Jordan weak amenability and orthogonal forms on JB*-algebras

We prove the existence of a linear isometric correspondence between the Banach space of all symmetric orthogonal forms on a JB$^*$-algebra $\mathcal{J}$ and the Banach space of all purely Jordan generalized derivations from $\mathcal{J}$ into $\mathcal{J}^*$. We also establish the existence of a similar linear isometric correspondence between the Banach spaces of all anti-symmetric orthogonal forms on $\mathcal{J}$, and of all Lie Jordan derivations from $\mathcal{J}$ into $\mathcal{J}^*$

The Microbiota-Gut-Brain Axis Heart Shunt Part I: The French Paradox, Heart Disease and the Microbiota

It has been well established that a vegetarian and polyphenol-rich diet, including fruits, vegetables, teas, juices, wine, indigestible fiber and whole grains, provide health-promoting phytochemicals and phytonutrients that are beneficial for the heart and brain. What is not well-characterized is the affect these foods have when co-metabolized within our dynamic gut and its colonizing flora. The concept of a heart shunt within the microbiota-gut-brain axis underscores the close association between brain and heart health and the so-called French paradox offers clues for understanding neurodegenerative and cerebrovascular diseases. Moreover, oxidation-redox reactions and redox properties of so-called brain and heart-protective foods are underappreciated as to their enhanced or deleterious mechanisms of action. Focusing on prodromal stages, and common mechanisms underlying heart, cerebrovascular and neurodegenerative diseases, we may unmask and understanding the means to better treat these related diseases