Fredholm Index Theory and the Trace Murphy, Gerard J.
Proceedings of the Royal Irish Academy. Section A: Mathematical and Physical Sciences,
12/1994, Volume:
94A, Issue:
2
Journal Article
New, simpler proofs are given of some of the main results concerning the Fredholm index of a Hilbert space operator. The principal tool used is Fedosov's trace formula for the index. It is indicated ...how the approach taken can be used to extend the indea of the Fredholm index to a more general setting.
Spectral inclusion results are obtained for the ordinary and the exponential spectrum, and various properties of the latter spectrum are derived. Applications are given, particularly to Toeplitz ...operators.
We study the concept of co-amenability for a compact quantum group. Several
conditions are derived that are shown to be equivalent to it. Some consequences
of co-amenability that we obtain are ...faithfulness of the Haar integral and
automatic norm-boundedness of positive linear functionals on the quantum
group's Hopf *-algebra (neither of these properties necessarily holds without
co-amenability).
Let $A$ be a Banach algebra containing an element $x$. Topological conditions on the spectrum of $x$ are given which are necessary and sufficient to ensure the continuity of the spectrum or spectral ...radius at $x$.
We study the concept of co-amenability for a compact quantum group. Several conditions are derived that are shown to be equivalent to it. Some consequences of co-amenability that we obtain are ...faithfulness of the Haar integral and automatic norm-boundedness of positive linear functionals on the quantum group's Hopf *-algebra (neither of these properties necessarily holds without co-amenability).
We say that a unital C*-algrebra A has the approximate positive factorization
property (APFP) if every element of A is a norm limit of products of positive
elements of A. (There is also a definition ...for the nonunital case.) T. Quinn
has recently shown that a unital AF algebra has the APFP if and only if it has
no finite dimensional quotients. This paper is a more systematic investigation
of C*-algebras with the APFP. We prove various properties of such algebras. For
example: They have connected invertible group, trivial K_1, and stable rank 1.
In the unital case, the K_0 group separates the tracial states. The APFP passes
to matrix algebras. and if I is an ideal in A such that I and A/I have the
APFP, then so does A. We also give some new examples of C*-algebras with the
APFP, including type II_1 factors and infinite-dimensional simple unital direct
limits with slow dimension growth, real rank zero, and trivial K_1 group. An
infinite- dimensional simple unital direct limit with slow dimension growth and
with the APFP must have real rank zero, but we also give examples of unital
algebras with the APFP which do not have real rank zero.
Our analysis also leads to the introduction of a new concept of rank for a
C*-algebra that may be of interest in the future.
IN an Encyclical dated December, 1925, at the close of the Jubilee Year, His Holiness Pius XI decreed that a special feast should be held each year on the last Sunday of October in honor of Christ ...the immortal King of the Ages. The doctrine of the Kingship of Christ had a long history behind it. Back in the remote beginnings of the Hebrew race God had told Jacob he would make his family a chosen people.
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