Dual operator algebras and the classes A(m,n).
Il Bong Jung · Deep Blue (University of Michigan) · 1989
In this dissertation we study the problem of solving systems of simultaneous equations in the preduals of certain dual operator algebras. In particular, we research the structure of dual algebras with property $(\\sk{10}\\sb{m,n})$ for some positive integers m and n. Let ${\\cal H}$ be a separable, infinite dimensional, complex Hilbert space, and let ${\\cal L}({\\cal H}$) be the algebra of all bounded linear operators on ${\\cal H}$. We obtain the following dilation theorem for operators in the classes $\\sk{10}\\sb{m,n}$. Suppose $T\\in\\sk{10}\\sb{m,n}({\\cal H}$), for some positive integers m and n. If A is any absolutely continuous contraction on a Hilbert space ${\\cal K}$, A possesses an m-cyclic set of vectors, and $A\\sp\\*$ has an n-cyclic set of vectors, then there exist invariant subspaces ${\\cal M}$ and ${\\cal N}$ for T with ${\\cal M} \\supset {\\cal N}$ and closed, one-to-one linear transformation $X :{\\cal D}(X) \o {\\cal M} \\ominus {\\cal N}$ such that (a) the domain ${\\cal D}(X)$ of X is dense in ${\\cal K}$ and contains the given m-cyclic set, (b) the range ${\\cal R}(X)$ of X is dense in ${\\cal M} \\ominus {\\cal N}$, and (c) ${\\cal A}{\\cal D}(X) \\subset {\\cal D}(X)$ and $T\\sb{{\\cal M}\\ominus{\\cal N}}Xz = XAz$ for all z in ${\\cal D}(X)$. Suppose $T\\in C\\sb{.0}\\cap\\sk{10}\\sb{n,1}({\\cal H})$ satisfying $d\\sb{T}\\* < \\infty$, for some positive integer n. Then using the dilation theorem we have a positive integer r with $n\\leq r + d\\sb{T}$ such that its Jordan model $J\\sb{T}$ = S($\heta\\sb1)\\oplus\\cdots\\oplus S(\heta\\sb{i})\\oplus S\\sp{(r)},$ and 0 $\\leq i \\leq d\\sb{T}$, in the sense that if i = 0, then $J\\sb{T} = S\\sp{(r)}.$ Moreover, if $S\\sp{(n)}$ is the unilateral shift of multiplicity n, where n is a positive integer, and $B\\in C\\sb0$ (i.e. there exists $u \\in H\\sp{\\infty}, u \ot\\equiv 0$, such that u(B) = 0) with $d\\sb{B\\*} < \\infty$, then we obtain $S\\sp{(n)} \\oplus B \\in \\sk{10}\\sb{n,\\aleph0}(1)\\\\\\sk{10}\\sb{n+1,1}.$ This result implies immediately that every Jordan operator of the form S($\heta\\sb1)\\oplus\\cdots\\oplus S(\heta\\sb{k})\\oplus S\\sp{(n)},$ $0 \\leq k \\leq \\infty, 1 \\leq n <\\infty$, belongs to $\\sk{10}\\sb{n,\\aleph 0}(1)\\\\\\sk{10}\\sb{n+1,1}$. Therefore we can distinguish the classes $\\sk{10}\\sb{m,n}$ one from another, in the sense that $\\sk{10}\\sb{m,n} \e \\sk{10} \\sb{p,q}$ if and only if $m \e p$ or $n \e q.$ Also we obtain some new sufficient conditions for membership in $\\sk{10}\\sb{1,\\aleph 0}$, and also some new results concerning operators in $\\sk{10}\\sb{m,n}$.