By a quasi-representation of a group G we mean an approximately multiplicative map of G to the unitary group of a unital C∗-algebra. A quasi-representation induces a partially defined map at the level K-theory. In the early 90s Exel and Loring associated two invariants to almostcommuting pairs of unitary matrices u and v: one a K-theoretic invariant, which may be regarded as the image of the Bo...

In this paper, we define invariant tensor product and study invariant tensor products associated with discrete series representations. Let G(V1) × G(V2) be a pair of classical groups diagonally embedded in G(V1 ⊕ V2). Suppose that dimV1 < dimV2. Let π be a discrete series representation of G(V1 ⊕ V2). We prove that the functor π⊗G(V1) maps unitary representations of G(V1) to unitary representat...

One of the central problems in representation theory is understanding irreducible unitary representations. The reason is that in many applications of linear algebra (like those of representation theory to harmonic analysis) the notion of length of vectors is fundamentally important. Unitary representations are exactly those preserving a good notion of length. The paper [4] provides an algorithm...

In this paper we consider a class of hierarchically rank structured matrices, including some of the hierarchical matrices occurring in the literature, such as hierarchically semiseparable (HSS) and certain H∈-matrices. We describe a fast O(rn log(n)) and stable algorithm to transform this hierarchical representation into a so-called unitary-weight representation, as introduced in an earlier wor...

Important developments in fault-tolerant quantum computation using the braiding of anyons have placed the theory of braid groups at the very foundation of topological quantum computing. Furthermore, the realization by Kauffman and Lomonaco that a specific braiding operator from the solution of the Yang-Baxter equation, namely the Bell matrix, is universal implies that in principle all quantum g...

We discuss the differing definitions of complex and quaternionic projective group representations employed by us and by Emch. The definition of Emch (termed here a strong projective representation) is too restrictive to accommodate quaternionic Hilbert space embeddings of complex projective representations. Our definition (termed here a weak projective representation) encompasses such embedding...

We discuss the differing definitions of complex and quaternionic projective group representations employed by us and by Emch. The definition of Emch (termed here a strong projective representation) is too restrictive to accommodate quaternionic Hilbert space embeddings of complex projective representations. Our definition (termed here a weak projective representation) encompasses such embedding...

Given a unitary representation U of an Abelian group G and a subgroup H, we characterise the positive operator valued measures based on the quotient group G/H and covariant with respect to U .

As quantum parallelism allows the effective co-representation of classical mutually exclusive states, the diagonalization method of classical recursion theory has to bemodified.Quantumdiagonalization involves unitary operatorswhose eigenvalues are different from one.

We give an elementary construction of the Fibonacci model, a unitary braid group representation that is universal for quantum computation. This paper is dedicated to Professor C. N. Yang, on his 85-th birthday.