Enflo finite-expansion and localized-trace system

Let $B$ be a Banach space generated by a sequence $(e_j{)}_{j\ge 1}$ that is linearly independent for finite sums. A linear operator $T$ on the algebraic span is a finite-expansion operator if

$T{e}_j={\sum }_i{a}_{ij}{e}_i$

where the sum is finite for every fixed $j$. The union of these finite supports need not be finite. A finite-expansion operator need not have finite rank; when bounded it extends from the dense algebraic span to $B$.

For a nonempty finite subset $M$ of the generators, define

$\mathrm{Tr}(M,T):={\sum }_{e_i\in M}{a}_{ii},\stackrel{}{\mathrm{Tr}}(M,T):=\frac{1}{\mathrm{\mid }M\mathrm{\mid }}\mathrm{Tr}(M,T)\mathrm{.}$

These are localized diagonal sums relative to the fixed independent generator; they are not asserted to be representation-independent tensor traces.

The generator has property A if for every finite sum $x={\sum }_j{a}_j{e}_j$ and every participating index $k$,

$\mathrm{\parallel }x\mathrm{\parallel }\ge \mathrm{\mid }{a}_k\mathrm{\mid }\mathrm{\parallel }{e}_k\mathrm{\parallel }\mathrm{.}$

For such a finite set $M$, write $[M]$ for its finite-dimensional span and $\mathrm{\parallel }T{\mathrm{\parallel }}_{(M)}:={\sup }_{0\mathrm{\ne }x\in [M]}\mathrm{\parallel }Tx\mathrm{\parallel }\mathrm{/}\mathrm{\parallel }x\mathrm{\parallel }$.