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Given an operator system S$\mathcal {S}$, we define the parameters rk(S)$r_k(\mathcal {S})$ (resp. dk(S)$d_k(\mathcal {S})$) defined as the maximal value of the completely bounded norm of a unital k$k$‐positive map from an arbitrary operator system into S$\mathcal {S}$ (resp. from S$\mathcal {S}$ into an arbitrary operator system). In the case of the matrix algebras Mn$\mathsf {M}_n$, for 1⩽k⩽n$1 \leqslant k \leqslant n$, we compute the exact value rk(Mn)=2n−kk$r_k(\mathsf {M}_n) = \frac{2n-k}{k}$ and show upper and lower bounds on the parameters dk(Mn)$d_k(\mathsf {M}_n)$. Moreover, when S$\mathcal {S}$ is a finite‐dimensional operator system, adapting results of Passer and the fourth author [J. Operator Theory 85 (2021), no. 2, 547–568], we show that the sequence (rk(S))$(r_k(\mathcal {S}))$ tends to 1 if and only if S$\mathcal {S}$ is exact and that the sequence (dk(S))$(d_k(\mathcal {S}))$ tends to 1 if and only if S$\mathcal {S}$ has the lifting property. [ABSTRACT FROM AUTHOR]

We define four 3×3 commuting contractions which do not dilate to commuting isometries. However, they do satisfy the scalar von Neumann inequality. These matrices are all nilpotent of order 2. We also show that any three 3×3 commuting contractions which are scalar plus nilpotent of order 2 do dilate to commuting isometries. [ABSTRACT FROM PUBLISHER]

If T=[T1 … Tn] is a row contraction with commuting entries, and the Arveson dilation is T̃=[T̃1 … T̃n], then any operator X commuting with each Ti dilates to an operator Y of the same norm that commutes with each T̃i. [ABSTRACT FROM PUBLISHER]

This paper is a comprehensive study of the nest representations for the free semigroupoid algebra ${\mathfrak{L}}_G$ of a countable directed graph $G$ as well as its norm-closed counterpart, the tensor algebra ${\mathcal{T}}^{+}(G)$.We prove that the finite-dimensional nest representations separate the points in ${\mathfrak{L}}_G$, and a fortiori, in ${\mathcal{T}}^{+}(G)$. The irreducible finite-dimensional representations separate the points in ${\mathfrak{L}}_G$ if and only if $G$ is transitive in components (which is equivalent to being semisimple). Also the upper triangular nest representations separate points if and only if for every vertex $x \in {\mathcal{T}}(G)$ supporting a cycle, $x$ also supports at least one loop edge.We also study faithful nest representations. We prove that ${\mathfrak{L}}_G$ (or ${\mathcal{T}}^{+}(G)$) admits a faithful irreducible representation if and only if $G$ is strongly transitive as a directed graph. More generally, we obtain a condition on $G$ which is equivalent to the existence of a faithful nest representation. We also give a condition that determines the existence of a faithful nest representation for a maximal type ${\mathbb{N}}$ nest. [ABSTRACT FROM AUTHOR]

A free semigroup algebra is the weak operator topology closed algebra generated by a set of isometries with pairwise orthogonal ranges. The most important example is the left regular free semigroup algebra generated by the left regular representation of the free semigroup on $n$ generators. This algebra is the appropriate non-commutative $n$-dimensional analogue of the analytic Toeplitz algebra. We develop a detailed picture of the invariant subspace structure analogous to Beurling's theorem and show that this algebra is hyper-reflexive with distance constant at most 51.The free semigroup algebras, known as atomic, for which the range projections of words in the generators lie in an atomic masa are completely classified. This provides a complete classification for a large class of representations of the Cuntz C*-algebras $\mathcal{O}_n$. This allows us to describe completely the invariant subspace structure of these algebras, and thereby show that these algebras are all hyper-reflexive. [ABSTRACT FROM AUTHOR]