International Journal of Mathematics

In the paper, we provide some examples of Blackadar and Kirchberg's MF algebras by considering minimal and maximal tensor products of MF algebras and crossed products of MF algebras by finite groups or an integer group. We also present some examples of C*-algebras, which BDF extension semigroups are not groups. These examples include, for example, , and with 2 ≤ |H₁| < ∞ and 3 ≤ |H₂| < ∞ where |H₁| and |H₂| are the orders of the groups H₁ and H₂ respectively, and several others similarly.

Let A be a unital separable simple infinite dimensional C*-algebra with tracial rank not more than one and with the tracial state space T(A) and let U(A) be the unitary group of A. Suppose that u ∈ U₀(A), the connected component of U(A) containing the identity. We show that, for any ϵ > 0, there exists a self-adjoint element h ∈ A_s.a such that

Denote by CU(A) the closure of the subgroup of unitary group of U(A) generated by its commutators. It is known that CU(A) ⊂ U₀(A). Denote by the affine function on T(A) defined by . We show that u can be approximated by unitaries in A with finite spectrum if and only if u ∈ CU(A) and for all n ≥ 1. Examples are given for which there are unitaries in CU(A) which cannot be approximated by unitaries with finite spectrum. Significantly these results are obtained in the absence of amenability.

We extend and unify several Hardy type inequalities to functions whose values are positive semi-definite operators. In particular, our methods lead to the operator versions of Hardy–Hilbert's and Godunova's inequalities. While classical Hardy type inequalities hold for parameter values p > 1, it is typical that the operator versions hold only for 1 < p ≤ 2, even for functions with values in 2 × 2 matrices.

The group Diff(S¹) of the orientation-preserving diffeomorphisms of the circle S¹ plays an important role in conformal field theory. We consider a subgroup B₀ of Diff(S¹) whose elements stabilize "the point at infinity." This subgroup is of interest for the actual physical theory that lives on the punctured circle, or the real line.

We investigate the unique central extension of the Lie algebra of that group. We determine the first and second cohomologies, its ideal structure and the automorphism group. We define a generalization of Verma modules and determine when these representations are irreducible. Its endomorphism semigroup is investigated and some unitary representations of the group which do not extend to Diff(S¹) are constructed.

Let X be a homogeneous tree of degree q + 1 (2 ≤ q ≤ ∞) and let ψ : X × X → ℂ be a function for which ψ(x, y) only depends on the distance between x, y ∈ X. Our main result gives a necessary and sufficient condition for such a function to be a Schur multiplier on X × X. Moreover, we find a closed expression for the Schur norm ||ψ||_S of ψ. As applications, we obtaina closed expression for the completely bounded Fourier multiplier norm ||⋅||_M0A(G) of the radial functions on the free (non-abelian) group 𝔽_N on N generators (2 ≤ N ≤ ∞) and of the spherical functions on the q-adic group PGL₂(ℚ_q) for every prime number q.

In this errata we correct several mistakes Secs. 1 and 3–6 that appear in the published version of our paper. The corrections have been implemented in the revised version [1]. In addition, in Sec. 2 we clarify an important point which was not adequately addressed in the published version; in Sec. 7 we point out an enhancement of the hyperbolization scheme included in [1]. The reader may also wish to consult [2].