[Paper Review] Convex polytopes and the index of Wiener-Hopf operators

Alexander Alldridge|arXiv (Cornell University)|Feb 18, 2011

Advanced Operator Algebra Research6 references3 citations

TL;DR

This paper establishes a deep connection between the K-theory of Wiener-Hopf C*-algebras associated with cones over convex polytopes and the cellular homology of the polytope itself. By constructing a filtration of the algebra via symbol maps and identifying the resulting E¹ term of the Atiyah-Hirzebruch spectral sequence with the cellular complex of the polytope, the authors prove that the algebra is KK-contractible and that its quotient modulo compact operators is KK-equivalent to the algebra of continuous functions vanishing at infinity on the real line.

ABSTRACT

We study the C$^*$-algebra of Wiener-Hopf operators $A_Ω$ on a cone $Ω$ with polyhedral base $P$. As is known, a sequence of symbol maps may be defined, and their kernels give a filtration by ideals of $A_Ω$, with liminary subquotients. One may define $K$-group valued 'index maps' between the subquotients. These form the $E^1$ term of the Atiyah-Hirzebruch type spectral sequence induced by the filtration. We show that this $E^1$ term may, as a complex, be identified with the cellular complex of $P$, considered as CW complex by taking convex faces as cells. It follows that $A_Ω$ is $KK$-contractible, and that $A_Ω/\mathbb K$ and $S$ are $KK$-equivalent. Moreover, the isomorphism class of $A_Ω$ is a complete invariant for the combinatorial type of $P$.

Motivation & Objective

• To understand the K-theory and KK-classification of Wiener-Hopf C*-algebras associated with cones over convex polytopes.
• To analyze the structure of these algebras via a filtration induced by symbol maps and their subquotients.
• To determine whether the isomorphism class of the algebra classifies the combinatorial type of the underlying polytope.
• To establish the KK-equivalence of the quotient algebra modulo compact operators and the algebra C₀(ℝ).

Proposed method

• Define a filtration of the Wiener-Hopf C*-algebra $A_{ ext{Ω}}$ using symbol maps, with kernels forming ideals and subquotients that are liminary.
• Identify the E¹ term of the Atiyah-Hirzebruch spectral sequence associated with this filtration as the cellular complex of the polytope $P$ when viewed as a CW complex with convex faces as cells.
• Use the fact that the polytope $P$ is contractible to show that its cellular complex is exact, implying the spectral sequence collapses at $E^2 = 0$.
• Apply the spectral sequence to compute $K$-theory, showing $K_*(A_{ ext{Ω}}) = 0$ and $K_*(A_{ ext{Ω}}/ ext{K}) = K_*( ext{C}_0( ext{ℝ}))$.
• Leverage the universal coefficient theorem (UCT) and bootstrap category membership to conclude $KK$-contractibility and $KK$-equivalence.
• Establish that the isomorphism class of $A_{ ext{Ω}}$ is a complete invariant for the combinatorial type of $P$ via the $f$-vector and index maps.

Experimental results

Research questions

• RQ1 How does the $K$-theory of the Wiener-Hopf algebra $A_{ ext{Ω}}$ over a cone with polyhedral base $P$ relate to the topology of $P$?
• RQ2 Can the $E^1$ term of the spectral sequence induced by the symbol filtration be identified with the cellular complex of $P$?
• RQ3 Is the algebra $A_{ ext{Ω}}$ $KK$-contractible, and what does this imply for its $KK$-class?
• RQ4 Is the isomorphism class of $A_{ ext{Ω}}$ a complete invariant for the combinatorial type of the polytope $P$?
• RQ5 What is the $KK$-class of $A_{ ext{Ω}}/ ext{K}$, and how does it relate to $C_0( ext{ℝ})$?

Key findings

• The $E^1$ term of the Atiyah-Hirzebruch spectral sequence for $A_{ ext{Ω}}$ is isomorphic to the cellular complex of the polytope $P$ when $P$ is viewed as a CW complex with convex faces as cells.
• The spectral sequence collapses at $E^2 = 0$, implying $K_*(A_{ ext{Ω}}) = 0$ and $K_*(A_{ ext{Ω}}/ ext{K}) = K_*( ext{C}_0( ext{ℝ}))$.
• The C*-algebra $A_{ ext{Ω}}$ is $KK$-contractible, meaning it is $KK$-equivalent to the zero algebra.
• The quotient algebra $A_{ ext{Ω}}/ ext{K}$ is $KK$-equivalent to $S = ext{C}_0( ext{ℝ})$.
• The isomorphism class of $A_{ ext{Ω}}$ is a complete invariant for the combinatorial type of the polytope $P$, as determined by the $f$-vector and index maps.
• The index map $K_1(A_{ ext{Ω}}/ ext{K}) o ext{Z}$ is an isomorphism, confirming that the Fredholm index fully classifies the $K_1$-group.