[Paper Review] Localization techniques in circle-equivariant KK-theory

Heath Emerson|arXiv (Cornell University)|Apr 17, 2010

Advanced Operator Algebra Research20 references3 citations

TL;DR

This paper develops localization techniques in circle-equivariant KK-theory, showing that certain $τ$-C*-algebras—including Cuntz-Krieger algebras—are not $K\sp{\mathbb{T}}$-equivalent to commutative $T$-C*-algebras despite being in the bootstrap category. It establishes an equivariant Lefschetz fixed-point formula in $K\sp{\mathbb{T}}$-theory, linking the $\mathbb{C}[X,X^{-1}]$-valued trace of a $K\sp{\mathbb{T}}$-endomorphism to the Atiyah-Singer $\mathbb{T}$-index of a geometric coincidence cycle.

ABSTRACT

Let T be the circle and A be a T-C*-algebra. Then the T-equivariant K-theory of A is a module over the representation ring of the circle. The latter is a Laurent polynomial ring. Using the support of the module as an invariant, and techniques of Atiyah, Bott and Segal, we deduce that there are examples of T-C*-algebras A not KK^T-equivalent to any commutative T-C*-algebra. This is in contrast to the non-equivariant situation, in which any C*-algebra in the boostrap category is KK-equivalent to a commutative one. Our examples arise from dynamics, and include Cuntz-Krieger algebras with their usual circle actions. Using similar techniques, we also prove an equivariant version of the Lefschetz fixed-point formula. This is a special case of a result with Ralf Meyer that applies to general compact connected groups. The Lefschetz theorem equates the module trace of the module map of the T-equivariant K-theory of a smooth compact manifold induced by an equivariant self-correspondence of the manifold, with an appropriate Kasparov product; the Kasparov product is the T-equivariant index of the Dirac operator on a suitable `coincidence manifold' of the correspondence. Finally, we prove several results related to localization and the Kunneth and universal coefficient theorems, and give an essentially complete description of the T-equivariant K-theory of compact spaces, by combining localization techniques of Atiyah and Segal and results of Paul Baum and Alain Connes for equivariant K-theory of finite group actions.

Motivation & Objective

• To identify obstructions in $K\sp{\mathbb{T}}$-theory that prevent certain $T$-C*-algebras from being $K\sp{\mathbb{T}}$-equivalent to commutative ones.
• To extend classical Lefschetz fixed-point theory to the equivariant setting using $K\sp{\mathbb{T}}$-theory and geometric correspondences.
• To demonstrate that the support of $K\sp{\mathbb{T}}(O_A)$ for Cuntz-Krieger algebras $O_A$ can be arbitrary algebraic integers, violating the support constraints of commutative $T$-C*-algebras.
• To establish a precise formula equating the $\mathbb{C}[X,X^{-1}]$-valued trace of a $K\sp{\mathbb{T}}$-endomorphism with the Atiyah-Singer $\mathbb{T}$-index of a geometric cycle.
• To provide a computational framework for the equivariant Lefschetz number in terms of fixed-point data and representation theory on smooth $T$-manifolds.

Proposed method

• Using Atiyah-Bott-Segal localization techniques to analyze the support of $K\sp{\mathbb{T}}(A)$ as a module over $\mathrm{Rep}(\mathbb{T}) \cong \mathbb{C}[X,X^{-1}]$, identifying obstructions to commutativity.
• Constructing a sheaf of modules over $\mathbb{C}^*$ from the $\mathbb{C}[X,X^{-1}]$-module $K\sp{\mathbb{T}}(A)$, and using Zariski localization to probe its support.
• Defining the $\mathbb{C}[X,X^{-1}]$-valued trace of a $K\sp{\mathbb{T}}$-endomorphism via compression to the free part of the module.
• Formulating an equivariant Lefschetz theorem by associating a $\mathbb{T}$-equivariant Baum-Douglas cycle to a geometric correspondence and computing its $\mathbb{T}$-index.
• Applying the Atiyah-Singer index theorem to relate the $\mathbb{T}$-index of the coincidence cycle to the $\mathbb{C}[X,X^{-1}]$-trace of the endomorphism.
• Computing the trace explicitly in examples via the minimal polynomial of the endomorphism operator on $K\sp{\mathbb{T}}(X)$, such as for $\mathbb{CP}^1$ with $\mathbb{T}$-action.

Experimental results

Research questions

• RQ1 Can $K\sp{\mathbb{T}}$-theory detect non-commutative obstructions to $K\sp{\mathbb{T}}$-equivalence in the bootstrap category?
• RQ2 What is the support of $K\sp{\mathbb{T}}(O_A)$ for a Cuntz-Krieger algebra $O_A$ under its standard $\mathbb{T}$-action, and how does it differ from that of commutative $\mathbb{T}$-C*-algebras?
• RQ3 How can the classical Lefschetz fixed-point formula be generalized to $\mathbb{T}$-equivariant geometric correspondences in $KK^\mathbb{T}$-theory?
• RQ4 What is the precise relationship between the $\mathbb{C}[X,X^{-1}]$-valued trace of a $K\sp{\mathbb{T}}$-endomorphism and the Atiyah-Singer $\mathbb{T}$-index of a geometric cycle?
• RQ5 How do the eigenvalues of the matrix $A$ in $O_A$ relate to the spectral data of $K\sp{\mathbb{T}}(O_A)$ as a $\mathbb{C}[X,X^{-1}]$-module?

Key findings

• The support of $K\sp{\mathbb{T}}(C_0(X))$ for a locally compact $\mathbb{T}$-space $X$ is always contained in the unit circle or is all of $\mathbb{C}^*$, a constraint violated by Cuntz-Krieger algebras.
• For Cuntz-Krieger algebras $O_A$, the support of $K\sp{\mathbb{T}}(O_A)$ is the set of nonzero eigenvalues of the $0$-$1$ matrix $A$, which can be arbitrary algebraic integers, proving they are not $K\sp{\mathbb{T}}$-equivalent to any commutative $\mathbb{T}$-C*-algebra.
• The equivariant Lefschetz formula states that $\mathrm{trace}_{\mathbb{C}[X,X^{-1}]}(f_*) = \mathrm{ind}_{\mathbb{T}}(\mathrm{Lef}(f))$, equating a homological invariant to a geometric $\mathbb{T}$-index.
• For a smooth $\mathbb{T}$-manifold $X$ with isolated fixed points, the trace $\mathrm{trace}_{\mathbb{C}[X,X^{-1}]}(\lambda_\xi)$ equals the sum of $\xi_P$ over fixed points $P$, where $\xi_P \in \mathbb{C}[X,X^{-1}]$ is the restriction of $\xi$ to $P$.
• In the case of $\mathbb{CP}^1$ with the standard $\mathbb{T}$-action, $\mathrm{trace}_{\mathbb{C}[X,X^{-1}]}(\lambda_{[H]}^k) = X^k + X^{-k}$, computed via the minimal polynomial of the endomorphism on $\mathrm{K}^0_{\mathbb{T}}(\mathbb{CP}^1)$.
• The formula $e_X([E]) = \chi(F)\sum_\lambda \dim_{\mathbb{C}}(E_\lambda) f_\lambda$ holds for $\mathbb{T}$-invariant bundles on a fixed-point manifold $F$, with $f_\lambda$ the character of the $\mathbb{T}$-action on the eigenspace $E_\lambda$.