Induction in stages for crossed products of C*-algebras by maximal coactions

Let B be a C*-algebra with a maximal coaction of a locally compact group G, and let N and H be closed normal subgroups of G with N contained in H. We show that the process Ind_(G/H)^G which uses Mansfield's bimodule to induce representations of the crossed product of B by G from those of the restricted crossed product of B by (G/H) is equivalent to the two-stage induction process: Ind_(G/N)^G composed with Ind_(G/H)^(G/N). The proof involves a calculus of symmetric imprimitivity bimodules which relates the bimodule tensor product to the fibred product of the underlying spaces.Comment: 38 pages, LaTeX, uses Xy-pic; significant reorganization of previous version; short section on regularity of induced representations adde

Fell bundles and imprimitivity theorems

Our goal in this paper and two sequels is to apply the Yamagami-Muhly-Williams equivalence theorem for Fell bundles over groupoids to recover and extend all known imprimitivity theorems involving groups. Here we extend Raeburn's symmetric imprimitivity theorem, and also, in an appendix, we develop a number of tools for the theory of Fell bundles that have not previously appeared in the literature.Comment: minor change

The Rieffel Correspondence for Equivalent Fell Bundles

We establish a generalized Rieffel correspondence for ideals in equivalent Fell bundles.Comment: 20 pages, fixed typ

Rieffel deformation via crossed products

We start from Rieffel data (A,f,X) where A is a C*-algebra, X is an action of an abelian group H on A and f is a 2-cocycle on the dual group. Using Landstad theory of crossed product we get a deformed C*-algebra A(f). In the case of H being the n-th Cartesian product of the real numbers we obtain a very simple proof of invariance of K-groups under the deformation. In the general case we also get a very simple proof that nuclearity is preserved under the deformation. We show how our approach leads to quantum groups and investigate their duality. The general theory is illustrated by an example of the deformation of SL(2,C). A description of it, in terms of noncommutative coordinates is given.Comment: 39 page

Two to Five Truths in Non-Negative Matrix Factorization

In this paper, we explore the role of matrix scaling on a matrix of counts when building a topic model using non-negative matrix factorization. We present a scaling inspired by the normalized Laplacian (NL) for graphs that can greatly improve the quality of a non-negative matrix factorization. The results parallel those in the spectral graph clustering work of \cite{Priebe:2019}, where the authors proved adjacency spectral embedding (ASE) spectral clustering was more likely to discover core-periphery partitions and Laplacian Spectral Embedding (LSE) was more likely to discover affinity partitions. In text analysis non-negative matrix factorization (NMF) is typically used on a matrix of co-occurrence ``contexts'' and ``terms" counts. The matrix scaling inspired by LSE gives significant improvement for text topic models in a variety of datasets. We illustrate the dramatic difference a matrix scalings in NMF can greatly improve the quality of a topic model on three datasets where human annotation is available. Using the adjusted Rand index (ARI), a measure cluster similarity we see an increase of 50\% for Twitter data and over 200\% for a newsgroup dataset versus using counts, which is the analogue of ASE. For clean data, such as those from the Document Understanding Conference, NL gives over 40\% improvement over ASE. We conclude with some analysis of this phenomenon and some connections of this scaling with other matrix scaling methods