Journal article
On the K-theory of some C∗-algebras of Toeplitz and singular integral operators
Journal of functional analysis, Vol.110(1), pp.161-225
1992
DOI: 10.1016/0022-1236(92)90046-L
Abstract
We study the
C
∗-algebras
GJ
(X,
R
) are
I
(X,
R
) of singular integral operators and Toeplitz operators (respectively) associated with a strictly ergodic flow (X,
R
). We show that the commutator ideals of these algebras,
CGJ
(X,
R
) and
CL
(X,
R
), are simple and are closely related to the transformation group
C
∗-algebra, C
∗(X,
R
). We calculate the
K-theory of
GJ
(X,
R
),
L
(X,
R
) and their commutator ideals. The main results of this calculation, Corollary 3.8.4 and Theorem 4.1.1, assert that C
∗(X,
R
) is contained in
CGJ
(X,
R
) and if j denotes the inclusion map, then
j
∗: K
0(C
∗(X,R)) → K
0(CGJ(X,R))
is an order isomorphism and there is a short exact sequence
0 → K
1(C
∗(R))
→
i
i
K
1C
∗(R))
→
j
i
K
1(CGJ(X,R)) → 0
where i is the canonical imbedding of C
∗(
R
) into C
∗(X
R
). We show also that, up to a change of scale, there is a unique trace on each of the commutator ideals. The key ingredient of our analysis is Theorem 3.1.1 which asserts a bijective correspondence between Silov representations of the algebra of analytic functions on the flow and
C
∗-representations of
GJ
(X,
R
) and
L
(X,
R
). This simultaneously generalizes Coburn's theorem on the uniqueness of the
C
∗-algebra generated by an isometry and Douglas's theorem on the uniqueness of the
C
∗-algebra generated by an isometric representation of a dense subsemigroup of
R
+.
Details
- Title: Subtitle
- On the K-theory of some C∗-algebras of Toeplitz and singular integral operators
- Creators
- Paul S Muhly - Department of Mathematics, University of Iowa, Iowa City, Iowa 52242 USAIan F Putnam - Department of Mathematics, Statistics, and Computing Science, Dalhousie University, Halifax, Nova Scotia B3H 3J5, CanadaJingbo Xia - Department of Mathematics, State University of New York at Buffalo, Buffalo, New York 14214 USA
- Resource Type
- Journal article
- Publication Details
- Journal of functional analysis, Vol.110(1), pp.161-225
- DOI
- 10.1016/0022-1236(92)90046-L
- ISSN
- 0022-1236
- eISSN
- 1096-0783
- Publisher
- Elsevier Inc
- Language
- English
- Date published
- 1992
- Academic Unit
- Statistics and Actuarial Science; Mathematics
- Record Identifier
- 9984083216802771
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