In this paper, we will study the theory of cyclic homology for regular multiplier Hopf algebras. We associate a cyclic module to a triple $(mathcal{R},mathcal{H},mathcal{X})$ consisting of a regular multiplier Hopf algebra $mathcal{H}$, a left $mathcal{H}$-comodule algebra $mathcal{R}$, and a unital left $mathcal{H}$-module $mathcal{X}$ which is also a unital algebra. First, we construct a para...

We first show that a bounded linear operator $ T $ on a real Banach space $ E $ is quasicompact (Riesz, respectively) if and only if $T': E_{mathbb{C}}longrightarrow E_{mathbb{C}}$ is quasicompact  (Riesz, respectively), where the complex Banach space $E_{mathbb{C}}$ is a suitable complexification of $E$ and $T'$ is the complex linear operator on $E_{mathbb{C}}$ associated with $T$. Next, we pr...

We show that if T is an isometry (as metric spaces) between the invertible groups of unital Banach algebras, then T is extended to a surjective real-linear isometry up to translation between the two Banach algebras. Furthermore if the underling algebras are closed unital standard operator algebras, (T (eA)) −1 T is extended to a surjective real algebra isomorphism; if T is a surjective isometry...

Let A,B be two unital C∗−algebras. We prove that every almost unital almost linear mapping h : A −→ B which satisfies h(3uy + 3yu) = h(3u)h(y) + h(y)h(3u) for all u ∈ U(A), all y ∈ A, and all n = 0, 1, 2, ..., is a Jordan homomorphism. Also, for a unital C∗−algebra A of real rank zero, every almost unital almost linear continuous mapping h : A −→ B is a Jordan homomorphism when h(3uy + 3yu) = h...

1.1. Definitions. 1) A (non-commutative) probability space consists of a pair (A, φ), where • A is a unital algebra • φ : A → C is a unital linear functional, i.e. in particular φ(1) = 1 2) Unital subalgebras A1, . . . ,An ⊂ A are called free, if we have φ(a1 . . . ak) = 0 whenever ai ∈ Aj(i) (i = 1, . . . , k) j(1) 6= j(2) 6= · · · 6 = j(k) φ(ai) = 0 (i = 1, . . . , k) 3) Random variables x1, ...

Parallel to the first two authors’ earlier classification of separable unital oneparameter continuous fields of Kirchberg algebras with torsion free K-groups supported in one dimension, one-parameter separable unital continuous fields of AF-algebras are classified by their ordered K0-sheaves. We prove Effros-Handelman-Shen type theorems for separable unital one-parameter continuous fields of AF...

We show that morphisms from n homotopy unital A∞-algebras to a single one are maps over an operad module with n + 1 commuting actions of the operad A ∞, whose algebras are homotopy unital A∞-algebras. The operad A∞ and modules over it have two useful gradings related by isomorphisms which change the degree. The composition of A ∞-morphisms with several entries is presented as a convolution of a...

Let C be the differential graded category of differential graded k-modules. We prove that the Yoneda A∞-functor Y : A op → A∞(A,C) is a full embedding for an arbitrary unital A∞-category A. Since A∞-algebras were introduced by Stasheff [Sta63, II] there existed a possibility to consider A∞-generalizations of categories. It did not happen until A∞-categories were encountered in studies of mirror...