We consider contractive homomorphisms of a planar algebra A(Ω) over a finitely connected bounded domain Ω ⊆ C and ask if they are necessarily completely contractive. We show that a homomorphism ρ : A(Ω) → B(H) for which dim(A(Ω)/ ker ρ) = 2 is the direct integral of homomorphisms ρT induced by operators on two dimensional Hilbert spaces via a suitable functional calculus ρT : f 7→ f(T ), f ∈ A(...

this paper continues the investigation of the rst author begun in part one. the hereditary properties of n-homomorphism amenability for banach algebras are investigated and the relations between n-homomorphism amenability of a banach algebra and its ide- als are found. analogous to the character amenability, it is shown that the tensor product of two unital banach algebras is n-homomorphism am...

This is a survey of the author's recent results on the Kadison and Halmos similarity problems and the closely connected notion of " length " of an operator algebra. We start by recalling a well known conjecture formulated by Kadison [Ka] in 1955. Kadison's similarity problem. Let A be a unital C *-algebra and let u: A → B(H) (H Hilbert) be a unital homomorphism (i.e. we have u(1) = 1 and u(ab) ...

Gallucio, Goddyn and Hell proved in 2001 that in any minor-closed class, graphs with large enough girth have a homomorphism to any given odd cycle. In this paper, we study the computational aspects of this problem. We show that for any minor-closed class F containing all planar graphs, and such that all minimal obstructions are 3-connected, the following holds: for any k there is a g = g(F , k)...

Motivated by the Lipschitz-lifting property of Banach spaces introduced Godefroy and Kalton, we consider lattice-lifting property, which is an analogous notion within category lattices lattice homomorphisms. Namely, a X satisfies if every homomorphism to having bounded linear right-inverse must have right-inverse. In terms free lattices, this can be rephrased into following question: embed they...

A homomorphism from a graph G to a graph H is a vertex mapping f : VG → VH such that f(u) and f(v) form an edge in H whenever u and v form an edge in G. The H-Coloring problem is to test if a graph G allows a homomorphism to a given graph H. A well-known result of Hell and Nešetřil determines the computational complexity of this problem for any fixed graph H. We study a natural variant of this ...

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(...

We prove that a discrete group G is amenable iff it is strongly unitarizable in the following sense: every unitarizable representation π on G can be unitarized by an invertible chosen in the von Neumann algebra generated by the range of π. Analogously a C *-algebra A is nuclear iff any bounded homomorphism u : A → B(H) is strongly similar to a *-homomorphism in the sense that there is an invert...

We consider the computational complexity of pure Nash equilibria in graphical games. It is known that the problem is NP-complete in general, but tractable (i.e., in P) for special classes of graphs such as those with bounded treewidth. It is then natural to ask: is it possible to characterize all tractable classes of graphs for this problem? In this work, we provide such a characterization for ...

A relational structure is homomorphism-homogeneous if any homomorphism between its finite substructures extends to an endomorphism of the structure in question. In this note, we characterise all permutations on a finite set enjoying this property. To accomplish this, we switch from the more traditional view of a permutation as a set endowed with two linear orders to a different representation b...