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 investigate birationality of the anti-pluricanonical map φ −m, the rational map defined by the anti-pluricanonical system | − mKX |, on weak Q-Fano 3-folds.

We first consider a cyclic φ-contraction map on a reflexive Banach space X and provide a positive answer to a question raised by Al-Thagafi and Shahzad on the existence of best proximity points for cyclic φ-contraction maps in reflexive Banach spaces in one of their works 2009 . In the second part of the paper, we will discuss the existence of best proximity points in the framework of more gene...

Let X be a locally compact Hausdorff topological space, let V = ( v n , k ) ∈ N $\mathcal {V}=(v_{n,k})_{n,k\in {\mathbb {N}}}$ system of positive continuous functions on and φ self-map X. The composition operators C : f ↦ ∘ $C_\varphi f\mapsto \ f\,\circ\, \varphi$ the weighted function (LF)-spaces {V}C(X)$ 0 {V}_0C(X)$ resp.) (PLB)-spaces A {A}C(X)$ {A}_0C(X)$ are studied. We characterize whe...

Notice that best proximity point results have been studied to find necessaryconditions such that the minimization problemminx∈A∪Bd(x,Tx)has at least one solution, where T is a cyclic mapping defined on A∪B.A point p ∈ A∪B is a best proximity point for T if and only if thatis a solution of the minimization problem (2.1). Let (A,B) be a nonemptypair in a normed...

In 2006, I. Beg and M. Abbas have studied the existence of coincidence and common fixed points for two mappings satisfying a weak contractive condition. Their results were extended in 2008 by A. Azam and M. Shakeel to the case of three mappings. Recently M. Abbas and D. Dorić employed the contractive conditions introduced in [Q. Zhang and Y. Song, Fixed point theory for generalized φ-weak contr...