Extending partial isometries

We show that a finite metric spaceA admits an extension to a finite metric spaceB so that each partial isometry ofA extends to an isometry ofB. We also prove a more precise result on extending a single partial isometry of a finite metric space. Both these results have consequences for the structure of the isometry groups of the rational Urysohn metric space and the Urysohn metric space. Research supported by NSF grant DMS-0400931. I would like to thank Ward Henson and Alekos Kechris for conversations and emails regarding the paper. Part of this work was done when I visited Caltech in May 2004. I thank the mathematics department there for support     

Semigroups of partial isometries

We study self-adjoint semigroups of partial isometries on a Hilbert space. These semigroups coincide precisely with faithful representations of abstract inverse semigroups. Groups of unitary operators are specialized examples of self-adjoint semigroups of partial isometries. We obtain a general structure result showing that every self-adjoint semigroup of partial isometries consists of “generalized weighted composition” operators on a space of square-integrable Hilbert-space valued functions. If the semigroup is finitely generated then the underlying measure space is purely atomic, so that the semigroup is represented as “zero-unitary” matrices. The same is true if the semigroup contains a compact operator, in which case it is not even required that the semigroup be self-adjoint.     

On the hyperreflexivity of power partial isometries

Necessary and sufficient conditions for hyperreflexivity of completely non-unitary power partial isometries are given.     

A note on roots and powers of partial isometries

Let T be a bounded operator and let \(k \ge 2\) be an integer. We study in this paper the following question: T is a partial isometry implies that \(T^k\) is a partial isometry and conversely.     

Extending partial isometries of antipodal graphs

We prove EPPA (extension property for partial automorphisms) for all antipodal classes from Cherlin’s list of metrically homogeneous graphs, thereby answering a question of Aranda et al. This paper should be seen as the first application of a new general method for proving EPPA which can bypass the lack of automorphism-preserving completions. It is done by combining the recent strengthening of the Herwig–Lascar theorem by Hubička, Nešetřil and the author with the ideas of the proof of EPPA for two-graphs by Evans et al.     

Representations of semigroups of partial isometries

We present a new proof that the irreducible representationsof the von Neumann algebra generated by a strongly continuoussemigroup of partial isometries of index 1 are unique up toequivalence, as well as a proof that when such an algebra isa factor, its representations are completely reducible. As anapplication, we show that the irreducible representations ofa strongly continuous semigroup of isometries {U(), 0} suchthat are equivalent.     

Multiplicative isometries on the Smirnov class

We show that T is a surjective multiplicative (but not necessarily linear) isometry from the Smirnov class on the open unit disk, the ball, or the polydisk onto itself, if and only if there exists a holomorphic automorphism Φ such that T(f)=f ○ Φ for every class element f or T(f) = [`(f<sup>°</sup> [`(j)] )]\overline {f^\circ \bar \varphi } for every class element f, where the automorphism Φ is a unitary transformation in the case of the ball and Φ(z ₁, ..., z _n ) = (l₁ z_i1 ,...,l_n z_in )(\lambda _1 z_{i_1 } ,...,\lambda _n z_{i_n } ) for |λ _j | = 1, 1 ≤ j ≤ n, and (i ₁; ..., i _n )is some permutation of the integers from 1through n in the case of the n-dimensional polydisk.     

A characterization of isometries of rational Euclidean spaces

We show that an injection of the rational Euclidean n-space, n5, which preserves the distances e, 1/2e, e an arbitrary non-zero rational number, is necessarily an isometry. Further, we show that the above characterization fails in case n=3 or 4.     

One class of C*-algebras generated by a family of partial isometries and multiplicators

We consider a C*-subalgebra of the algebra of all bounded operators on the Hilbert space of square-summable functions defined on some countable set. The algebra under consideration is generated by a family of partial isometries and the multiplier algebra isomorphic to the algebra of all bounded functions defined on the mentioned set. The partial isometry operators satisfy correlations defined by a prescribed map on the set. We show that the considered algebra is ?-graduated. After that we construct the conditional expectation from the latter onto the subalgebra responding to zero. Using this conditional expectation, we prove that the algebra under consideration is nuclear.     

On a class of C*-algebras generated by a countable family of partial isometries

The paper investigates the properties of an operator T _φ on the Hilbert space l ²(?), which are induced by the mapping φ of the set ? into itself. It is shown if the mapping φ is such that every preimage has finite, but not equipotentionally bounded cardinality, then the operator T _φ allows a closure and can be represented as a countable sum of partial isometries. The C*-algebras U _φ , P _φ and U _φ associated with given mappings and generated by the mentioned partial isometries are considered. Some properties of these algebras and some relations between them are given.