In this paper we construct an action of a compact matrix quantum group on a Cuntz algebra or a UHF-algebra, and investigate the fixed point subalgebra of the algebra under the action. Especially we consider the action of μ U(2) on the Cuntz algebra <&i and the action of S μ U(2) on the UHF-algebra of type 2°° . We show that these fixed point subalgebras are generated by a sequence of Jones 9 projections.

Abstract. We prove a number of results about the stable and particularly the real ranks of tensor products of C * -algebras under the assumption that one of the factors is commutative. In particular, we prove the following:(1) If X is any locally compact σ-compact Hausdorff space and A is any C * -algebra, then

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An ESS model to better understand the evolutionary dynamics of a primitive non-mating type gamete size was developed with reference to the PBS (Parker, Baker and Smith's) theory, which was based on total numbers of zygotes formed and the zygote survival rates. We did not include mating types since it has been suggested that primitive mating systems did not have mating types. As input parameters, we used experimental data on gamete motility of marine green algae. Based on hard sphere collision mechanics, we detailed the fertilization kinetics of gametes that swim in water prior to fusing with their partners through a set of coupled, non-linear differential equations. These equations were integrated numerically using typical values of the constant parameters. To estimate the relative zygote survival rate, we used a function that is sigmoid in shape and examined some evolutionarily stable strategies in mating systems that depend on optimizing values of the invasion success ratio.

We compute the real rank and the stable rank of full group C*-algebras. Main result is (i) rr (C*(Fn)) = ∞, (ii) sr (C*(G1 * G2)) = ∞(|G1| ≥ 2, |G2| ≥ 2 and |G1| + |G2| ≥ 5), (iii) sr (C*(G1 * G2)) = 1(|G1| = |G2| = 2), where Fn is the free group with n generators, G1 and G2 are finite groups and |G| means the order of the group G.