We show that a regular locally compact quantum group $\mathbb{G}$ is discrete if and only if $\mathcal{L}^{\infty}(\mathbb{G})$ contains non-zero compact operators on $\mathcal{L}^{2}(\mathbb{G})$. As a corollary we classify all discrete quantum groups among regular locally compact quantum groups $\mathbb{G}$ where $\mathcal{L}^{1}(\mathbb{G})$ has the Radon--Nikodym property.

In this short note we introduce a notion called ``quantum injectivity'' of locally compact quantum groups, and prove that it is equivalent to amenability of the dual. Particularly, this provides a new characterization of amenability of locally compact groups.

If $A$ is a $\sigma$-unital $C^*$-algebra and $a$ is a strictly positive element of $A$ then for every compact subset $K$ of the complete regularization $\mathrm{Glimm}(A)$ of $\mathrm{Prim}(A)$ there exists $\alpha \gt 0$ such that $K\subset \{G\in \mathrm{Glimm}(A) \mid \Vert a + G\Vert \geq \alpha\}$. This extends a result of J. Dauns to all $\sigma$-unital $C^*$-algebras. However, there are a $C^*$-algebra $A$ and a compact subset of $\mathrm{Glimm}(A)$ that is not contained in any set of the form $\{G\in \mathrm{Glimm}(A) \mid \Vert a + G\Vert \geq \alpha\}$, $a\in A$ and $\alpha \gt 0$.

We show that for $p$-operator spaces, there are natural notions of minimal and maximal structures. These are useful for dealing with tensor products.

Using the Dadarlat isomorphism, we give a characterization for the Kasparov product of $C^*$-algebra extensions. A certain relation between $KK(A, \mathcal q(B))$ and $KK(A, \mathcal q(\mathcal k B))$ is also considered when $B$ is not stable and it is proved that $KK(A, \mathcal q(B))$ and $KK(A, \mathcal q(\mathcal k B))$ are not isomorphic in general.

In this paper, we give a different proof of the fact that the odd dimensional quantum spheres are groupoid $C^{*}$-algebras. We show that the $C^{*}$-algebra $C(S_{q}^{2\ell+1})$ is generated by an inverse semigroup $T$ of partial isometries. We show that the groupoid $\mathcal{G}_{tight}$ associated with the inverse semigroup $T$ by Exel is exactly the same as the groupoid considered by Sheu.

We show that the following $K_0$-monoid properties of $C^*$-algebras in the class $\Omega$ are inherited by simple unital $C^*$-algebras in the class $TA\Omega$: (1) weak comparability, (2) strictly unperforated, (3) strictly cancellative.

Product type equivalence relations are hyperfinite measured equivalence relations, which, up to orbit equivalence, are generated by product type odometer actions. We give a concrete example of a hyperfinite equivalence relation of non-product type, which is the tail equivalence on a Bratteli diagram. In order to show that the equivalence relation constructed is not of product type we will use a criterion called property A. This property, introduced by Krieger for non-singular transformations, is defined directly for hyperfinite equivalence relations in this paper.

For a locally compact group $G$, let $A(G)$ be its Fourier algebra, let $M_{cb}A(G)$ denote the completely bounded multipliers of $A(G)$, and let $A_{\mathit{Mcb}}(G)$ stand for the closure of $A(G)$ in $M_{cb}A(G)$. We characterize the norm one idempotents in $M_{cb}A(G)$: the indicator function of a set $E \subset G$ is a norm one idempotent in $M_{cb}A(G)$ if and only if $E$ is a coset of an open subgroup of $G$. As applications, we describe the closed ideals of $A_{\mathit{Mcb}}(G)$ with an approximate identity bounded by $1$, and we characterize those $G$ for which $A_{\mathit{Mcb}}(G)$ is $1$-amenable in the sense of B. E. Johnson. (We can even slightly relax the norm bounds.)

It is shown that a separable $C^*$-algebra is inner quasidiagonal if and only if it has a separating family of quasidiagonal irreducible representations. As a consequence, a separable $C^*$-algebra is a strong NF algebra if and only if it is nuclear and has a separating family of quasidiagonal irreducible representations. We also obtain some permanence properties of the class of inner quasidiagonal $C^*$-algebras.

Recently, Beattie, Bulacu ,and Torrecillas proved Radford's formula for the fourth power of the antipode for a co-Frobenius Hopf algebra. In this note, we show that this formula can be proved for any regular multiplier Hopf algebra with integrals (algebraic quantum groups). This, of course, not only includes the case of a finite-dimensional Hopf algebra, but also that of any Hopf algebra with integrals (co-Frobenius Hopf algebras). Moreover, it turns out that the proof in this more general situation, in fact, follows in a few lines from well-known formulas obtained earlier in the theory of regular multiplier Hopf algebras with integrals. We discuss these formulas and their importance in this theory. We also mention their generalizations, in particular to the (in a certain sense) more general theory of locally compact quantum groups. Doing so, and also because the proof of the main result itself is very short, the present note becomes largely of an expository nature.

We generalize Löwner's method for proving that matrix monotone functions are operator monotone. The relation $x\leq y$ on bounded operators is our model for a definition of $C^{*}$-relations being residually finite dimensional. Our main result is a meta-theorem about theorems involving relations on bounded operators. If we can show there are residually finite dimensional relations involved and verify a technical condition, then such a theorem will follow from its restriction to matrices. Applications are shown regarding norms of exponentials, the norms of commutators, and "positive" noncommutative $*$-polynomials.

In this paper we present a classification, up to equivariant isomorphism, of $C^*$-dynamical systems $(A,{\mathbb R},\alpha )$ arising as inductive limits of directed systems $\{ (A_n,{\mathbb R},\alpha_n),\varphi_{nm}\}$, where each $A_n$ is a finite direct sum of matrix algebras over the continuous functions on the unit circle, and the $\alpha_n$s are outer actions generated by rotation of the spectrum.

We will give a characterization of stable real $C^*$-algebras analogous to the one given for complex $C^*$-algebras by Hjelmborg and Rørdam. Using this result, we will prove that any real $C^*$-algebra satisfying the corona factorization property is stable if and only if its complexification is stable. Real $C^*$-algebras satisfying the corona factorization property include AF-algebras and purely infinite $C^*$-algebras. We will also provide an example of a simple unstable $C^*$-algebra, the complexification of which is stable.

We shall present examples of Schauder bases in the preduals to the hyperfinite factors of types~$\hbox{II}_1$, $\hbox{II}_\infty$, $\hbox{III}_\lambda$, $0 < \lambda \leq 1$. In the semifinite (respectively, purely infinite) setting, these systems form Schauder bases in any associated separable symmetric space of measurable operators (respectively, in any non-commutative $L^p$-space).

Using Poincar\'e duality, we obtain a formula of Lefschetz type that computes the Lefschetz number of an endomorphism of a separable nuclear $C^*$-algebra satisfying Poincar\'e duality and the Kunneth theorem. (The Lefschetz number of an endomorphism is the graded trace of the induced map on $\textrm{K}$-theory tensored with $\mathbb{C}$, as in the classical case.) We then examine endomorphisms of Cuntz--Krieger algebras $O_A$. An endomorphism has an invariant, which is a permutation of an infinite set, and the contracting and expanding behavior of this permutation describes the Lefschetz number of the endomorphism. Using this description, we derive a closed polynomial formula for the Lefschetz number depending on the matrix $A$ and the presentation of the endomorphism.

A. Bonkat obtained a universal coefficient theorem in the setting of Kirchberg's ideal-related $KK$-theory in the fundamental case of a $C^*$-algebra with one specified ideal. The universal coefficient sequence was shown to split, unnaturally, under certain conditions. Employing certain $K$-theoretical information derivable from the given operator algebras using a method introduced here, we shall demonstrate that Bonkat's UCT does not split in general. Related methods lead to information on the complexity of the $K$-theory which must be used to classify $*$-isomorphisms for purely infinite $C^*$-algebras with one non-trivial ideal.

We investigate the behavior of the quasi-Baer and the right FI-extending right ring hulls under various ring extensions including group ring extensions, full and triangular matrix ring extensions, and infinite matrix ring extensions. As a consequence, we show that for semiprime rings $R$ and $S$, if $R$ and $S$ are Morita equivalent, then so are the quasi-Baer right ring hulls $\widehat{Q}_{\mathfrak{qB}}(R)$ and $\widehat{Q}_{\mathfrak{qB}}(S)$ of $R$ and $S$, respectively. As an application, we prove that if unital $C^*$-algebras $A$ and $B$ are Morita equivalent as rings, then the bounded central closure of $A$ and that of $B$ are strongly Morita equivalent as $C^*$-algebras. Our results show that the quasi-Baer property is always preserved by infinite matrix rings, unlike the Baer property. Moreover, we give an affirmative answer to an open question of Goel and Jain for the commutative group ring $A[G]$ of a torsion-free Abelian group $G$ over a commutative semiprime quasi-continuous ring $A$. Examples that illustrate and delimit the results of this paper are provided.

In this paper we propose a new technical tool for analyzing representations of Hilbert $C^*$-product systems. Using this tool, we give a new proof that every doubly commuting representation over $\mathbb{N}^k$ has a regular isometric dilation, and we also prove sufficient conditions for the existence of a regular isometric dilation of representations over more general subsemigroups of $\mathbb R_{+}^k$.

Let $\Gamma$ be a discrete group and let $f \in \ell^{1}(\Gamma)$. We observe that if the natural convolution operator $\rho_f: \ell^{\infty}(\Gamma)\to \ell^{\infty}(\Gamma)$ is injective, then $f$ is invertible in $\ell^{1}(\Gamma)$. Our proof simplifies and generalizes calculations in a preprint of Deninger and Schmidt by appealing to the direct finiteness of the algebra $\ell^{1}(\Gamma)$. We give simple examples to show that in general one cannot replace $\ell^{\infty}$ with $\ell^{p}$, $1\leq p< \infty$, nor with $L^{\infty}(G)$ for nondiscrete $G$. Finally, we consider the problem of extending the main result to the case of weighted convolution operators on $\Gamma$, and give some partial results.

We prove the equivalence of three definitions given by different comparison relations for infiniteness of positive elements in simple $C^*$-algebras.

In this paper, we give two characterizations of the exactness of operator spaces.

We describe the representation theory of $C^*$-crossed-products of a unital $C^*$-algebra A by the cyclic group of order~2. We prove that there are two main types of irreducible representations for the crossed-product: those whose restriction to A is irreducible and those who are the sum of two unitarily unequivalent representations of~A. We characterize each class in term of the restriction of the representations to the fixed point $C^*$-subalgebra of~A. We apply our results to compute the K-theory of several crossed-products of the free group on two generators.

We say that a numerical semigroup is \emph{$d$-squashed} if it can be written in the form $$ S=\frac 1 N \langle a_1,\dots,a_d \rangle \cap \mathbb{Z}$$ for $N,a_1,\dots,a_d$ positive integers with $\gcd(a_1,\dots, a_d)=1$. Rosales and Urbano have shown that a numerical semigroup is 2-squashed if and only if it is proportionally modular. Recent works by Rosales \emph{et al.} give a concrete example of a numerical semigroup that cannot be written as an intersection of $2$-squashed semigroups. We will show the existence of infinitely many numerical semigroups that cannot be written as an intersection of $2$-squashed semigroups. We also will prove the same result for $3$-squashed semigroups. We conjecture that there are numerical semigroups that cannot be written as the intersection of $d$-squashed semigroups for any fixed $d$, and we prove some partial results towards this conjecture.

We present a new approach to noncommutative real algebraic geometry based on the representation theory of $C^\ast$-algebras. An important result in commutative real algebraic geometry is Jacobi's representation theorem for archimedean quadratic modules on commutative rings. We show that this theorem is a consequence of the Gelfand--Naimark representation theorem for commutative $C^\ast$-algebras. A noncommutative version of Gelfand--Naimark theory was studied by I. Fujimoto. We use his results to generalize Jacobi's theorem to associative rings with involution.