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We consider a family of dynamical systems (A,alpha,L) in which alpha is an endomorphism of a C*-algebra A and L is a transfer operator for alpha. We extend Exels construction of a crossed product to cover non-unital algebras A, and show that the C*-algebra of a locally finite graph can be realised as one of these crossed products. When A is commutative, we find criteria for the simplicity of the crossed product, and analyse the ideal structure of the crossed product.
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Starting from a discrete $C^*$-dynamical system $(mathfrak{A}, theta, omega_o)$, we define and study most of the main ergodic properties of the crossed product $C^*$-dynamical system $(mathfrak{A}rtimes_alphamathbb{Z}, Phi_{theta, u},om_ocirc E)$, $E:mathfrak{A}rtimes_alphamathbb{Z}rightarrowga$ being the canonical conditional expectation of $mathfrak{A}rtimes_alphamathbb{Z}$ onto $mathfrak{A}$, provided $ainaut(ga)$ commute with the $*$-automorphism $th$ up tu a unitary $uinga$. Here, $Phi_{theta, u}inaut(mathfrak{A}rtimes_alphamathbb{Z})$ can be considered as the fully noncommutative generalisation of the celebrated skew-product defined by H. Anzai for the product of two tori in the classical case.
We investigate the notion of tracial $mathcal Z$-stability beyond unital C*-algebras, and we prove that this notion is equivalent to $mathcal Z$-stability in the class of separable simple nuclear C*-algebras.
Let $mathcal{H}$ be an infinite dimensional Hilbert space and $mathcal{B}(mathcal{H})$ be the C*-algebra of all bounded linear operators on $mathcal{H}$, equipped with the operator-norm. By improving the Brown-Pearcy construction, Terence Tao in 2018, extended the result of Popa [1981] which reads as : For each $0<varepsilonleq 1/2$, there exist $D,X in mathcal{B}(mathcal{H})$ with $|[D,X]-1_{mathcal{B}(mathcal{H})}|leq varepsilon$ such that $|D||X|=Oleft(log^5frac{1}{varepsilon}right)$, where $[D,X]:= DX-XD$. In this paper, we show that Taos result still holds for certain class of unital C*-algebras which include $mathcal{B}(mathcal{H})$ as well as the Cuntz algebra $mathcal{O}_2$.
A cosystem consists of a possibly nonselfadoint operator algebra equipped with a coaction by a discrete group. We introduce the concept of C*-envelope for a cosystem; roughly speaking, this is the smallest C*-algebraic cosystem that contains an equivariant completely isometric copy of the original one. We show that the C*-envelope for a cosystem always exists and we explain how it relates to the usual C*-envelope. We then show that for compactly aligned product systems over group-embeddable right LCM semigroups, the C*-envelope is co-universal, in the sense of Carlsen, Larsen, Sims and Vittadello, for the Fock tensor algebra equipped with its natural coaction. This yields the existence of a co-universal C*-algebra, generalizing previous results of Carlsen, Larsen, Sims and Vittadello, and of Dor-On and Katsoulis. We also realize the C*-envelope of the tensor algebra as the reduced cross sectional algebra of a Fell bundle introduced by Sehnem, which, under a mild assumption of normality, we then identify to the quotient of the Fock algebra by the image of Sehnems strong covariance ideal. In another application, we obtain a reduced Hao-Ng isomorphism theorem for the co-universal algebras.
Let $(G, P)$ be an abelian, lattice ordered group and let $X$ be a compactly aligned product system over $P$. We show that the C*-envelope of the Nica tensor algebra $mathcal{N}mathcal{T}^+_X$ coincides with both Sehnems covariance algebra $mathcal{A} times_X P$ and the co-universal C*-algebra $mathcal{N}mathcal{O}^r_X$ for injective, gauge compatible, Nica-covariant representations of Carlsen, Larsen, Sims and Vittadello. We give several applications of this result on both the selfadjoint and non-selfadjoint operator algebra theory. First we guarantee the existence of $mathcal{N}mathcal{O}^r_X$, thus settling a problem of Carlsen, Larsen, Sims and Vittadello which was open even for abelian, lattice ordered groups. As a second application, we resolve a problem posed by Skalski and Zacharias on dilating isometric representations of product systems to unitary representations. As a third application we characterize the C*-envelope of the tensor algebra of a finitely aligned higher-rank graph which also holds for topological higher-rank graphs. As a final application we prove reduced Hao-Ng isomorphisms for generalized gauge actions of discrete groups on C*-algebras of product systems. This generalizes recent results that were obtained by various authors in the case where $(G, P) =(mathbb{Z},mathbb{N})$.
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