International Journal of Mathematics

We introduce a class of generalized universal irrational rotation C*-algebras A_{θ, γ} = C*(x, w) which is characterized by the relations w*w = ww* = 1, x*x = γ(w), xx* = γ(e^-2πiθw), and xw = e^-2πiθwx, where θ is an irrational number and γ(z) ∈ C(𝕋) is a positive function. We characterize tracial linear functionals, simplicity, and K-groups of A_{θ, γ} in terms of zero points of γ(z). We show that if A_{θ, γ} is simple then A_{θ, γ} is an A𝕋-algebra of real rank zero. We classify A_{θ, γ} in terms of θ and zero points of γ(z). Let A_θ = C*(u, v) be the universal irrational rotation C*-algebra with vu = e^2πiθuv. Then C*(u + v) ≅ A_θ,|1+z|². As an application, we show that C*(u + v) is a proper simple C*-subalgebra of A_θ which has a unique trace, K₁(C*(u + v)) ≅ ℤ, and there is an order isomorphism of K₀(C*(u + v)) onto ℤ + ℤθ. Moreover, C*(u + v) is a unital simple A𝕋-algebra of real rank zero. We also show that u + v is strongly irreducible relative to the hyperfinite type II₁ factor.

In this paper, we build Wiener measure for the path space on H-type group by using the heat kernel corresponding to the sub-Laplacian and give the definition of the Wiener integral. Then we give the Feynman–Kac formula.

We reduce the problem of classifying all maximal antipodal sets in the oriented real Grassmann manifold to that of classifying all maximal subsets satisfying certain conditions in the set consisting of subsets of cardinality k in {1, …, n}. Using this reduction we classify all maximal antipodal sets in for k ≤ 4. We construct some maximal antipodal subsets for higher k.

This article is retracted with immediate effect from IJM. The original submission had only Pan as the author and Qiao was a referee of this paper. After acceptance, Qiao was added as a coauthor without approval. The editors have also found that Theorem 1 is mathematically incorrect.

We explicitly calculate the universal character ring of the (-2, 2m + 1, 2n)-pretzel link and show that it is reduced for all integers m and n.

We discuss a family , with n ≥ 2, t > 1, of real hypersurfaces in a complex affine n-dimensional quadric arising in connection with the classification of homogeneous compact simply connected real-analytic hypersurfaces in ℂⁿ due to Morimoto and Nagano. To finalize their classification, one needs to resolve the problem of the embeddability of in ℂⁿ for n = 3, 7. We show that is not embeddable in ℂ⁷ for every t and that is embeddable in ℂ³ for all 1 < t < 1 + 10^-6. As a consequence of our analysis of a map constructed by Ahern and Rudin, we also conjecture that the embeddability of takes place for all .

We show certain symmetries for Borcherds lifts on the Hilbert modular group over a real quadratic field. We give two different proofs, the one analytic and the other arithmetic. The latter proof yields an explicit description of the action of Hecke operators on Borcherds lifts.

In order to give numerical characterizations of the notion of "mutual orthogonality", we introduce two kinds of family of positive definite matrices for a unitary u in a finite von Neumann algebra M. They are arising from u naturally depending on the decompositions of M. One corresponds to the tensor product decomposition and the other does to the crossed product decomposition. By using the von Neumann entropy for these positive definite matrices, we characterize the notion of mutual orthogonality between subalgebras.