The chain group $C(G)$ of a locally compact group $G$ has one generator $g_{rho}$ for each irreducible unitary $G$-representation $rho$, a relation $g_{rho}=g_{rho}g_{rho}$ whenever $rho$ is weakly contained in $rhootimes rho$, and $g_{rho^*}=g_{rho}
^{-1}$ for the representation $rho^*$ contragredient to $rho$. $G$ satisfies chain-center duality if assigning to each $g_{rho}$ the central character of $rho$ is an isomorphism of $C(G)$ onto the dual $widehat{Z(G)}$ of the center of $G$. We prove that $G$ satisfies chain-center duality if it is (a) a compact-by-abelian extension, (b) connected nilpotent, (c) countable discrete icc or (d) connected semisimple; this generalizes M. M{u}gers result compact groups satisfy chain-center duality.
This paper introduces the notion of Rota-Baxter $C^{ast}$-algebras. Here a Rota-Baxter $C^{ast}$-algebra is a $C^{ast}$-algebra with a Rota-Baxter operator. Symmetric Rota-Baxter operators, as special cases of Rota-Baxter operators on $C^{ast}$-algeb
ra, are defined and studied. A theorem of Rota-Baxter operators on concrete $C^{ast}$-algebras is given, deriving the relationship between two kinds of Rota-Baxter algebras. As a corollary, some connection between $ast$-representations and Rota-Baxter operators is given. The notion of representations of Rota-Baxter $C^{ast}$-algebras are constructed, and a theorem of representations of direct sums of Rota-Baxter representations is derived. Finally using Rota-Baxter operators, the notion of quasidiagonal operators on $C^{ast}$-algebra is reconstructed.
We introduced the quasicentral modulus to study normed ideal perturbations of operators. It is a limit of condenser quasicentral moduli in view of a recently noticed analogy with capacity in nonlinear potential theory. We prove here some basic proper
ties of the condenser quasicentral modulus and compute a simple example. Some of the results are in the more general setting of a semifinite von Neumann algebra.
We introduce noncommutative weak Orlicz spaces associated with a weight and study their properties. We also define noncommutative weak Orlicz-Hardy spaces and characterize their dual spaces.
Starting with a vertex-weighted pointed graph $(Gamma,mu,v_0)$, we form the free loop algebra $mathcal{S}_0$ defined in Hartglass-Penneys article on canonical $rm C^*$-algebras associated to a planar algebra. Under mild conditions, $mathcal{S}_0$ is
a non-nuclear simple $rm C^*$-algebra with unique tracial state. There is a canonical polynomial subalgebra $Asubset mathcal{S}_0$ together with a Dirac number operator $N$ such that $(A, L^2A,N)$ is a spectral triple. We prove the Haagerup-type bound of Ozawa-Rieffel to verify $(mathcal{S}_0, A, N)$ yields a compact quantum metric space in the sense of Rieffel. We give a weighted analog of Benjamini-Schramm convergence for vertex-weighted pointed graphs. As our $rm C^*$-algebras are non-nuclear, we adjust the Lip-norm coming from $N$ to utilize the finite dimensional filtration of $A$. We then prove that convergence of vertex-weighted pointed graphs leads to quantum Gromov-Hausdorff convergence of the associated adjusted compact quantum metric spaces. As an application, we apply our construction to the Guionnet-Jones-Shyakhtenko (GJS) $rm C^*$-algebra associated to a planar algebra. We conclude that the compact quantum metric spaces coming from the GJS $rm C^*$-algebras of many infinite families of planar algebras converge in quantum Gromov-Hausdorff distance.
It is shown that any bundle of KMS state spaces which can occur for a flow on a unital separable C*-algebra with a trace state can also be realized by a flow on any given unital infinite-dimensional simple AF algebra.
Let $Gamma$ be a discrete group freely acting via homeomorphisms on the compact Hausdorff space $X$ and let $C(X)rtimes_eta Gamma$ be the completion of the convolution algebra $C_c(Gamma, C(X))$ with respect to a $C^*$-norm $eta$. A non-zero ideal $J
unlhd C(X)rtimes_etaGamma$ is exotic if $Jcap C(X) =(0)$. We give examples showing that exotic ideals are present whenever $Gamma$ is non-amenable and there is an invariant probability measure on $X$. This fact, along with the recent theory of exotic crossed product functors, allows us to provide negative answers to two questions of K. Thomsen. Let $mathfrak{A}$ be a non-atomic MASA on a separable Hilbert space and let $mathcal B_0$ be the linear span of the unitary operators $Uinmathcal B(mathcal H)$ such that $Umathfrak{A} U^*=mathfrak{A}$. We observe that while $mathcal B_0$ contains no compact operators, the norm-closure of $mathcal B_0$ contains all compact operators. This gives a positive answer to a question of A. Katavolos and V. Paulsen. For a free action of $Gamma$ on a compact Hausdorff space $X$, the opaque and grey ideals in $C(X)rtimes_eta Gamma$ coincide. We conclude with an example of a free action of $Gamma$ on a compact Hausdorff space $X$ along with a norm $eta$ for which these ideals are non-trivial.
Let $G$ be a locally compact abelian group with a Haar measure, and $Y$ be a measure space. Suppose that $H$ is a reproducing kernel Hilbert space of functions on $Gtimes Y$, such that $H$ is naturally embedded into $L^2(Gtimes Y)$ and is invariant u
nder the translations associated with the elements of $G$. Under some additional technical assumptions, we study the W*-algebra $mathcal{V}$ of translation-invariant bounded linear operators acting on $H$. First, we decompose $mathcal{V}$ into the direct integral of the W*-algebras of bounded operators acting on the reproducing kernel Hilbert spaces $widehat{H}_xi$, $xiinwidehat{G}$, generated by the Fourier transform of the reproducing kernel. Second, we give a constructive criterion for the commutativity of $mathcal{V}$. Third, in the commutative case, we construct a unitary operator that simultaneously diagonalizes all operators belonging to $mathcal{V}$, i.e., converts them into some multiplication operators. Our scheme generalizes many examples previously studied by Nikolai Vasilevski and other authors.
In this paper, we derive, from a general Simonenkos local principle, Fredholm criteria for restriction to isotypical components. More precisely, we gave a full proof, of the equivariant local principle for restriction to isotypical components of inva
riant pseudodifferential operators announced in cite{BCLN2}. Furthermore, we extend this result by relaxing the hypothesis made in the preceding quoted paper.
We define a notion of tracial $mathcal{Z}$-absorption for simple not necessarily unital C*-algebras. This extends the notion defined by Hirshberg and Orovitz for unital (simple) C*-algebras. We provide examples which show that tracially $mathcal{Z}$-
absorbing C*-algebras need not be $mathcal{Z}$-absorbing. We show that tracial $mathcal{Z}$-absorption passes to hereditary C*-subalgebras, direct limits, matrix algebras, minimal tensor products with arbitrary simple C*-algebras. We find sufficient conditions for a simple, separable, tracially $mathcal{Z}$-absorbing C*-algebra to be $mathcal{Z}$-absorbing. We also study the Cuntz semigroup of a simple tracially $mathcal{Z}$-absorbing C*-algebra and prove that it is almost unperforated and weakly almost divisible.
