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...

The descent algebra of the symmetric group, over a field of non-zero characteristic p, is studied. A homomorphism into the algebra of generalised p-modular characters of the symmetric group is defined. This is then used to determine the radical, and its nilpotency index. It also allows the irreducible representations of the descent algebra to be described.

This encyclopedia article briefly reviews without proofs some of the main results in Poisson reduction. The article recalls most the necessary prerequisites to understand the main results. The Poisson reduction techniques allow the construction of new Poisson structures out of a given one by combination of two operations: restriction to submanifolds that satisfy certain compatibility assumption...

In this article, it is proved that under some conditions every bijective Jordan triple product homomorphism from generalized matrix algebras onto rings is additive. As a corollary, we obtain that every bijective Jordan triple product homomorphism from Mn(A) (A is not necessarily a prime algebra) onto an arbitrary ring R is additive.

However it can just as well be considered a fundamental representation theorem of universal algebra, via the connection between the homfunctor of a category and free algebras for the theory represented by that category. This connection is not generally appreciated outside category theoretic circles, which this section endeavors to correct by presenting the relevant concepts from an algebraic pe...

Let A be a unital operator algebra. Let us assume that every bounded unital homomorphism u: A → B(H) is similar to a contractive one. Let Sim(u) = inf{S S −1 } where the infimum runs over all invertible operators S: H → H such that the " conjugate " homomorphism a → S −1 u(a)S is contractive. Now for all c > 1, let Φ(c) = sup Sim(u) where the supremum runs over all unital homomorphism u: A → B(...

In this note we clarify the relationship between the local and global definitions of dual pairs in Poisson geometry. It turns out that these are not equivalent. For the passage from local to global one needs a connected fiber hypothesis (this is well known), while the converse requires a dimension condition (which appears not to be known). We also provide examples illustrating the necessity of ...

The paper attempts a systematic study of homomorphisms of relational structures. Such structures are modeled as multialgebras (i.e., relation is represented as a set-valued function). The rst, main, result is that, under reasonable restrictions on the form of the deenition of homomorphism, there are exactly nine compositional homomorphisms of multialgebras. Then the comparison of the obtained c...

In this paper, we investigate the generalized Hyers-Ulam-Rassias and the Isac and Rassias-type stability of the conditional of orthogonally ring $*$-$n$-derivation and orthogonally ring $*$-$n$-homomorphism on $C^*$-algebras. As a consequence of this, we prove the hyperstability of orthogonally ring $*$-$n$-derivation and orthogonally ring $*$-$n$-homomorphism on $C^*$-algebras.