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We introduce the notion of additive units, or `addits, of a pointed Arveson system, and demonstrate their usefulness through several applications. By a pointed Arveson system we mean a spatial Arveson system with a fixed normalised reference unit. We show that the addits form a Hilbert space whose codimension-one subspace of `roots is isomorphic to the index space of the Arveson system, and that the addits generate the type I part of the Arveson system. Consequently the isomorphism class of the Hilbert space of addits is independent of the reference unit. The addits of a pointed inclusion system are shown to be in natural correspondence with the addits of the generated pointed product system. The theory of amalgamated products is developed using addits and roots, and an explicit formula for the amalgamation of pointed Arveson systems is given, providing a new proof of its independence of the particular reference units. (This independence justifies the terminology `spatial product of spatial Arveson systems). Finally a cluster construction for inclusion subsystems of an Arveson system is introduced and we demonstrate its correspondence with the action of the Cantor--Bendixson derivative in the context of the random closed set approach to product systems due to Tsirelson and Liebscher.
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Suppose that $X_{1}$ and $X_{2}$ are two selfadjoint random variables that are freely independent over an operator algebra $mathcal{B}$. We describe the possible operator atoms of the distribution of $X_{1}+X_{2}$ and, using linearization, we determine the possible eigenvalues of an arbitrary polynomial $p(X_{1},X_{2})$ in case $mathcal{B}=mathbb{C}$.
We introduce the notion of additive units and roots of a unit in a spatial product system. The set of all roots of any unit forms a Hilbert space and its dimension is the same as the index of the product system. We show that a unit and all of its roots generate the type I part of the product system. Using properties of roots, we also provide an alternative proof of the Powers problem that the cocycle conjugacy class of Powers sum is independent of the choice of intertwining isometries. In the last section, we introduce the notion of cluster of a product subsystem and establish its connection with random sets in the sense of Tsirelson and Liebscher.
We consider C*-algebras of finite higher-rank graphs along with their rotational action. We show how the entropy theory of product systems with finite frames applies to identify the phase transitions of the dynamics. We compute the positive inverse temperatures where symmetry breaks, and in particular we identify the subharmonic parts of the gauge-invariant equilibrium states. Our analysis applies to positively weighted rotational actions through a recalibration of the entropies.
We study the structure of C*-algebras associated with compactly aligned product systems over group embeddable right LCM-semigroups. Towards this end we employ controlled maps and a controlled elimination method that associates the original cores to those of the controlling pair, and we combine with applications of the C*-envelope theory for cosystems of nonselfadjoint operator algebras recently produced. We derive several applications of these methods that generalize results on single C*-correspondences. First we show that if the controlling group is exact then the co-universal C*-algebra of the product system coincides with the quotient of the Fock C*-algebra by the ideal of strong covariance relations. We show that if the controlling group is amenable then the product system is amenable. In particular if the controlling group is abelian then the co-universal C*-algebra is the C*-envelope of the tensor algebra. Secondly we give necessary and sufficient conditions for the Fock C*-algebra to be nuclear and exact. When the controlling group is amenable we completely characterize nuclearity and exactness of any equivariant injective Nica-covariant representation of the product system. Thirdly we consider controlled maps that enjoy a saturation property. In this case we induce a compactly aligned product system over the controlling pair that shares the same Fock representation, and preserves injectivity. By using co-universality, we show that they share the same reduced covariance algebras. If in addition the controlling pair is a total order then the fixed point algebra of the controlling group induces a super-product system that has the same reduced covariance algebra and is moreover reversible.
Let $mathcal{M}$ be a von Neumann algebra with a normal semifinite faithful trace $tau$. We prove that every continuous $m$-homogeneous polynomial $P$ from $L^p(mathcal{M},tau)$, with $0<p<infty$, into each topological linear space $X$ with the property that $P(x+y)=P(x)+P(y)$ whenever $x$ and $y$ are mutually orthogonal positive elements of $L^p(mathcal{M},tau)$ can be represented in the form $P(x)=Phi(x^m)$ $(xin L^p(mathcal{M},tau))$ for some continuous linear map $Phicolon L^{p/m}(mathcal{M},tau)to X$.
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