We extend some classical results of Cowling and Meda to the noncommutative setting. Let (Tt)t>0 be a symmetric contraction semigroup on a noncommutative space Lp(M), and let the functions φ and ψ be regularly related. We prove that the semigroup (Tt)t>0 is φ-ultracontractive, i.e. ‖Ttx‖∞ ≤ Cφ(t)‖x‖1 for all x ∈ L1(M) and t > 0 if and only if its infinitesimal generator L has the Sobolev embeddi...

Notes taken by Stuart White during the lectures and not proof read. Errors will abound. 1. Nuclearity: Approximation and perturbation Recall: Definition 1.1. A C∗-algebra. Definition 1.2. A cp map Theorem 1.3. Stinespring’s theorem. Every cp map is a compression of a ∗-homomorphism. Definition 1.4. φ : A → B cp has order zero if a ⊥ b =⇒ φ(a) ⊥ φ(b). 1 Theorem 1.5. (i) φ : A→ B has order zero i...

We prove the existence and uniqueness of fixed point belonging to the zero-set of a given function. The results are established in the setting of metric spaces and partial metric spaces. Our approach combines the recent notions of (F,φ)contraction and Z-contraction. The main result allows to deduce, as a particular case, some of the most known results in the literature. An example supports the ...

This is the content of the talk given at the conference “Effective Aspects of Complex Hyperbolic Varieties”, Aver Wrac’h, France, September 10-14, 07. We present two methods of constructing low degree Kobayashi hyperbolic hypersurfaces in Pn: • The projection method • The deformation method The talk is based on joint works of the speaker with B. Shiffman and C. Ciliberto. 1. DIGEST on KOBAYASHI...

1. Geometric Derivative of Clear Raindrop Continuous mapping As shown in Fig. 1 (a), the appearance of each raindrop is a contracted image of the background, as if it is taken from a catadioptric camera. The numeric values indicated in Fig. 1 (c) are the contraction ratios between the original image and the image inside the raindrops, calculated from the black and white patterns. The contractio...

Let X be a smooth complex projective variety with nef ⋀2TX and dim⁡X≥3. We prove that, up to finite étale cover X˜→X, the Albanese map X˜→Alb(X˜) is locally trivial fibration whose fibers are isomorphic Fano F ⋀2TF. As bi-product, we see that TX or variety. Moreover study contraction of KX-negative extremal ray φ:X→Y. In particular, blow-up space at point if φ birational type. also morphism fib...

Recent results have shown that any closed operator A commuting with the backwards shift S∗ restricted to K u := H 2 ⊖ uH, where u is an inner function, can be realized as a Nevanlinna function of S∗ u := S ∗| K2 u , A = φ(S∗ u), where φ belongs to a certain class of Nevanlinna functions which depend on u. In this paper this result is generalized to show that given any contraction T of class C0(...