東華大學圖書館 |

語系: 繁體中文

說明(常見問題)

回圖書館首頁

手機版館藏查詢

登入

回首頁

切換: 標籤 | MARC模式 | ISBD

Finitely Summable K-Homology, the In...

Lorenzo, Levi,

FindBook

Google Book

Amazon

博客來

Finitely Summable K-Homology, the Index Pairing, and Cantor Minimal Systems /

紀錄類型:	書目-電子資源 : Monograph/item
正題名/作者:	Finitely Summable K-Homology, the Index Pairing, and Cantor Minimal Systems // Levi Lorenzo.
作者:	Lorenzo, Levi,
面頁冊數:	1 electronic resource (125 pages)
附註:	Source: Dissertations Abstracts International, Volume: 86-11, Section: B.
Contained By:	Dissertations Abstracts International86-11B.
標題:	Mathematics. -
電子資源:	https://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=31936774
ISBN:	9798315701958

Finitely Summable K-Homology, the Index Pairing, and Cantor Minimal Systems /
Lorenzo, Levi,

Finitely Summable K-Homology, the Index Pairing, and Cantor Minimal Systems /Levi Lorenzo. - 1 electronic resource (125 pages)

Source: Dissertations Abstracts International, Volume: 86-11, Section: B.

K-theory is an invariant that has proven incredibly effective at classifying C∗ -algebras. It has a dual theory, K-homology. K-homology and K-theory admit a bilinear pairing to the integers, called the index pairing. The pairing allows one to learn about K-theory using K-homology and vice-versa. In general, this pairing is difficult to compute. However, if a C∗ -algebra admits a dense subalgebra on which its K-homology is finitely summable, Connes has constructed formulas that are often more computable. Given a smooth manifold, M, of dimension n, its K-homology is p-summable on the algebra of smooth functions on M for all p > n. It is natural to ask whether similar results hold in the noncommutative setting. Emerson-Nica, Goffeng-Mesland, Rave Gerontogiannis and Connes, for example, have done so. We examine the question of index pairings in this thesis. Connes showed that irrational rotation algebras have uniformly finitely summable K-homology, and we seek to build on this. Irrational rotation algebras are examples of crossed product algebras arising from minimal actions on a topological space. In this thesis, we examine the case when the space is a Cantor set and study the K-homology of crossed product algebras arising from minimal actions on it. Cantor minimal systems have been extensively studied in the framework of C∗ -algebras, for example [34]. Using the work of [31], [20], [36], [23] we show uniform summability for a nice class of Cantor minimal systems, called odometers. In lieu of uniform summability for Cantor minimal systems generally, we utilize the sequences of [34] and the AF-algebra results of [36] to show we can use summability to compute any index pairing for C(X)⋊φZ when (X, φ) is a Cantor minimal system. Specifically, we can compute pairings between K0 (C(X) ⋊φZ) and K0(C(X) ⋊φZ) by pulling back cycles to an AF-algebra, on which we can take cycles to be finitely summable, and pushing forward projections to that algebra. For the part of K1(C(X) ⋊φZ) that is detectable by index pairings, we construct finitely summable cycles.

English

ISBN: 9798315701958Subjects--Topical Terms:

515831
Mathematics.
Subjects--Index Terms:

Cantor minimal systems

Finitely Summable K-Homology, the Index Pairing, and Cantor Minimal Systems /
LDR:03570nmm a22004333i 4500 001 2433437
005 20250825062332.5
006 m o d
007 cr|nu||||||||
008 261011s2025 miu||||||m |||||||eng d
020 $a 9798315701958
035 $a (MiAaPQD)AAI31936774
035 $a AAI31936774
040 $a MiAaPQD $b eng $c MiAaPQD $e rda
100 1 $a Lorenzo, Levi, $e author. $3 3823048
245 1 0 $a Finitely Summable K-Homology, the Index Pairing, and Cantor Minimal Systems / $c Levi Lorenzo.
264 1 $a Ann Arbor : $b ProQuest Dissertations & Theses, $c 2025
300 $a 1 electronic resource (125 pages)
336 $a text $b txt $2 rdacontent
337 $a computer $b c $2 rdamedia
338 $a online resource $b cr $2 rdacarrier
500 $a Source: Dissertations Abstracts International, Volume: 86-11, Section: B.
500 $a Advisors: Deeley, Robin Committee members: Goffeng, Magnus; Packer Jesudason, Judith; Farsi, Carla; Gorokhovsky, Alexander.
502 $b Ph.D. $c University of Colorado at Boulder $d 2025.
520 $a K-theory is an invariant that has proven incredibly effective at classifying C∗ -algebras. It has a dual theory, K-homology. K-homology and K-theory admit a bilinear pairing to the integers, called the index pairing. The pairing allows one to learn about K-theory using K-homology and vice-versa. In general, this pairing is difficult to compute. However, if a C∗ -algebra admits a dense subalgebra on which its K-homology is finitely summable, Connes has constructed formulas that are often more computable. Given a smooth manifold, M, of dimension n, its K-homology is p-summable on the algebra of smooth functions on M for all p > n. It is natural to ask whether similar results hold in the noncommutative setting. Emerson-Nica, Goffeng-Mesland, Rave Gerontogiannis and Connes, for example, have done so. We examine the question of index pairings in this thesis. Connes showed that irrational rotation algebras have uniformly finitely summable K-homology, and we seek to build on this. Irrational rotation algebras are examples of crossed product algebras arising from minimal actions on a topological space. In this thesis, we examine the case when the space is a Cantor set and study the K-homology of crossed product algebras arising from minimal actions on it. Cantor minimal systems have been extensively studied in the framework of C∗ -algebras, for example [34]. Using the work of [31], [20], [36], [23] we show uniform summability for a nice class of Cantor minimal systems, called odometers. In lieu of uniform summability for Cantor minimal systems generally, we utilize the sequences of [34] and the AF-algebra results of [36] to show we can use summability to compute any index pairing for C(X)⋊φZ when (X, φ) is a Cantor minimal system. Specifically, we can compute pairings between K0 (C(X) ⋊φZ) and K0(C(X) ⋊φZ) by pulling back cycles to an AF-algebra, on which we can take cycles to be finitely summable, and pushing forward projections to that algebra. For the part of K1(C(X) ⋊φZ) that is detectable by index pairings, we construct finitely summable cycles.
546 $a English
590 $a School code: 0051
650 4 $a Mathematics. $3 515831
650 4 $a Theoretical mathematics. $3 3173530
650 4 $a Applied mathematics. $3 2122814
653 $a Cantor minimal systems
653 $a Index pairing
653 $a K-homology
653 $a Summability
653 $a Finitely summable
690 $a 0405
690 $a 0642
690 $a 0364
710 2 $a University of Colorado at Boulder. $b Mathematics. $e degree granting institution. $3 3823049
720 1 $a Deeley, Robin $e degree supervisor.
773 0 $t Dissertations Abstracts International $g 86-11B.
790 $a 0051
791 $a Ph.D.
792 $a 2025
856 4 0 $u https://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=31936774