[Paper Review] Bivariant $K$-theory and the Weyl algebra

Joachim Cuntz|ArXiv.org|Jan 22, 2004

Algebraic structures and combinatorial models5 references7 citations

TL;DR

This paper introduces a new bivariant $K$-theory, $kk^{\rm alg}$, for locally convex algebras, including the Weyl algebra, using noncommutative stable homotopy and classifying maps for extensions. The key result is that the Weyl algebra $W$ is isomorphic to $\mathbb{C}$ in $kk^{\rm alg}$, implying its invariants are determined by the coefficient ring $R = kk^{\rm alg}(\mathbb{C},\mathbb{C})$, with applications to matrix algebras and $C^*$-algebraic isomorphisms.

ABSTRACT

We introduce a new version $kk^{ m alg}$ of bivariant $K$-theory that is defined on the category of all locally convex algebras. A motivating example is the Weyl algebra $W$, i.e. the algebra generated by two elements satisfying the Heisenberg commutation relation, with the fine locally convex topology. We determine its $kk^{ m alg}$-invariants using a natural extension for $W$. Using similar methods the $kk^{ m alg}$-invariants can be determined for many other algebras of similar type.

Motivation & Objective

• To develop a bivariant $K$-theory $kk^{\rm alg}$ for the category of all locally convex algebras, extending beyond submultiplicative seminorms.
• To provide a framework for computing $kk^{\rm alg}$-invariants of algebras like the Weyl algebra, which cannot be topologized via submultiplicative seminorms.
• To establish a Chern-Connes character from $kk^{\rm alg}(A,B)$ to periodic cyclic homology $HP_*(A,B)$, linking $K$-theoretic and homological invariants.
• To demonstrate that $kk^{\rm alg}(W, W) \cong R$ and that $W$ is isomorphic to $\mathbb{C}$ in $kk^{\rm alg}$, using a natural extension of the Weyl algebra.
• To compare $kk^{\rm alg}$ with the classical $kk$-theory, showing that while $kk^{\rm alg}$ loses control over coefficients, it retains strong computational power for certain algebras.

Proposed method

• Define $kk^{\rm alg}$ as a noncommutative stable homotopy theory via inductive limits over suspensions of both variables, replacing inverse Bott maps.
• Use classifying maps for extensions of higher length to construct the theory, inspired by Thom's thesis and triangulated category formalism.
• Construct a natural extension $0 \to I \to W' \to W \to 0$ for the Weyl algebra $W$, which is linearly split and admits completion to $0 \to \bar{I} \to \bar{W}' \to W \to 0$ with $\bar{I}$ an $m$-algebra.
• Apply long exact sequences in $kk$, $kk^{\rm alg}$, and $HP_*$ to the completed extension, comparing invariants via a commutative diagram.
• Use the Chern-Connes character to relate $kk^{\rm alg}$ to periodic cyclic homology, noting that $R = kk^{\rm alg}(\mathbb{C},\mathbb{C})$ is a unital ring mapping to $\mathbb{C}$, with $R = \mathbb{Z}$ shown later.
• Leverage the fact that $kk^\text{alg}_*(\mathbb{C}, \bar{W}') = HP_*(\mathbb{C}, \bar{W}') = 0$ and $kk_1(\mathbb{C}, \bar{I}) = \mathbb{Z}$ to deduce $kk_0(\mathbb{C}, W) \cong \mathbb{Z}$.

Experimental results

Research questions

• RQ1 Can a bivariant $K$-theory be defined for all locally convex algebras, including those not admitting submultiplicative seminorms?
• RQ2 What is the $kk^{\rm alg}$-invariant of the Weyl algebra $W$, and how does it compare to $\mathbb{C}$?
• RQ3 How does the $kk^{\rm alg}$-theory relate to periodic cyclic homology and classical $kk$-theory?
• RQ4 Can $kk^{\rm alg}$ detect non-isomorphisms between matrix algebras over $W$, such as $M_k(W) \not\cong M_l(W)$ for $k \neq l$?
• RQ5 What diffotopy invariants can be derived from $kk^{\rm alg}$ for $C^*$-algebraic tensor products like $W \hat{\otimes} \mathcal{O}_n$?

Key findings

• The Weyl algebra $W$ is isomorphic to $\mathbb{C}$ in the category $kk^{\rm alg}$, meaning $kk^{\rm alg}(A, W) \cong kk^{\rm alg}(A, \mathbb{C})$ for all $A$.
• The coefficient ring $R = kk^{\rm alg}(\mathbb{C}, \mathbb{C})$ is isomorphic to $\mathbb{Z}$, as shown in a later update, ensuring $kk^{\rm alg}(A,B)$ is a $\mathbb{Z}$-module.
• For $D = \mathbb{C}$, the long exact sequence in $kk^\text{alg}$ shows $kk^\text{alg}_0(\mathbb{C}, W) \cong \mathbb{Z}$, with the $\mathbb{Z}$-summand arising from the unit map $\mathbb{C} \to W$.
• The map $kk_0(\mathbb{C}, \mathbb{C}) \to kk_0(\mathbb{C}, W)$ is injective and maps $1 \in \mathbb{Z}$ to the image of the unit in $W$, confirming the $\mathbb{Z}$-summand.
• It is shown that $M_k(W) \not\cong M_l(W)$ for $k \neq l$, and that any $\varphi: M_k(\mathbb{C}) \to \mathcal{K} \hat{\otimes} W$ can only map the unit to a unit in $M_l(W)$ if $l$ is a multiple of $k$, via diffotopy invariants.
• For $C^*$-algebras $\mathcal{O}_n$, $W \hat{\otimes} \mathcal{O}_n \cong W \hat{\otimes} \mathcal{O}_m$ only if $n = m$, as $kk_1(\mathbb{C}, \bar{I} \hat{\otimes} \mathcal{O}_n) = \mathbb{Z}$ implies $n = m$.