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Cocycles and positive functionals in higher cohomology

Antonio López Neumann, Piotr W. Nowak

Abstract

We establish and explore the correspondence between positive functionals and cocycles in higher unitary cohomology. We generalize the classical cocycle version of the Gelfand-Naimark-Segal construction to higher degrees and apply it to characterize vanishing of higher unitary cohomology as an extension property for positive functionals. We also prove that under mild conditions the algebraic spectral gap for the one sided Laplacian characterizes cohomological vanishing instead of reducedness of unitary cohomology

Cocycles and positive functionals in higher cohomology

Abstract

We establish and explore the correspondence between positive functionals and cocycles in higher unitary cohomology. We generalize the classical cocycle version of the Gelfand-Naimark-Segal construction to higher degrees and apply it to characterize vanishing of higher unitary cohomology as an extension property for positive functionals. We also prove that under mild conditions the algebraic spectral gap for the one sided Laplacian characterizes cohomological vanishing instead of reducedness of unitary cohomology
Paper Structure (10 sections, 19 theorems, 97 equations)

This paper contains 10 sections, 19 theorems, 97 equations.

Table of Contents

  1. Matrices over group algebras
  2. Group cohomology and finiteness properties
  3. Split conjugation spaces
  4. The higher dimensional GNS construction
  5. Reducedness of cohomology and sums of squares
  6. Higher Property $H_T$
  7. Remarks and questions
  8. Matricial homology and ideals
  9. Algebraic characterizations of reducedness
  10. Split conjugation spaces and matricial homology

Key Result

Theorem A

(Theorem theorem: Higher Ozawa equation is equivalent to Higher (T)) Let $\Gamma$ be a group of type $F$ and $n \geq (\mathrm{cdim}_\mathbb{Q}(\Gamma)+1) /2$. The following conditions are equivalent:

Theorems & Definitions (39)

  • Theorem A
  • Theorem B
  • Proposition 1.1
  • Proposition 1.2: GNS construction
  • proof
  • Remark 1
  • Lemma 1.3
  • proof
  • Lemma 2.1
  • proof
  • ...and 29 more