on a separable infinite
dimensional Hilbert space and $X=[0,1],[0,\infty),(-\infty,\infty)$,
or $[0,1]/\{ 0,1 \}$. Using BDF's essential codimension, we determine
conditions for a projection in the corona algebra to be liftable to a
projection in the multiplier algebra. We also determine the
conditions for two projections to be equal in $K_0$, Murray-von
Neumann equivalent, unitarily equivalent, or homotopic. In light of
these characterizations, we construct examples showing that the
equivalence notions above are all distinct.
Keywords:essential codimension, continuous field of Hilbert spaces, Corona algebra Categories:46L05, 46L80 |