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Effective Representations of the Space of Linear Bounded Operators
Vasco Brattka,
Applied General Topology
4:1 (2003) 115-131
Abstract
Representations of topological spaces by infinite sequences of symbols are used in computable analysis
to describe computations in topological spaces with the help of Turing machines.
From the computer science point of view such representations can be considered as data structures of
topological spaces. Formally, a representation of a topological space is a surjective
mapping from Cantor space onto the corresponding space. Typically, one is interested in admissible,
i.e. topologically well-behaved representations which are continuous and characterized
by a certain maximality condition.
We discuss a number of representations of the space of linear bounded operators on Banach spaces.
Since the operator norm topology of the operator space is non-separable in typical cases, the operator
space cannot be represented admissibly with respect to this topology. However, other topologies,
like the compact open topology and the Fell topology (on the operator graph) give rise to a number
of promising representations of operator spaces which can partially replace the operator norm topology.
These representations reflect the information which is included in certain data structures for operators,
such as programs or enumerations of graphs.
We investigate the sublattice of these representations with respect to continuous and computable
reducibility. Certain additional conditions, as finite dimensionality, let some classes of
representations collapse, and thus, change the corresponding graph. Altogether, a precise picture of
possible data structures for operator spaces and their mutual relation can be drawn.
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