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Having in view the study of a version of Gel'fand-Neumark duality adapted to the context of Alain Connes' spectral triples, in this very preliminary review, we first present a description of the relevant categories of geometrical spaces,…

Let $f:X\to X$ be a non-isomorphic (i.e., $\text{deg } f>1$) surjective endomorphism of a smooth projective threefold $X$. We prove that any birational minimal model program becomes $f$-equivariant after iteration, provided that $f$ is…

Let $X$ be a (real or complex) rearrangement-in\-va\-riant function space on $\Om$ (where $\Om = [0,1]$ or $\Om \subseteq \bbN$) whose norm is not proportional to the $L_2$-norm. Let $H$ be a separable Hilbert space. We characterize…

This paper studies coarse compactifications and their boundary. We introduce two alternative descriptions to Roe's original definition of coarse compactification. One approach uses bounded functions on $X$ that can be extended to the…

A $\mathbb{C}^{*}$-action on a projective variety $X$ is said to be of Euler type at a nonsingular fixed point $x$ if the isotropy action of $\mathbb{C}^{*}$ on $T_{x}X$ is by scalar multiplication. In this paper, it's proven that a smooth…

Johnson and Zippin recently showed that if $X$ is a weak^*-closed subspace of $\ell_1$ and T:X-> C(K) is any bounded operator then T can extended to a bounded operator $\tilde T:\ell_1\to C(K).$ We give a converse result: if X is a subspace…

Let $G/P$ be a generalized flag variety, where $G$ is a complex semisimple connected Lie group and $P\subset G$ a parabolic subgroup. Let also $X\subset G/P$ be a Schubert variety. We consider the canonical embedding of $X$ into a…

In a recent paper it was shown that all the Hilbert space formulas for quantum probabilities can be realized as functions of geometric properties of the associated projective space, but those functions were expressed using the structures of…

We show that a reflexive subspace of the predual of a von Neumann algebra embeds into a noncommutative Lp space for some p>1. This is a noncommutative version of Rosenthal's result for commutative Lp spaces. Similarly for 1 < q < 2, an…

Let X be a simply connected space and k a commutative ring. Goodwillie, Burghelea and Fiedorowiscz proved that the Hochschild cohomology of the singular chains on the pointed loop space HH^{*}S_*(\Omega X) is isomorphic to the free loop…

Using twisted and formal-local methods, we prove that every separated algebraic space which is the (open/flat) pushout of affine schemes has enough Azumaya algebras. As a corollary we show that, under mild hypothesis, every cohomological…

We prove that the predual, $M_*$, of a JBW$^*$-triple $M$ is a 1-Plichko space (i.e. it admits a countably 1-norming Markushevich basis or, equivalently, it has a commutative 1-projectional skeleton), and obtain a natural description of the…

We define a notion of nonassociative $\mathrm{L}^p$-space associated to a $\mathrm{JBW}^*$-algebra (Jordan von Neumann algebra) equipped with a normal faithful state $\varphi$. In the particular case of $\mathrm{JW}^*$-algebras underlying…

We prove that for every JBW$^*$-triple $E$ of rank $>1$, the symmetric part of its predual reduces to zero. Consequently, the predual of every infinite dimensional von Neumann algebra $A$ satisfies the linear biholomorphic property, that…

According to Horrocks (1966), a vector bundle E on the projective n-space extends stably to the projective N-space, N>n, if there exists a vector bundle on the larger space whose restriction to the smaller one is isomorphic to E plus a…

A subalgebra $\mathcal{A}$ of $\mathbb{M}_n(\mathbb{C})$ is said to be projection compressible if $P\mathcal{A}P$ is an algebra for all orthogonal projections $P\in\mathbb{M}_n(\mathbb{C})$. Likewise, $\mathcal{A}$ is said to be idempotent…

For a smooth projective complex variety $X$, we prove that there exists a birational morphism $X\times X\to Y$ to a projective variety $Y$ contracting the diagonal $\Delta_X\subset X\times X$ to a point if and only if $X$ has maximal…

Motivated by Ball, Li, Timotin and Trent's Schur-Agler class version of commutant lifting theorem, we introduce a class, denoted by $\mathcal{P}_n(\mathcal{H})$, of $n$-tuples of commuting contractions on a Hilbert space $\mathcal{H}$. We…

In 1997, Bauschke, Borwein, and Lewis have stated a trichotomy theorem that characterizes when the convergence of the method of alternating projections can be arbitrarily slow. However, there are two errors in their proof of this theorem.…

We prove a conjecture of Brevig, Ortega-Cerd\`a, Seip and Zhao about contractive inequalities between Dirichlet and Hardy spaces and discuss its consequent connection with the Riesz projection.