Relative position between a pair of spin model subfactors, I

In this article, we investigate relative position between a pair of spin model subfactors of the hyperfinite type $II_1$ factor $R$ arising from two complex Hadamard matrices of order $2$ as well as order $4$. More precisely, we characterize when the two subfactors are equal, compute the Pimsner-Popa probabilistic constant and the Connes-St{\o}rmer relative entropy between them. To the best of our knowledge, this article is the first instance in the literature that the exact value of the Connes-St{\o}rmer relative entropy for a pair of (non-trivial) subfactors has been obtained. We construct en route a family of potentially new subfactors of $R$. All these subfactors are irreducible with Jones index $4n,n\in\mathbb{N}$. As a corollary, a rigidity of the angle between the two subfactors is established. Finally, as a pleasant application of the relative entropy, we characterize when the pair of spin model subfactors form a commuting square.Comment: 103 pages, 10 figures, substantial improvement, new results added, final versio

Fourier theoretic inequalities for inclusion of simple C*-algebras

This paper originates from a naive attempt to establish various non-commutative Fourier theoretic inequalities for an inclusion of simple C*-algebras equipped with a conditional expectation of index-finite type. In this setting, we discuss the Hausdorff-Young inequality and Young's inequality. As a consequence, we prove the Hirschman-Beckner uncertainty principle and Donoho-Stark uncertainty principle. Our results generalize some of the results of Jiang, Liu and Wu [Noncommutative uncertainty principle, J. Funct. Anal., 270(1): 264--311, 2016].Comment: 27 pages, 2 figures, minor revisions, few references added, To appear in New York J. Mat

A few remarks on intermediate subalgebras of an inclusion of C∗C^*-algebras

In this short note, we prove that the angle between intermediate $C^*$-subalgebras of an inclusion of simple $C^*$-algebras with finite Watatani index is stable. The notion of angle is instrumental in providing a bound for the cardinality of the lattice of intermediate subalgebras for an irreducible inclusion of simple $C^*$-algebras. We improve the existing upper bound for the cardinality of this set.Comment: Minor modification, few typos correcte

Temperature-Dependent Gentamicin Resistance of Francisella tularensis is Mediated by Uptake Modulation

Gentamicin (Gm) is an aminoglycoside commonly used to treat bacterial infections such as tularemia - the disease caused by Francisella tularensis. In addition to being pathogenic, F. tularensis is found in environmental niches such as soil where this bacterium likely encounters Gm producers (Micromonospora sp.). Here we show that F. tularensis exhibits increased resistance to Gm at ambient temperature (26°C) compared to mammalian body temperature (37°C). To evaluate whether F. tularensis was less permeable to Gm at 26°C, a fluorescent marker [Texas Red (Tr)] was conjugated with Gm, yielding Tr-Gm. Bacteria incubated at 26°C showed reduced fluorescence compared to those at 37°C when exposed to Tr-Gm suggesting that uptake of Gm was reduced at 26°C. Unconjugated Gm competitively inhibited uptake of Tr-Gm, demonstrating that this fluorescent compound was taken up similarly to unconjugated Gm. Lysates of F. tularensis bacteria incubated with Gm at 37°C inhibited the growth of Escherichia coli significantly more than lysates from bacteria incubated at 26°C, further indicating reduced uptake at this lower temperature. Other facultative pathogens (Listeria monocytogenes and Klebsiella pneumoniae) exhibited increased resistance to Gm at 26°C suggesting that the results generated using F. tularensis may be generalizable to diverse bacteria. Regulation of the uptake of antibiotics provides a mechanism by which facultative pathogens survive alongside antibiotic-producing microbes in nature