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Search: MSC category 47L80
( Algebras of specific types of operators (Toeplitz, integral, pseudodifferential, etc.) )
1. CJM 2010 (vol 62 pp. 889)
 Xia, Jingbo

Singular Integral Operators and Essential Commutativity on the Sphere
Let ${\mathcal T}$ be the $C^\ast $algebra generated by the Toeplitz operators $\{T_\varphi : \varphi \in L^\infty (S,d\sigma )\}$ on the Hardy space $H^2(S)$ of the unit sphere in $\mathbf{C}^n$. It is well known that ${\mathcal T}$ is contained in the essential commutant of $\{T_\varphi : \varphi \in \operatorname{VMO}\cap L^\infty (S,d\sigma )\}$. We show that the essential commutant of $\{T_\varphi : \varphi \in \operatorname{VMO}\cap L^\infty (S,d\sigma )\}$ is strictly larger than ${\mathcal T}$.
Categories:32A55, 46L05, 47L80 

2. CJM 2001 (vol 53 pp. 506)
 Davidson, Kenneth R.; Kribs, David W.; Shpigel, Miron E.

Isometric Dilations of NonCommuting Finite Rank $n$Tuples
A contractive $n$tuple $A=(A_1,\dots,A_n)$ has a minimal joint
isometric dilation $S=\break
(S_1,\dots,S_n)$ where the $S_i$'s are
isometries with pairwise orthogonal ranges. This determines a
representation of the CuntzToeplitz algebra. When $A$ acts on a
finite dimensional space, the $\wot$closed nonselfadjoint algebra
$\fS$ generated by $S$ is completely described in terms of the
properties of $A$. This provides complete unitary invariants for the
corresponding representations. In addition, we show that the algebra
$\fS$ is always hyperreflexive. In the last section, we describe
similarity invariants. In particular, an $n$tuple $B$ of $d\times d$
matrices is similar to an irreducible $n$tuple $A$ if and only if
a certain finite set of polynomials vanish on $B$.
Category:47L80 
