11.2 K0 and K1 groups and their properties

K0 and K1 groups are fundamental tools in C*-algebra K-theory. They measure equivalence classes of projections and homotopy classes of unitaries, forming abelian groups that capture important structural information about C*-algebras.

These groups are invariant under -isomorphisms and exhibit cyclic relationships through Bott periodicity. For commutative C-algebras, K0 and K1 connect to vector bundles and homotopy classes of functions, providing a bridge between algebra and topology.

K0 and K1 Groups

Fundamental K-Theory Groups

• K0 group measures equivalence classes of projections in a C*-algebra
• K1 group captures homotopy classes of unitaries in a C*-algebra
• Both K0 and K1 form abelian groups under direct sum operations
• K0(A) consists of formal differences [p] - [q] of equivalence classes of projections
• K1(A) comprises homotopy classes of invertible elements in the unitization of A

Properties and Calculations

• K0 and K1 groups are invariant under -isomorphisms between C-algebras
• Bott periodicity theorem establishes a cyclic relationship: K0(SA) ≅ K1(A) and K1(SA) ≅ K0(A)
• For commutative C*-algebras C(X), K0(C(X)) relates to vector bundles over X
• K1(C(X)) connects to homotopy classes of continuous functions from X to GL(n,C)
• Calculations often involve six-term exact sequences and Mayer-Vietoris sequences

Abelian Group Structure

• K0 and K1 groups inherit abelian group properties from projection and unitary operations
• Addition in K0 corresponds to direct sum of projections: [p] + [q] = [p ⊕ q]
• K1 group addition derives from multiplication of unitaries: [u] + [v] = [u ⊕ v]
• Zero elements in K0 and K1 represented by classes of zero projection and identity unitary
• Inverse elements exist due to complementary projections and unitary inverses

Generators of K-Groups

Idempotents and Projections

• Idempotents (elements satisfying e^2 = e) generate K0 groups
• Projections (self-adjoint idempotents) form a special class of idempotents
• Murray-von Neumann equivalence relates projections: p ~ q if p = vv and q = vv for some v
• K0 group elements arise from differences of equivalence classes of projections
• Stabilization allows consideration of projections in matrix algebras over A: M∞(A)

Unitaries and Their Properties

• Unitaries (elements satisfying uu = uu = 1) generate K1 groups
• Homotopy equivalence of unitaries defines K1 group elements
• Path-connectedness of unitary group U(n) impacts K1 group structure
• Determinant function on unitaries relates to K1 group homomorphisms
• Exponential map connects self-adjoint elements to unitaries: exp(ih) for self-adjoint h

Connected Components and Topology

• Connected components of projection and unitary spaces relate to K-theory
• Topology of projection space impacts K0 group structure
• Path-connectedness of unitary group influences K1 group calculations
• Continuous fields of projections and unitaries connect to higher K-groups
• Spectral theory of normal elements relates to connected components in C*-algebras

Classification Tools

Murray-von Neumann Equivalence

• Defines equivalence relation on projections: p ~ q if p = vv and q = vv for some v
• Extends to matrix algebras over A for K0 group calculations
• Generalizes to partial isometries for non-unital C*-algebras
• Relates to trace functions and dimension theory in von Neumann algebras
• Provides foundation for comparison theory in C*-algebras (Cuntz semigroup)

Stable Rank and K-Theory

• Stable rank measures algebraic dimension of C*-algebras
• Low stable rank (≤ 2) simplifies K1 group calculations
• Relates to topological dimension for commutative C*-algebras
• Impacts cancellation properties in K0 groups
• Interacts with real rank in classification of simple C*-algebras

Dimension Functions and States

• Dimension functions assign "sizes" to projections compatible with Murray-von Neumann equivalence
• States on K0 groups provide order structure and connect to traces on C*-algebras
• Range of dimension function impacts structure of K0 group (gap projections)
• Relates to classification of AF algebras via Bratteli diagrams
• Extends to quasitraces and dimension functions on Cuntz semigroup for more general C*-algebras