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Register a new userCorrigendum to: Essential normality, essential norms and hyperrigidity
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In our paper Essential normality, essential norms and hyperrigidity we claimed that the restriction of the identity representation of a certain operator system (constructed from a polynomial ideal) has the unique extension property, however the justification we gave was insufficient. In this note we provide the required justification under some additional assumptions. Fortunately, homogeneous ideals that are sufficiently non-trivial are covered by these assumptions. This affects the section of our paper relating essential normality and hyperrigidity. We show here that Proposition 4.11 and Theorem 4.12 hold under the additional assumptions. We do not know if they hold in the generality considered in our paper.
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Let $S = (S_1, ldots, S_d)$ denote the compression of the $d$-shift to the complement of a homogeneous ideal $I$ of $mathbb{C}[z_1, ldots, z_d]$. Arveson conjectured that $S$ is essentially normal. In this paper, we establish new results supporting this conjecture, and connect the notion of essential normality to the theory of the C*-envelope and the noncommutative Choquet boundary. The unital norm closed algebra $mathcal{B}_I$ generated by $S_1,ldots,S_d$ modulo the compact operators is shown to be completely isometrically isomorphic to the uniform algebra generated by polynomials on $overline{V} := overline{mathcal{Z}(I) cap mathbb{B}_d}$, where $mathcal{Z}(I)$ is the variety corresponding to $I$. Consequently, the essential norm of an element in $mathcal{B}_I$ is equal to the sup norm of its Gelfand transform, and the C*-envelope of $mathcal{B}_I$ is identified as the algebra of continuous functions on $overline{V} cap partial mathbb{B}_d$, which means it is a complete invariant of the topology of the variety determined by $I$ in the ball. Motivated by this determination of the C*-envelope of $mathcal{B}_I$, we suggest a new, more qualitative approach to the problem of essential normality. We prove the tuple $S$ is essentially normal if and only if it is hyperrigid as the generating set of a C*-algebra, which is a property closely connected to Arvesons notion of a boundary representation. We show that most of our results hold in a much more general setting. In particular, for most of our results, the ideal $I$ can be replaced by an arbitrary (not necessarily homogeneous) invariant subspace of the $d$-shift.
Given a C$^*$-correspondence $X$, we give necessary and sufficient conditions for the tensor algebra $mathcal T_X^+$ to be hyperrigid. In the case where $X$ is coming from a topological graph we obtain a complete characterization.
Let $mathcal{H}_d^{(t)}$ ($t geq -d$, $t>-3$) be the reproducing kernel Hilbert space on the unit ball $mathbb{B}_d$ with kernel [ k(z,w) = frac{1}{(1-langle z, w rangle)^{d+t+1}} . ] We prove that if an ideal $I triangleleft mathbb{C}[z_1, ldots, z_d]$ (not necessarily homogeneous) has what we call the approximate stable division property, then the closure of $I$ in $mathcal{H}_d^{(t)}$ is $p$-essentially normal for all $p>d$. We then show that all quasi homogeneous ideals in two variables have the stable division property, and combine these two results to obtain a new proof of the fact that the closure of any quasi homogeneous ideal in $mathbb{C}[x,y]$ is $p$-essentially normal for $p>2$.
We characterise, in several complementary ways, etale groupoids with locally compact Hausdorff space of units whose essential groupoid C*-algebra has the ideal intersection property, assuming that the groupoid is either Hausdorff or $sigma$-compact. This leads directly to a characterisation of the simplicity of this C*-algebra which, for Hausdorff groupoids, agrees with the reduced groupoid C*-algebra. Specifically, we prove that the ideal intersection property is equivalent to the absence of essentially confined amenable sections of isotropy groups. For groupoids with compact space of units we moreover show that is equivalent to the uniqueness of equivariant pseudo-expectations and in the minimal case to an appropriate generalisation of Powers averaging property. A key technical idea underlying our results is a new notion of groupoid action on C*-algebras that includes the essential groupoid C*-algebra itself. By considering a relative version of Powers averaging property, we obtain new examples of C*-irreducible inclusions in the sense of R{o}rdam. These arise from the inclusion of the C*-algebra generated by a suitable group representation into a simple groupoid C*-algebra. This is illustrated by the example of the C*-algebra generated by the quasi-regular representation of Thompsons group T with respect to Thompsons group F, which is contained C*-irreducibly in the Cuntz algebra $mathcal{O}_2$.
A sequential of concepts developed in last decade has enabled a resolution to multiple anomalies of water ice and its low-dimensionality, particularly. Developed concepts include the coupled hydrogen bond oscillator pair, segmental specific heat, three-body coupling potentials, quasisolidity, and supersolidity. Resolved anomalies include ice buoyancy, ice slipperiness, water skin toughness, supercooling and superheating at the nanoscale, etc. Evidence shows consistently that molecular undercoordination shortens the HO bond and stiffens its phonon while undercoordination does the OH nonbond contrastingly associated with strong lone pair polarization, which endows the low-dimensional water ice with supersolidity. The supersolid phase is hydrophobic, less dense, viscoelastic, thermally more diffusive and stable, having longer electron and phonon lifetime. The equal number of lone pairs and protons reserves the configuration and orientation of the coupled hydrogen bond bonds and restricts molecular rotation and proton hopping, which entitles water the simplest, ordered, tetrahedrally-coordinated, fluctuating molecular crystal covered with a supersolid skin. The hydrogen bond segmental cooperativity and specific-heat disparity form the soul dictating the extraordinary adaptivity, reactivity, recoverability, sensitivity of water ice when subjecting to physical perturbation. It is recommended that the premise of hydrogen bonding and electronic dynamics would deepen the insight into the core physics and chemistry of water ice.
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