Glossary

For a side-by-side translation from tiling language to operator-algebraic language, see the Dictionary for tiling theorists.

Bisection normalizer. A partial isometry supported on a compact-open groupoid bisection. Conjugation by it carries the diagonal algebra on its source to the diagonal algebra on its range. This is stronger than merely having finite Fourier support.

Busby--Smith action. A twisted action consisting of automorphisms \(\alpha_g\) and multiplier unitaries \(\nu(g,h)\). The automorphisms compose up to inner conjugation by \(\nu\), and \(\nu\) satisfies a cocycle identity. After the GPS step, the residual magnetic phase has this form on the diagonal corner.

Crossed product. The operator algebra generated by a coefficient algebra and unitaries implementing a group action. Here the coefficient algebra records local patterns and the unitaries record translations. A magnetic multiplier changes the multiplication law of those unitaries.

Exact couple. A diagram of exact maps that generates a spectral sequence. In the curvature comparison, filtered mapping-torus and PV data give exact couples whose compatibility identifies the concrete transfer staircase with Barlak's \(d_2\).

Full corner. A corner \(PAP\) cut out by a projection \(P\) whose ideal is all of \(A\). A full corner is Morita equivalent to \(A\), so it retains the stable module and \(K\)-theory information while permitting a more concrete colored-orbit model.

Holonomy. The mismatch obtained by transporting around a closed loop. The square holonomy in this book is the unitary \(\Omega_{ij}\).

Kasparov product. The composition operation in \(KK\)-theory. The mixed class can be viewed abstractly as a product. Chapter 6 gives the conceptual balanced Hilbert-module construction, and Technical Engine B supplies its complete analytic verification.

\(KK\)-equivalence. An invertible generalized morphism in Kasparov's bivariant \(K\)-theory. It implies isomorphisms on \(K\)-theory and permits homotopy-coherent replacements that need not be literal algebra isomorphisms.

Mapping torus. An algebra of paths whose endpoints are related by an automorphism. Its cellular filtration encodes group-cohomological data used in the PV/Barlak comparison.

Morita equivalence. An equivalence of module categories implemented by an imprimitivity bimodule. Morita-equivalent algebras need not be literally isomorphic, but have canonically related \(K\)-theory and trace pairings.

Pimsner--Voiculescu sequence. The six-term exact sequence computing the \(K\)-theory of a crossed product by \(\mathbb Z\). For the height algebra it identifies \(K_0\) with integer coinvariants and \(K_1\) with integer invariants.

Strict equivariant module. A module equipped with semilinear maps \(V_g\) satisfying the group law exactly. Equality of the underlying \(K_0\)-class with all its translates does not by itself provide such maps.

AF kernel. The index-zero subgroup \(\Gamma_0\) of the topological full group of a minimal Cantor system. Its elements are generated inside finite Kakutani--Rokhlin models.

Block-supported multiplier. A multiplier whose skew form has entries only among the transverse generators. Equivalently, \(\Theta_{ih}=0\) for every transverse \(i\).

Clopen coinvariant class. An element of \(C(\Sigma,\mathbb Z)_{T_h}\), usually represented by an integer-valued locally constant function or a difference of clopen characteristic functions.

Finite PE. Finite pattern-equivariant: determined by a finite clopen partition, or geometrically by a bounded patch radius.

Graded module. A formal difference of two finite projective modules, implemented concretely by a \(\mathbb Z/2\)-graded projection. This permits an arbitrary integer-valued class, not only a positive one.

Height direction. The distinguished generator \(h\) for which \(T_h\) is a minimal Cantor homeomorphism and \(B_h=C(\Sigma)\rtimes_{T_h}\mathbb Z\).

Heisenberg module. Rieffel's projective module obtained from an embedding of a lattice into a phase space over a locally compact abelian group. Its smooth vectors form a Gabor-module model. See the introductory construction.

Mixed class. A \(K_0\)-class combining a coefficient coinvariant class with a transverse noncommutative-torus class.

Pattern frequency. The invariant-measure value of a clopen patch cylinder. It is the coefficient factor \(\mu(f)\) in the mixed trace.

Square defect. The unitary \[ \Omega_{ij}=W_i\alpha_i(W_j) \bigl(W_j\alpha_j(W_i)\bigr)^{\ast} \] measuring failure of two semilinear transports to commute.

Fixed-corner magnetic phase. The cohomology class of the diagonal Busby--Smith cocycle after fixing the corner, commuting symbols, coefficient action, and lift convention. Its vanishing is exactly the diagonal strictification criterion in that corner.

Finite-stable phase hypothesis. The existential assertion that the fixed-corner phase vanishes after some allowed finite stabilization and index change. The book does not promote this hypothesis to an invariant of the underlying \(K_0\)-class.

Transfer function. A locally constant integer-valued function \(h_i\) solving \((\alpha_i-1)g=(\alpha_h-1)h_i\). It encodes a bounded clopen flow.

Transverse directions. The quotient group \(G\cong\mathbb Z^{p-1}\) complementary to the chosen height direction.

Twisted absorption. The construction of a strict finite PE \(G\)-equivariant representative of a coefficient class so that it can be descended together with a transverse twisted module.