arXiv · 0705.4681
We compare the random group model of Gromov and the model of generic groups of Arzhantseva and Ol'shanskii.
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Ilya Kapovich, Paul Schupp. 2010-02-08. On group-theoretic models of randomness and genericity. https://arxiv.org/abs/0705.4681
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We show that the quantum uniformity norms introduced by Bu, Gu, and Jaffe are the pullbacks of the matrix-valued uniformity norms of Gowers and Hatami under the map $S\mapsto(a\mapsto W_aSW_a^*)$, which we call the Weyl-orbit embedding. This identification yields the Gowers--Cauchy--Schwarz inequality and the triangle inequality in all orders, answering a question of Bu, Gu, and Jaffe, and characterizes the Clifford hierarchy in terms of unitary-valued Leibman polynomial Weyl orbit maps. We then study the structure of the Weyl-orbit embedding. Its range is the fixed-point algebra of a natural isometric action, and the associated norm-one projection is a trace-preserving conditional expectation. We also establish the corresponding dual pullback identity. We further prove Hanner's inequality for both uniformity norms and derive uniform convexity with explicit lower bounds for their moduli of convexity. This yields a variational characterization of the quantum uniformity norms, with the Weyl-orbit map as the unique norm-minimizing representative of each fiber, and an isometric quotient realization of the quantum uniformity spaces. Finally, we decompose the ambient matrix-valued uniformity space canonically into isometric $1$-complemented twisted copies of the quantum uniformity space.
An $\mathfrak{N}$-semigroup is an archimedean, idempotent-free, cancellative and commutative semigroup. For example, the additive semigroup $\mathbb{R}_{>0}$ of strictly positive real numbers is a $\mathfrak{N}$-semigroup. Hewitt-Zuckerman showed that every $\mathfrak{N}$-semigroup admits a semigroup homomorphism to $\mathbb{R}_{>0}$. We show that every uniquely divisible $\mathfrak{N}$-semigroup $S$ admits enough semigroup homomorphisms to $\mathbb{R}_{>0}$ to detect divisibility in $S$. %They were studied by Tamura and Kobayashi. The main ingredient is a version of the Eidelheit and Kakutani hyperplane separation theorem for vector spaces over $\mathbb{Q}$. A counterexample due to Ravsky explains why this result does not generalise to the larger class of $\mathfrak{N}$-semigroups.
This paper is the fourth in a series devoted to the finite basis problem for $4$-element additively idempotent semirings. These algebras are divided into five types according to their additive reducts; we study the largest of these types, namely those whose additive reducts are chains. Up to isomorphism, there are $386$ such algebras, denoted by $S_{(4, k)}$, $481 \leq k \leq 866$. We show that all but three of them, namely $S_{(4,545)}$, $S_{(4,634)}$, and $S_{(4,710)}$, are finitely based. This completes the classification, with respect to the finite basis property, of all $4$-element additively idempotent semirings with chain additive reducts.