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...

Given two graphs G and H, there is a bi-resolving (or bi-covering) graph homomorphism from G to H if and only if their adjacency matrices satisfy certain matrix relations. We investigate the bi-covering extensions of bi-resolving homomorphisms and give several sufficient conditions for a bi-resolving homomorphism to have a bi-covering extension with an irreducible domain. Using these results, w...

We give a complete classification of homomorphisms from the commutator subgroup braid group on n $n$ strands to when is at least 7. In particular, we show that each non-trivial homomorphism extends an automorphism strands. This answers four questions Vladimir Lin. Our main new tool theory totally symmetric sets.

Effect algebras have been introduced in the 1990s in the study of the foundations of quantum mechanics, as part of a quantum-theoretic version of probability theory. This paper is part of that programme and gives a systematic account of Lebesgue integration for [0, 1]-valued functions in terms of effect algebras and effect modules. The starting point is the ‘indicator’ function for a measurable...

A result of Fiz Pontiveros shows that if A is a random subset of ZN where each element is chosen independently with probability N−1/2+o(1), then with high probability every Freiman homomorphism defined on A can be extended to a Freiman homomorphism on the whole of ZN . In this paper we improve the bound to CN−2/3(logN)1/3, which is best possible up to the constant factor.

In this paper, we consider the hidden subgroup problem (HSP) over the class of semi-direct product groups Zn⋊Zq. The definition of the semi-direct product depending on the choice of an homomorphism, we first analyze the different possibilities for this homomorphism in function of n and q. Then, we present a polynomial-time quantum algorithm for the case Zpr ⋊ Zp when p is an odd prime.

We show that certain canonical realizations of the complexes Hom(G,H) and Hom+(G,H) of (partial) graph homomorphisms studied by Babson and Kozlov are in fact instances of the polyhedral Cayley trick. For G a complete graph, we then characterize when a canonical projection of these complexes is itself again a complex, and exhibit several well-known objects that arise as cells or subcomplexes of ...

[2] The notation L(G) is meant to suggest the presence of the inner product on this space of functions. On a general space X with an integral, the iner product is 〈f1, f2〉 = ∫ X f1f 2. [3] The term character has different meanings in different contexts. The simplest sense is a group homomorphism to C×. However, an equally important use is for the trace of a group homomorphism ρ : G → GLn(k) fro...

A graph G is said to be hom-idempotent if there is a homomorphism from G2 to G, and weakly hom-idempotent if for some n ≥ 1 there is a homomorphism from Gn+1 to Gn . We characterize both classes of graphs in terms of a special class of Cayley graphs called normal Cayley graphs. This allows us to construct, for any integer n, a Cayley graph G such that Gn+1 → Gn 6→ Gn−1, answering a question of ...