In this paper, we study the structure of the reduced C *algebras and von Neumann algebras associated to the free orthogonal and free unitary quantum groups. We show that the reduced von Neumann algebras of these quantum groups always have the Haagerup approximation property. Combining this result with a Haagerup-type inequality due to Vergnioux [35], we also show that the reduced C *algebras always have the metric approximation property.

We study the graph isomorphism game that arises in quantum information theory from the perspective of bigalois extensions of compact quantum groups. We show that every algebraic quantum isomorphism between a pair of (quantum) graphs X and Y arises as a quotient of a certain measured bigalois extension for the quantum automorphism groups G X and G Y of the graphs X and Y . In particular, this implies that the quantum groups G X and G Y are monoidally equivalent. We also establish a converse to this result, which says that every compact quantum group G monoidally equivalent to G X is of the form G Y for a suitably chosen quantum graph Y that is quantum isomorphic to X. As an application of these results, we deduce that the * -algebraic, C * -algebraic, and quantum commuting (qc) notions of a quantum isomorphism between classical graphs X and Y all coincide. Using the notion of equivalence for non-local games, we deduce the same result for other synchronous non-local games, including the synBCS game and certain related graph homomorphism games. Universiteit Leuven, mateusz.wasilewski@kuleuven.be 2. Conversely, for any compact quantum group G monoidally equivalent to G X , one can construct from this monoidal equivalence a quantum graph Y , an isomorphism of quantum groups G ∼ = G Y , and an algebraic quantum isomorphism X ∼ = A * Y .Recasting all of the above in the context of the (classical) graph isomorphism game, our results show that the condition A(Iso(X, Y )) = 0 is sufficient to ensure the existence of perfect quantum strategies for this game (Corollary 4.8 and Theorem 4.9):Theorem Two classical graphs X and Y are algebraically quantum isomorphic if and only if the graph isomorphism game has a perfect quantum-commuting (qc)-strategy.We mention that a weaker version of the above theorem (that assumed the existence of a non-zero C * -algebra representation of A(Iso(X, Y ))) was recently proved in [18].

ABSTRACT. For a compact quantum group G of Kac type, we study the existence of a Haar tracepreserving embedding of the von Neumann algebra L ∞ (G) into an ultrapower of the hyperfinite II 1 -factor (the Connes embedding property for L ∞ (G)). We establish a connection between the Connes embedding property for L ∞ (G) and the structure of certain quantum subgroups of G, and use this to prove that the II 1 -factorsassociated to the free orthogonal and free unitary quantum groups have the Connes embedding property for all N ≥ 4. As an application, we deduce that the free entropy dimension of the standard generators of L ∞ (O + N ) equals 1 for all N ≥ 4. We also mention an application of our work to the problem of classifying the quantum subgroups of O + N .

Let G be a locally compact group, and let A(G) and VN(G) be its Fourier algebra and group von Neumann algebra, respectively. In this paper we consider the similarity problem for A(G): Is every bounded representation of A(G) on a Hilbert space H similar to a * -representation? We show that the similarity problem for A(G) has a negative answer if and only if there is a bounded representation of A(G) which is not completely bounded. For groups with small invariant neighborhoods (i.e. SIN groups) we show that a representation π : A(G) → B(H ) is similar to a * -representation if and only if it is completely bounded. This, in particular, implies that corepresentations of VN(G) associated to non-degenerate completely bounded representations of A(G) are similar to unitary corepresentations. We also show that if G is a SIN, maximally almost periodic, or totally disconnected group, then a representation of A(G) is a * -representation if and only if it is a complete contraction. These results partially answer questions posed in Effros and Ruan (2003) [7] and Spronk (2002) [25].

Motivated by non-local games and quantum coloring problems, we introduce a graph homomorphism game between quantum graphs and classical graphs. This game is naturally cast as a "quantumclassical game"-that is, a non-local game of two players involving quantum questions and classical answers. This game generalizes the graph homomorphism game between classical graphs. We show that winning strategies in the various quantum models for the game directly generalize the notion of non-commutative graph homomorphisms due to D. Stahlke [44]. Moreover, we present a game algebra in this context that generalizes the game algebra for graph homomorphisms given by J.W. Helton, K. Meyer, V.I. Paulsen and M. Satriano [22]. We also demonstrate explicit quantum colorings of all quantum complete graphs, yielding the surprising fact that the algebra of the 4-coloring game for a quantum graph is always non-trivial, extending a result of [22].