We characterize mappings$S$_{$i$}and$T$_{$i$}, not necessarily linear, from sets $\mathcal {J}_{i}$ ,$i$=1,2, onto multiplicative subsets of function algebras, subject to the following conditions on the peripheral spectra of their products:$σ$_{$π$}($S$_{1}($a$)$S$_{2}($b$))⊂$σ$_{$π$}($T$_{1}($a$)$T$_{2}($b$)) and$σ$_{$π$}($S$_{1}($a$)$S$_{2}($b$))∩$σ$_{$π$}($T$_{1}($a$)$T$_{2}($b$))≠∅, $a\in \mathcal {J}_{1}$ , $b\in \mathcal {J}_{2}$ . As a direct consequence we obtain a large number of previous results about mappings subject to various spectral conditions.

For any unital separable simple infinite-dimensional nuclear$C$^{∗}-algebra with finitely many extremal traces, we prove that $$ \mathcal{Z} $$ -absorption, strict comparison and property (SI) are equivalent. We also show that any unital separable simple nuclear$C$^{∗}-algebra with tracial rank zero is approximately divisible, and hence is $$ \mathcal{Z} $$ -absorbing.

We calculate the local Fourier transforms for connections on the formal punctured disk, reproducing the results of J. Fang and C. Sabbah using a different method. Our method is similar to Fang’s, but more direct.

We present a 3D topological picture-language for quantum information. Our approach combines charged excitations carried by strings, with topological properties that arise from embedding the strings in the interior of a 3D manifold with boundary. A quon is a composite that acts as a particle. Specifically, a quon is a hemisphere containing a neutral pair of open strings with opposite charge. We interpret multiquons and their transformations in a natural way. We obtain a type of relation, a string–genus “joint relation,” involving both a string and the 3D manifold. We use the joint relation to obtain a topological interpretation of the C* Hopf algebra relations, which are widely used in tensor networks. We obtain a 3D representation of the controlled NOT (CNOT) gate that is considerably simpler than earlier work, and a 3D topological protocol for teleportation.

In this paper, we prove the DonohoStark uncertainty principle for locally compact quantum groups and characterize the minimizer which are bi-shifts of group-like projections. We also prove the HirschmanBeckner uncertainty principle for compact quantum groups and discrete quantum groups. Furthermore, we show Hardy's uncertainty principle for locally compact quantum groups in terms of bi-shifts of group-like projections.