# Journal of Operator Theory

Volume 77, Issue 1, Winter 2017 pp. 191-203.

Continuous fields of projections and orthogonality relations

**Authors**:
Sam Walters

**Author institution:**Department of Mathematics and Statistics, University of Northern B.C., Prince George, B.C. V2N 4Z9, Canada

**Summary: **From a continuous field of Fourier invariant projections of the continuous field of rotation $C^*$-algebras, we obtain a characteristic equation which fully determines the orthogonality of naturally arising projections from the field.
The continuous field turns out to be the support projection of a
noncommutative version of a 2-dimensional Theta function.
Further, we compute the K-theoretical topological invariants of the projection
field. The noncommutative Fourier transform is the canonical order 4
automorphism $\sigma$ of the rotation $C^*$-algebra $A_\theta$ defined by the
relations $\sigma(U) = V^{-1}, \sigma(V) = U$, where $U,V$ are the canonical
unitary generators of $A_\theta$ satisfying $VU = \mathrm e^{2\pi \mathrm i\theta}UV$.

**DOI: **http://dx.doi.org/10.7900/jot.2016mar19.2130

**Keywords: **$C^*$-algebra, automorphism, projection, topological invariant, K-theory, continuous field, Jacobi-Theta function

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