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This paper focuses on the approximation of continuous functions on the unit sphere by spherical polynomials of degree n via hyperinterpolation. Hyperinterpolation of degree n is a discrete approximation of the L 2 -orthogonal projection of the same degree with its Fourier coefficients evaluated by a positive-weight quadrature rule that exactly integrates all spherical polynomials of degree at most 2 n. This paper aims to bypass this quadrature exactness assumption by replacing it with the Marcinkiewicz–Zygmund property proposed in a previous paper. Consequently, hyperinterpolation can be constructed by a positive-weight quadrature rule (not necessarily with quadrature exactness). This scheme is referred to as unfettered hyperinterpolation. This paper provides a reasonable error estimate for unfettered hyperinterpolation. The error estimate generally consists of two terms: a term representing the error estimate of the original hyperinterpolation of full quadrature exactness and another introduced as compensation for the loss of exactness degrees. A guide to controlling the newly introduced term in practice is provided. In particular, if the quadrature points form a quasi-Monte Carlo (QMC) design, then there is a refined error estimate. Numerical experiments verify the error estimates and the practical guide. [ABSTRACT FROM AUTHOR]

We will show that group exactness is a von Neumann equivalence invariant. This result generalizes the previously known fact stating that group exactness is stable under measure equivalence and W*-equivalence. [ABSTRACT FROM AUTHOR]

Kirchberg's Embedding Problem (KEP) asks whether every separable C ⁎ algebra embeds into an ultrapower of the Cuntz algebra O 2 . In this paper, we use model theory to show that this conjecture is equivalent to a local approximate nuclearity condition that we call the existence of good nuclear witnesses . In order to prove this result, we study general properties of existentially closed C ⁎ algebras. Along the way, we establish a connection between existentially closed C ⁎ algebras, the weak expectation property of Lance, and the local lifting property of Kirchberg. The paper concludes with a discussion of the model theory of O 2 . Several results in this last section are proven using some technical results concerning tubular embeddings, a notion first introduced by Jung for studying embeddings of tracial von Neumann algebras into the ultrapower of the hyperfinite II 1 factor. [ABSTRACT FROM AUTHOR]

Abstract: We continue our study of tensor products in the operator system category. We define operator system quotients and exactness in this setting and refine the notion of nuclearity by studying operator systems that preserve various pairs of tensor products. One of our main goals is to relate these refinements of nuclearity to the Kirchberg conjecture. In particular, we prove that the Kirchberg conjecture is equivalent to the statement that every operator system that is (min,er)-nuclear is also (el,c)-nuclear. We show that operator system quotients are not always equal to the corresponding operator space quotients and then study exactness of various operator system tensor products for the operator system quotient. We prove that an operator system is exact for the min tensor product if and only if it is (min,el)-nuclear. We give many characterizations of operator systems that are (min,er)-nuclear, (el,c)-nuclear, (min,el)-nuclear and (el,max)-nuclear. These characterizations involve operator system analogues of various properties from the theory of C*-algebras and operator spaces, including the WEP and LLP. [Copyright &y& Elsevier]

Abstract: How do children as young as 2years of age know that numerals, like one, have exact interpretations, while quantifiers and words like a do not? Previous studies have argued that only numerals have exact lexical meanings. Children could not use scalar implicature to strengthen numeral meanings, it is argued, since they fail to do so for quantifiers [Papafragou, A., & Musolino, J. (2003). Scalar implicatures: Experiments at the semantics–pragmatics interface. Cognition, 86, 253–282]. Against this view, we present evidence that children’s early interpretation of numerals does rely on scalar implicature, and argue that differences between numerals and quantifiers are due to differences in the availability of the respective scales of which they are members. Evidence from previous studies establishes that (1) children can make scalar inferences when interpreting numerals, (2) children initially assign weak, non-exact interpretations to numerals when first acquiring their meanings, and (3) children can strengthen quantifier interpretations when scalar alternatives are made explicitly available. [Copyright &y& Elsevier]

Abstract: We show that the Hilbert space compression of any (unbounded) finite-dimensional CAT(0) cube complex is 1 and deduce that any finitely generated group acting properly, co-compactly on a CAT(0) cube complex is exact, and hence has Yu''s Property A. The class of groups covered by this theorem includes free groups, finitely generated Coxeter groups, finitely generated right angled Artin groups, finitely presented groups satisfying the B(4)–T(4) small cancellation condition and all those word-hyperbolic groups satisfying the B(6) condition. Another family of examples is provided by certain canonical surgeries defined by link diagrams. [Copyright &y& Elsevier]

Given a C ⁎ -dynamical system (A , G , α) , with G a discrete group, Schur A -multipliers and Herz–Schur (A , G , α) -multipliers are used to implement approximation properties, namely exactness and the strong operator approximation property (SOAP), of A ⋊ α , r G. The resulting characterisations of exactness and SOAP of A ⋊ α , r G generalise the corresponding statements for the reduced group C ⁎ -algebra. [ABSTRACT FROM AUTHOR]