Suppose $A$ is a separable unital $𝒞\left(X\right)$-algebra each fibre of which is isomorphic to the same strongly self-absorbing and $K_1$-injective $C^{*}$-algebra $𝒟$. We show that $A$ and $𝒞\left(X\right)\otimes 𝒟$ are isomorphic as $𝒞\left(X\right)$-algebras provided the compact Hausdorff space $X$ is finite-dimensional. This statement is known not to extend to the infinite-dimensional case.

Associated to an Hadamard matrix $H\in M_N\left(ℂ\right)$ is the spectral measure μ ∈ [0,N] of the corresponding Hopf image algebra, A = C(G) with $G\subset S{⁺}_N$. We study a certain family of discrete measures ${\mu }^r\in [0,N]$, coming from the idempotent state theory of G, which converge in Cesàro limit to μ. Our main result is a duality formula of type ${\int }_0^N{(x/N)}^{p}d{\mu }^r\left(x\right)={\int }_0^N{(x/N)}^{r}d{\nu }^p\left(x\right)$, where ${\mu }^r,{\nu }^r$ are the truncations of the spectral measures μ,ν associated to $H,H^t$. We also prove, using these truncations ${\mu }^r,{\nu }^r$, that for any deformed Fourier matrix $H=F_M{\otimes }_Q{F}_N$ we have μ = ν.

Connes and Moscovici recently studied "twisted" spectral triples (A,H,D) in which the commutators [D,a] are replaced by D∘a - σ(a)∘D, where σ is a second representation of A on H. The aim of this note is to point out that this yields representations of arbitrary covariant differential calculi over Hopf algebras in the sense of Woronowicz. For compact quantum groups, H can be completed to a Hilbert space and the calculus is given by bounded operators. At the end, we discuss an explicit example of...

Let X be an infinite-dimensional Banach space, and let ϕ be a surjective linear map on B(X) with ϕ(I) = I. If ϕ preserves injective operators in both directions then ϕ is an automorphism of the algebra B(X). If X is a Hilbert space, then ϕ is an automorphism of B(X) if and only if it preserves surjective operators in both directions.

In the present paper we define and study the properties of a deformation of measures and convolutions that works in a similar way to the $U_t$ deformation of Bożejko and Wysoczański, but in its definition operates on two levels of Jacobi coefficients of a measure, rather than on one.

In 1992, Speicher showed the fundamental fact that the probability measures playing the role of the classical Gaussian in the various non-commutative probability theories (viz. fermionic probability, Voiculescu’s free probability, and $q$-deformed probability of Bożejko and Speicher) all arise as the limits in a generalized Central Limit Theorem. The latter concerns sequences of non-commutative random variables (elements of a $*$-algebra equipped with a state) drawn from an ensemble of pair-wise commuting...

Given an ultragraph in the sense of Tomforde, we construct a topological quiver in the sense of Muhly and Tomforde in such a way that the universal C*-algebras associated to the two objects coincide. We apply results of Muhly and Tomforde for topological quiver algebras and of Katsura for topological graph C*-algebras to study the K-theory and gauge-invariant ideal structure of ultragraph C*-algebras.

The existence of unbounded *-representations of (locally convex) tensor product *-algebras is investigated, in terms of the existence of unbounded *-representations of the (locally convex) factors of the tensor product and vice versa.

Let G be an infinite locally compact abelian group and X be a Banach space. We show that if every bounded Fourier multiplier T on L²(G) has the property that $T\otimes I{d}_X$ is bounded on L²(G,X) then X is isomorphic to a Hilbert space. Moreover, we prove that if 1 < p < ∞, p ≠ 2, then there exists a bounded Fourier multiplier on $L^p\left(G\right)$ which is not completely bounded. Finally, we examine unconditionality from the point of view of Schur multipliers. More precisely, we give several necessary and sufficient conditions...