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Pentagon subspace lattices on Banach spaces. (English) Zbl 0998.47045
Summary: If $K$, $L$ and $M$ are (closed) subspaces of a Banach space $X$ satisfying $K\cap M=\{0\}$, $K\vee L=X$ and $L\subset M$, then ${\Cal P}=\{(0), K, L, M, X\}$ is a pentagon subspace lattice on $X$. If ${\Cal P}_{1}$ and ${\Cal P}_{2}$ are pentagons, every (algebraic) isomorphism $\varphi:\text{Alg }{\Cal P}_{1}\to\text{Alg }{\Cal P}_{2}$ is quasi-spatial.The SOT-closure of the finite rank subalgebra of $\text{Alg }{\Cal P}$ is $\{T\in \text{Alg }{\Cal P}:T(M)\subseteq L\}$. On separable Hilbert space $H$ every positive, injective, non-invertible operator $A$ and every non-zero subspace $M$ satisfying $M\cap \text{Ran} A=(0)$ give rise to a pentagon ${\Cal P}(A,M)$. $\text{Alg }{\Cal P}(A;M)$ and $\text{Alg }{\Cal P}(B;N)$ are spatially isomorphic if and only if $T \text{Ran}(A)=\text{Ran}(B)$ and $T(M)=N$ for an invertible operator $T\in B(H)$. If ${\Cal A}(A)$ is the set of operators leaving $\text{Ran}(A)$ invariant, every isomorphism $\varphi:{\Cal A}(A)\to {\Cal A}(B)$ is implemented by an invertible operator $T$ satisfying $T \text{Ran}(A)=\text{Ran}(B)$.

MSC:
47L10Algebras of operators on Banach spaces and other topological linear spaces
47C05Operators in topological algebras
47A15Invariant subspaces of linear operators