In Dong et al. (2022, Journal of Operator Theory 88, 365–406), the authors addressed the question of whether surjective maps preserving the norm of a symmetric Kubo-Ando mean can be extended to Jordan $\ast $-isomorphisms. The question was affirmatively answered for surjective maps between the positive definite cones of unital $C^{*}$-algebras for certain specific classes of symmetric Kubo-Ando means. Here, we give a comprehensive answer to this question for surjective maps between the positive definite cones of $AW^{*}$-algebras preserving the norm of any symmetric Kubo-Ando mean.

This paper considers the large N limit of Wilson loops for the two-dimensional Euclidean Yang–Mills measure on all orientable compact surfaces of genus larger or equal to $1$, with a structure group given by a classical compact matrix Lie group. Our main theorem shows the convergence of all Wilson loops in probability, given that it holds true on a restricted class of loops, obtained as a modification of geodesic paths. Combined with the result of [20], a corollary is the convergence of all Wilson loops on the torus. Unlike the sphere case, we show that the limiting object is remarkably expressed thanks to the master field on the plane defined in [3, 39], and we conjecture that this phenomenon is also valid for all surfaces of higher genus. We prove that this conjecture holds true whenever it does for the restricted class of loops of the main theorem. Our result on the torus justifies the introduction of an interpolation between free and classical convolution of probability measures, defined with the free unitary Brownian motion but differing from t-freeness of [5] that was defined in terms of the liberation process of Voiculescu [67]. In contrast to [20], our main tool is a fine use of Makeenko–Migdal equations, proving uniqueness of their solution under suitable assumptions, and generalising the arguments of [21, 33].

Given a unital $C^*$-algebra and a faithful trace, we prove that the topology on the associated density space induced by the $C^*$-norm is finer than the Bures metric topology. We also provide an example when this containment is strict. Next, we provide a metric on the density space induced by a quantum metric in the sense of Rieffel and prove that the induced topology is the same as the topology induced by the Bures metric and $C^*$-norm when the $C^*$-algebra is assumed to be finite dimensional. Finally, we provide an example where the Bures metric and induced quantum metric are not metric equivalent. Thus, we provide a bridge between these aspects of quantum information theory and noncommutative metric geometry.

Let G be a compact group, let $\mathcal {B}$ be a unital C$^*$-algebra, and let $(\mathcal {A},G,\alpha )$ be a free C$^*$-dynamical system, in the sense of Ellwood, with fixed point algebra $\mathcal {B}$. We prove that $(\mathcal {A},G,\alpha )$ can be realized as the G-continuous part of the invariants of an equivariant coaction of G on a corner of $\mathcal {B} \otimes {\mathcal {K}}({\mathfrak {H}})$ for a certain Hilbert space ${\mathfrak {H}}$ that arises from the freeness of the action. This extends a result by Wassermann for free and ergodic C$^*$-dynamical systems. As an application, we show that any faithful $^*$-representation of $\mathcal {B}$ on a Hilbert space ${\mathfrak {H}}_{\mathcal {B}}$ gives rise to a faithful covariant representation of $(\mathcal {A},G,\alpha )$ on some truncation of ${\mathfrak {H}}_{\mathcal {B}} \otimes {\mathfrak {H}}$.

Let R be a ring and let $n\ge 2$. We discuss the question of whether every element in the matrix ring $M_n(R)$ is a product of (additive) commutators $[x,y]=xy-yx$, for $x,y\in M_n(R)$. An example showing that this does not always hold, even when R is commutative, is provided. If, however, R has Bass stable rank one, then under various additional conditions every element in $M_n(R)$ is a product of three commutators. Further, if R is a division ring with infinite center, then every element in $M_n(R)$ is a product of two commutators. If R is a field and $a\in M_n(R)$, then every element in $M_n(R)$ is a sum of elements of the form $[a,x][a,y]$ with $x,y\in M_n(R)$ if and only if the degree of the minimal polynomial of a is greater than $2$.

Consider a possibly unsaturated Fell bundle $\mathcal {A}\to G$ over a locally compact, possibly non-Hausdorff, groupoid G. We list four notions of continuity of representations of $\mathit {C_c}(G;\mathcal {A})$ on a Hilbert space and prove their equivalence. This allows us to define the full $\mathit {C}^*$-algebra of the Fell bundle in different ways.

Let P be a pointed, closed convex cone in $\mathbb {R}^d$. We prove that for two pure isometric representations $V^{(1)}$ and $V^{(2)}$ of P, the associated CAR flows $\beta ^{V^{(1)}}$ and $\beta ^{V^{(2)}}$ are cocycle conjugate if and only if $V^{(1)}$ and $V^{(2)}$ are unitarily equivalent. We also give a complete description of pure isometric representations of P with commuting range projections that give rise to type I CAR flows. We show that such an isometric representation is completely reducible with each irreducible component being a pullback of the shift semigroup $\{S_t\}_{t \geq 0}$ on $L^2[0,\infty )$. We also compute the index and the gauge group of the associated CAR flows and show that the action of the gauge group on the set of normalized units need not be transitive.

For an integral domain R satisfying certain conditions, we characterise the primitive ideal space and its Jacobson topology for the semigroup crossed product $C^*(R_+) \rtimes R^\times $. We illustrate the result by the example $R=\mathbb {Z}[\sqrt {-3}]$.

We show that for $\mathrm {C}^*$-algebras with the global Glimm property, the rank of every operator can be realized as the rank of a soft operator, that is, an element whose hereditary sub-$\mathrm {C}^*$-algebra has no nonzero, unital quotients. This implies that the radius of comparison of such a $\mathrm {C}^*$-algebra is determined by the soft part of its Cuntz semigroup.

Under a mild additional assumption, we show that every Cuntz class dominates a (unique) largest soft Cuntz class. This defines a retract from the Cuntz semigroup onto its soft part, and it follows that the covering dimensions of these semigroups differ by at most $1$.

Given a Fell bundle $\mathcal {B}=\{B_t\}_{t\in G}$ over a locally compact group G and a closed subgroup $H\subset G,$ we construct quotients $C^{*}_{H\uparrow \mathcal {B}}(\mathcal {B})$ and $C^{*}_{H\uparrow G}(\mathcal {B})$ of the full cross-sectional C*-algebra $C^{*}(\mathcal {B})$ analogous to Exel–Ng’s reduced algebras $C^{*}_{\mathop {\mathrm {r}}}(\mathcal {B})\equiv C^{*}_{\{e\}\uparrow \mathcal {B}}(\mathcal {B})$ and $C^{*}_R(\mathcal {B})\equiv C^{*}_{\{e\}\uparrow G}(\mathcal {B}).$ An absorption principle, similar to Fell’s one, is used to give conditions on $\mathcal {B}$ and H (e.g., G discrete and $\mathcal {B}$ saturated, or H normal) ensuring $C^{*}_{H\uparrow \mathcal {B}}(\mathcal {B})=C^{*}_{H\uparrow G}(\mathcal {B}).$ The tools developed here enable us to show that if the normalizer of H is open in G and $\mathcal {B}_H:=\{B_t\}_{t\in H}$ is the reduction of $\mathcal {B}$ to $H,$ then $C^{*}(\mathcal {B}_H)=C^{*}_{\mathop {\mathrm {r}}}(\mathcal {B}_H)$ if and only if $C^{*}_{H\uparrow \mathcal {B}}(\mathcal {B})=C^{*}_{\mathop {\mathrm {r}}}(\mathcal {B});$ the last identification being implied by $C^{*}(\mathcal {B})=C^{*}_{\mathop {\mathrm {r}}}(\mathcal {B}).$ We also prove that if G is inner amenable and $C^{*}_{\mathop {\mathrm {r}}}(\mathcal {B})\otimes _{\max } C^{*}_{\mathop {\mathrm {r}}}(G)=C^{*}_{\mathop {\mathrm { r}}}(\mathcal {B})\otimes C^{*}_{\mathop {\mathrm {r}}}(G),$ then $C^{*}(\mathcal {B})=C^{*}_{\mathop {\mathrm {r}}}(\mathcal {B}).$

By employing the external Kasparov product, in [18], Hawkins, Skalski, White, and Zacharias constructed spectral triples on crossed product C$^\ast $-algebras by equicontinuous actions of discrete groups. They further raised the question for whether their construction turns the respective crossed product into a compact quantum metric space in the sense of Rieffel. By introducing the concept of groups separated with respect to a given length function, we give an affirmative answer in the case of virtually Abelian groups equipped with certain orbit metric length functions. We further complement our results with a discussion of natural examples such as generalized Bunce-Deddens algebras and higher-dimensional noncommutative tori.

We establish a Central Limit Theorem for tensor product random variables $c_k:=a_k \otimes a_k$, where $(a_k)_{k \in \mathbb {N}}$ is a free family of variables. We show that if the variables $a_k$ are centered, the limiting law is the semi-circle. Otherwise, the limiting law depends on the mean and variance of the variables $a_k$ and corresponds to a free interpolation between the semi-circle law and the classical convolution of two semi-circle laws.

Let $\mathfrak{C}$ be the smallest class of countable discrete groups with the following properties: (i) $\mathfrak{C}$ contains the trivial group, (ii) $\mathfrak{C}$ is closed under isomorphisms, countable increasing unions and extensions by $\mathbb{Z}$. Note that $\mathfrak{C}$ contains all countable discrete torsion-free abelian groups and poly-$\mathbb{Z}$ groups. Also, $\mathfrak{C}$ is a subclass of the class of countable discrete torsion-free elementary amenable groups. In this article, we show that if $\Gamma\in \mathfrak{C}$, then all strongly outer actions of Γ on the Razak–Jacelon algebra $\mathcal{W}$ are cocycle conjugate to each other. This can be regarded as an analogous result of Szabó’s result for strongly self-absorbing C$^*$-algebras.

In Caspers et al. (Can. J. Math. 75[6] [2022], 1–18), transference results between multilinear Fourier and Schur multipliers on noncommutative $L_p$-spaces were shown for unimodular groups. We propose a suitable extension of the definition of multilinear Fourier multipliers for non-unimodular groups and show that the aforementioned transference results also hold in this more general setting.

We discuss representations of product systems (of $W^*$-correspondences) over the semigroup $\mathbb{Z}^n_+$ and show that, under certain pureness and Szegö positivity conditions, a completely contractive representation can be dilated to an isometric representation. For $n=1,2$ this is known to hold in general (without assuming the conditions), but for $n\geq 3$, it does not hold in general (as is known for the special case of isometric dilations of a tuple of commuting contractions). Restricting to the case of tuples of commuting contractions, our result reduces to a result of Barik, Das, Haria, and Sarkar (Isometric dilations and von Neumann inequality for a class of tuples in the polydisc. Trans. Amer. Math. Soc. 372 (2019), 1429–1450). Our dilation is explicitly constructed, and we present some applications.

The goal of this paper is to show that the theory of curvature invariant, as introduced by Arveson, admits a natural extension to the framework of ${\mathcal U}$-twisted polyballs $B^{\mathcal U}({\mathcal H})$ which consist of k-tuples $(A_1,\ldots, A_k)$ of row contractions $A_i=(A_{i,1},\ldots, A_{i,n_i})$ satisfying certain ${\mathcal U}$-commutation relations with respect to a set ${\mathcal U}$ of unitary commuting operators on a Hilbert space ${\mathcal H}$. Throughout this paper, we will be concerned with the curvature of the elements $A\in B^{\mathcal U}({\mathcal H})$ with positive trace class defect operator $\Delta_A(I)$. We prove the existence of the curvature invariant and present some of its basic properties. A distinguished role as a universal model among the pure elements in ${\mathcal U}$-twisted polyballs is played by the standard $I\otimes{\mathcal U}$-twisted multi-shift S acting on $\ell^2({\mathbb F}_{n_1}^+\times\cdots\times {\mathbb F}_{n_k}^+)\otimes {\mathcal H}$. The curvature invariant $\mathrm{curv} (A)$ can be any non-negative real number and measures the amount by which A deviates from the universal model S. Special attention is given to the $I\otimes {\mathcal U}$-twisted multi-shift S and the invariant subspaces (co-invariant) under S and $I\otimes {\mathcal U}$, due to the fact that any pure element $A\in B^{\mathcal U}({\mathcal H})$ with $\Delta_A(I)\geq 0$ is the compression of S to such a co-invariant subspace.

Given any unital, finite, classifiable $\mathrm{C}^*$-algebra A with real rank zero and any compact simplex bundle with the fibre at zero being homeomorphic to the space of tracial states on A, we show that there exists a flow on A realizing this simplex. Moreover, we show that given any unital $\mathrm{UCT}$ Kirchberg algebra A and any proper simplex bundle with empty fibre at zero, there exists a flow on A realizing this simplex.

In [CDD22], we investigated the structure of $\ast $-isomorphisms between von Neumann algebras $L(\Gamma )$ associated with graph product groups $\Gamma $ of flower-shaped graphs and property (T) wreath-like product vertex groups, as in [CIOS21]. In this follow-up, we continue the structural study of these algebras by establishing that these graph product groups $\Gamma $ are entirely recognizable from the category of all von Neumann algebras arising from an arbitrary nontrivial graph product group with infinite vertex groups. A sharper $C^*$-algebraic version of this statement is also obtained. In the process of proving these results, we also extend the main $W^*$-superrigidity result from [CIOS21] to direct products of property (T) wreath-like product groups.

The subject of this article is operators represented on a Fock space which act only on the two leading components of the tensor. We unify the constructions from [Ans07, BL09, BL11, LS08] and extend a number of results from these articles to our more general setting. The results include the quadratic relation satisfied by the kernel of the free cumulant generating function, the resolvent form of the generating function for the Wick polynomials, and classification results for the case when the vacuum state on the operator algebra is tracial. We handle the generating functions in infinitely many variables by considering their matrix-valued versions.

We introduce and study the weak Glimm property for $\mathrm{C}^{*}$-algebras, and also a property we shall call (HS$_0$). We show that the properties of being nowhere scattered and residual (HS$_0$) are equivalent for any $\mathrm{C}^{*}$-algebra. Also, for a $\mathrm{C}^{*}$-algebra with the weak Glimm property, the properties of being purely infinite and weakly purely infinite are equivalent. It follows that for a $\mathrm{C}^{*}$-algebra with the weak Glimm property such that the absolute value of every nonzero, square-zero, element is properly infinite, the properties of being (weakly, locally) purely infinite, nowhere scattered, residual (HS$_0$), residual (HS$_{\text {t}}$), and residual (HI) are all equivalent, and are equivalent to the global Glimm property. This gives a partial affirmative answer to the global Glimm problem, as well as certain open questions raised by Kirchberg and Rørdam.