[Paper Review] On Coarse Spectral Geometry in Even Dimension

Robert Yuncken|arXiv (Cornell University)|Mar 9, 2010

Advanced Operator Algebra Research5 references3 citations

TL;DR

This paper establishes an even-dimensional analogue of Roe's coarse spectral geometry by introducing a symmetric Roe algebra $ C^*|\mathbb{R}|^\sigma $, where $ \sigma $ is the reflection involution on $ \mathbb{R} $. It proves $ K_0(C^*|\mathbb{R}|^\sigma) \cong \mathbb{Z} $, $ K_1 = 0 $, and shows that the induced map on $ K $-theory recovers the index pairing between $ K $-homology and $ K $-theory, offering a geometrically more natural framework than previous balancing constructions.

ABSTRACT

Let $σ$ be the involution of the Roe algebra $\Roe{\RR}$ which is induced from the reflection $\RR o\RR; x\mapsto -x$. A graded Fredholm module over a separable $C^*$-algebra $A$ gives rise to a homomorphism $ ildeρ:A o\Roe{\RR}^σ$ to the fixed-point subalgebra. We use this observation to give an even-dimensional analogue of a result of Roe. Namely, we show that the $K$-theory of this symmetric Roe algebra is $K_0(\Roe{\RR}^σ)\cong\ZZ$, $K_1(\Roe{\RR})=0$, and that the induced map $ ildeρ_*:K_0(A) o \ZZ$ on $K$-theory gives the index pairing of $K$-homology with $K$-theory.

Motivation & Objective

• To provide a geometrically more natural framework for coarse spectral geometry in even dimensions, avoiding the computationally heavy 'balancing' procedure required in Luu's approach.
• To define a symmetric Roe algebra $ C^*|\mathbb{R}|^\sigma $ via the fixed-point subalgebra under the reflection involution $ \sigma $ on $ \mathbb{R} $.
• To show that the $ K $-theory of this symmetric Roe algebra is $ K_0 \cong \mathbb{Z} $, $ K_1 = 0 $, matching the $ K $-theory of $ \mathbb{Z} $ in degree 0.
• To prove that the induced map $ \tilde{\rho}_*: K_0(A) \to K_0(C^*|\mathbb{R}|^\sigma) \cong \mathbb{Z} $ computes the index pairing between $ K $-homology and $ K $-theory.

Proposed method

• Construct a $ \sigma $-invariant $ C^* $-algebra $ C^*|\mathbb{R}|^\sigma $ as the fixed-point algebra of the Roe algebra under the reflection $ x \mapsto -x $.
• Use a graded Fredholm module $ (H, \rho, D) $ over a $ C^* $-algebra $ A $ to induce a $ \ast $-homomorphism $ \tilde{\rho}: A \to C^*|\mathbb{R}|^\sigma $.
• Apply the six-term exact sequence in $ K $-theory to the extension $ 0 \to \mathcal{K}(\mathcal{H})^\sigma \to C^*|\mathbb{R}|^\sigma \to C^*|Y_+|/\mathcal{K}(\mathcal{H}_+) \to 0 $, leveraging known $ K $-theory of $ C^*|Y_+| $.
• Use the trace argument on projections to show that the boundary map $ \partial: K_1(C^*|\mathbb{R}|^\sigma / \mathcal{K}(\mathcal{H})^\sigma) \to K_0(\mathcal{K}(\mathcal{H})^\sigma) $ has image $ \{(n,n)\} \subset \mathbb{Z} \oplus \mathbb{Z} $.
• Apply the six-term exact sequence to deduce $ K_0(C^*|\mathbb{R}|^\sigma) \cong \mathbb{Z} $, $ K_1(C^*|\mathbb{R}|^\sigma) = 0 $.
• Show that the index pairing $ (\theta, [p]) $ equals $ \tilde{\rho}_*([p]) $ by replacing $ \tilde{\rho}(p) $ with a finite-rank projection and computing the difference in dimensions of even and odd spectral subspaces.

Experimental results

Research questions

• RQ1 Can a symmetric Roe algebra be constructed in even dimensions that avoids the need for balancing Fredholm modules?
• RQ2 What is the $ K $-theory of the fixed-point subalgebra $ C^*|\mathbb{R}|^\sigma $ under the reflection involution?
• RQ3 Does the induced map $ \tilde{\rho}_*: K_0(A) \to K_0(C^*|\mathbb{R}|^\sigma) $ recover the index pairing in even-dimensional coarse spectral geometry?
• RQ4 How does the symmetric Roe algebra relate to Luu's $ KC^n $-picture of $ KK $-theory in even degrees?

Key findings

• The $ K $-theory of the symmetric Roe algebra $ C^*|\mathbb{R}|^\sigma $ is $ K_0 \cong \mathbb{Z} $ and $ K_1 = 0 $, as computed via the six-term exact sequence in $ K $-theory.
• The induced map $ \tilde{\rho}_*: K_0(A) \to K_0(C^*|\mathbb{R}|^\sigma) \cong \mathbb{Z} $ agrees with the index pairing between the $ K $-homology class of a graded Fredholm module and $ K $-theory classes in $ K_0(A) $.
• The index pairing $ (\theta, [p]) $ is equal to the difference in dimensions of the even and odd spectral subspaces of the projection $ \tilde{\rho}(p) $, which equals $ \tilde{\rho}_*([p]) $.
• The construction avoids the need for 'balancing' Fredholm modules, which otherwise require infinite direct sums and obscure spectral relationships.
• The symmetric Roe algebra provides a geometrically more natural framework for even-dimensional coarse spectral geometry than previous approaches.
• The result supports the existence of a reformulation of $ KK^0(A, \mathbb{C}) $ via $ KC^0_\sigma(A, \mathbb{C}) $, defined using $ \sigma $-equivariant morphisms into $ C^*|\mathbb{R}|^\sigma $.