Entire Cyclic CohomologyEdit
Entire Cyclic Cohomology is a refinement of cyclic cohomology designed to work smoothly in analytic and topological settings that arise in noncommutative geometry. Developed to handle algebras that carry a natural bornology or topological structure, it provides a robust home for the Chern character from K-theory to a cohomology theory that respects analytic growth conditions. In practical terms, Entire Cyclic Cohomology allows one to pair K-theory classes with cohomology classes to extract numerical indices even for infinite-dimensional or noncommutative spaces. See Noncommutative geometry for the broader program in which this theory sits, and Cyclic cohomology for its algebraic precursor.
Historically, cyclic cohomology was introduced as a noncommutative analogue of de Rham cohomology. Entire cyclic cohomology adds analytic control by restricting cochains to grow in an “entire” fashion, which yields pleasant convergence properties when one passes to smooth subalgebras of larger operator algebras. The resulting theory is particularly well-suited to the index problems that arise in mathematical physics and in the study of elliptic operators on noncommutative spaces. For standard references, see the discussions surrounding the JLO cocycle and the formulation of the Chern character in the setting of a spectral triple.
Background
Cyclic cohomology and the (b,B) bicomplex. The classical theory assigns to an algebra A a chain complex whose cohomology captures invariants analogous to those found in differential geometry. In many treatments, the periodic version HP^*(A) is a frequently used target for index-theoretic results.
The analytic motivation. When A carries a topology or a Bornology (a way to talk about controlled growth of multilinear maps), a purely algebraic theory can fail to converge or to behave well under natural constructions. Entire cyclic cohomology enforces growth conditions so that cochains behave nicely under deformations and under passage to smooth subalgebras.
The nerve of the construction. The entire cochain complex consists of sequences of multilinear functionals φ_n on A^{⊗(n+1)} that satisfy a growth bound compatible with the chosen topology on A. The differential (b) and the Connes operator (B) act as in cyclic cohomology, but the domain is restricted to cochains with the required entire behavior. The resulting cohomology is denoted HE^(A) or sometimes ECC^(A) in the literature.
The Chern character and pairings. A central feature is the natural Chern character from K-theory, ch: K_(A) → HE^(A), which mirrors the classical index-theoretic bridge between topology and analysis. In suitable situations, this pairing recovers numerical indices and connects to differential-geometric invariants through the analytic framework.
Construction and basic formalism
The underlying objects. A is typically a bornological algebra or a smooth subalgebra of a larger topological algebra (for example, a dense subalgebra of a C*-algebra that carries a compatible Fréchet or Banach space structure). The choice of bornology governs what counts as an admissible cochain.
The entire cochain complex. A cochain φ is given by a consistent family (φ_n) of multilinear maps φ_n: A^{⊗(n+1)} → ℂ. The growth condition requires that the series ∑ ||φ_n|| z^n / n! converges for all z ∈ ℂ (i.e., φ is entire). The differential and the cyclic symmetries are enforced by the standard (b,B) structure adapted to the entire setting.
Functoriality and products. Entire cyclic cohomology enjoys functorial behavior with respect to suitable homomorphisms and supports product structures that mirror the cup product in ordinary cohomology. This makes it compatible with the standard operations one expects in index theory and in the study of bundles and modules over algebras.
Relation to HP^(A). There is a natural bridge to periodic cyclic cohomology, and in many situations an isomorphism or a natural map exists between HE^(A) and HP^*(A) after appropriate completion or under growth restrictions. The analytic flavor of ECC often makes it preferable when working with noncompact or infinite-dimensional data.
Properties and examples
Morita invariance and stability. ECC respects Morita equivalence under the right hypotheses, which aligns with the philosophy that invariants of algebras should see the same “noncommutative space” up to stabilization. This mirrors similar properties in topological K-theory and in other cyclic theories.
Smooth commutative algebras. For A ≈ C^∞(M) on a smooth manifold M, the entire cohomology recovers familiar topological invariants in a way compatible with the C*-algebraic picture. In favorable cases, the Chern character lands in the ordinary de Rham-type invariants that encode the geometry of M and its vector bundles.
Noncommutative examples. For a noncommutative space described by a dense smooth subalgebra A∞ of a C*-algebra A, ECC provides a robust receptacle for the Chern character of projections and unitaries, allowing one to compute indices of elliptic-type operators represented in a noncommutative framework. The JLO cocycle gives explicit entire cocycles associated with a spectral triple (A, H, D), tying spectral data to cohomological invariants.
Spectral triples and the JLO cocycle. In a spectral triple, the Jaffe-Lesniewski-Osterwalder cocycle yields an explicit, entire cocycle that represents the Chern character in ECC. This connects the analytic data of a Dirac-type operator D to topological invariants of the algebra A.
Applications and significance
Index theory in noncommutative geometry. ECC plays a central role in formulating and proving index theorems for operators naturally living in noncommutative settings. The general theory provides a framework in which numerical indices can be extracted from operator-algebraic data.
Local index formulas. The Connes-Moscovici local index formula expresses the pairing between K-theory and ECC in a way that resembles residues and heat-kernel asymptotics. This mirrors the classical Atiyah-Singer index theorem while extending it to the noncommutative arena.
Connections to physics. Noncommutative geometry has been used to model spaces that arise in quantum physics and to formalize aspects of gauge theory in a purely operator-algebraic language. Entire cyclic cohomology supplies the analytic backbone for invariants that survive the noncommutative generalization of geometric spaces.
Computational considerations. While ECC is technically demanding, its growth controls help ensure that index pairings behave predictably under deformations. This makes the theory attractive for problems where one wants stable invariants under perturbations or smooth deformations of algebras.
Controversies and debates
Depth versus accessibility. Critics point out that the analytic overhead required to work with entire cyclic cohomology can be substantial. From a viewpoint that prioritizes concrete computations and broad accessibility, one may prefer more algebraic or topological approaches when possible. Proponents argue that the analytic framework is precisely what allows ECC to handle smooth subalgebras of operator algebras and to produce robust Chern characters in noncommutative settings.
Scope and comparisons with other theories. There is discussion about when ECC provides advantages over periodic cyclic cohomology or local cyclic homology, and how these theories compare in terms of functoriality, excision, and stability under deformation. In many contexts, ECC and HP are complementary, with ECC offering finer analytic control in settings where growth conditions matter.
Practical impact on physics and geometry. Some critics worry that the level of abstraction required by entire cyclic cohomology may outpace the needs of concrete problems in physics or geometry. Supporters contend that ECC is part of a coherent framework that unifies geometry, analysis, and topology in noncommutative spaces, and that this unification yields results inaccessible to more naive approaches.
The role of smooth subalgebras. The necessity of choosing a particular smooth subalgebra A∞ inside a larger C*-algebra A can be debated. While this choice is often guided by natural geometric or analytic structures, it also introduces a level of non-uniqueness. Advocates emphasize that the essential invariants do not depend on the exact smooth model once the right conditions are in place.