Toeplitz Extensions in Noncommutative Topology and Mathematical Physics

Arici, Francesca; Mesland, Bram

doi:10.1007/978-3-030-53305-2_1

Cited by 3 publications

(1 citation statement)

References 38 publications

(48 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…If we look at C * (A), the C * -algebra generated by A, we get something similar to the Toeplitz algebra short exact sequence. See [4,8,18] for some recent results on the Toeplitz algebra, and [3,10,15,16] for some applications. The lollipop algebra is replaced by the functions that are continuous on the boundary.…”

Section: Introductionmentioning

confidence: 99%

The Hardy-Weyl algebra

Agler¹,

McCarthy²

2022

Preprint

View full text Add to dashboard Cite

We study the algebra A generated by the Hardy operator H and the operator M x of multiplication by x on L 2 [0, 1]. We call A the Hardy-Weyl algebra. We show that its quotient by the compact operators is isomorphic to the algebra of functions that are continuous on Λ and analytic on the interior of Λ for a planar set Λ = [−1, 0] ∪ D(1, 1), which we call the lollipop. We find a Toeplitz-like short exact sequence for the C *algebra generated by A.We study the operator Z = H −M x , show that its point spectrum is (−1, 0]∪D(1, 1), and that the eigenvalues grow in multiplicity as the points move to 0 from the left.

show abstract

Section: Introductionmentioning

confidence: 99%

The Hardy-Weyl algebra

Agler¹,

McCarthy²

2022

Preprint

View full text Add to dashboard Cite

show abstract

The Atiyah–Singer index theorem

Freed

2021

Bull. Amer. Math. Soc.

View full text Add to dashboard Cite

The Atiyah-Singer index theorem, a landmark achievement of the early 1960s, brings together ideas in analysis, geometry, and topology. We recount some antecedents and motivations, various forms of the theorem, and some of its implications, which extend to the present. Contents 1. Introduction 1 2. Antecedents and motivations from algebraic geometry and topology 2 3. Antecedents in analysis 9 4. The index theorem and proofs 12 5. Variations on the theme 17 6. Heat equation proof 24 7. Geometric invariants of Dirac operators 29 8. Anomalies and index theory 38 About the author 45 References 45

show abstract

The Atiyah-Singer index theorem

Freed

1993

The Atiyah-Patodi-Singer Index Theorem

View full text Add to dashboard Cite

The Atiyah-Singer Index Theorem This is arguably one of the deepest and most beautiful results in modern geometry, and in my view is a must know for any geometer/topologist. It has to do with elliptic partial differential operators on a compact manifold, namely those operators P with the property that dim ker P, dim coker P < ∞. In general these integers are very difficult to compute without some very precise information about P. Remarkably, their difference, called the index of P , is a "soft" quantity in the sense that its determination can be carried out relying only on topological tools. You should compare this with the following elementary situation. Suppose we are given a linear operator A : C m → C n. From this information alone we cannot compute the dimension of its kernel or of its cokernel. We can however compute their difference which, according to the rank-nullity theorem for n×m matrices must be dim ker A−dim coker A = m − n. Michael Atiyah and Isadore Singer have shown in the 1960s that the index of an elliptic operator is determined by certain cohomology classes on the background manifold. These cohomology classes are in turn topological invariants of the vector bundles on which the differential operator acts and the homotopy class of the principal symbol of the operator. Moreover, they proved that in order to understand the index problem for an arbitrary elliptic operator it suffices to understand the index problem for a very special class of first order elliptic operators, namely the Dirac type elliptic operators. Amazingly, most elliptic operators which are relevant in geometry are of Dirac type. The index theorem for these operators contains as special cases a few celebrated results: the Gauss-Bonnet theorem, the Hirzebruch signature theorem, the Riemann-Roch-Hirzebruch theorem. In this course we will be concerned only with the index problem for the Dirac type elliptic operators. We will adopt an analytic approach to the index problem based on the heat equation on a manifold and Ezra Getzler's rescaling trick. Prerequisites: Working knowledge of smooth manifolds, and algebraic topology (especially cohomology). Some familiarity with basic notions of functional analysis: Hilbert spaces, bounded linear operators, L 2-spaces. Syllabus: Part I. Foundations: connections on vector bundles and the Chern-Weil construction, calculus on Riemann manifolds, partial differential operators on manifolds, Dirac operators, [21]. Part II. The statement and some basic applications of the index theorem, [27]. Part III. The proof of the index theorem, [27]. About the class There will be a few homeworks containing routine exercises which involve the basic notions introduced during the course. We will introduce a fairly large number of new objects and ideas and solving these exercises is the only way to gain something form this class and appreciate the rich flavor hidden inside this theorem. Contents Contents §3.1. The statement of the index theorem 93 §3.

show abstract

Toeplitz Extensions in Noncommutative Topology and Mathematical Physics

Cited by 3 publications

References 38 publications

The Hardy-Weyl algebra

The Hardy-Weyl algebra

The Atiyah–Singer index theorem

The Atiyah-Singer index theorem

Contact Info

Product

Resources

About