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Terry A. Loring

Publications and source records attributed to Terry A. Loring.

48 records · Page 3Linked to original sources
arXiv2013

Computing a logarithm of a unitary matrix with general spectrum

We analyze an algorithm for computing a skew-Hermitian logarithm of a unitary matrix. This algorithm is very easy to implement using standard software and it works well even for unitary matrices with no spectral conditions assumed. Certain examples, with many eigenvalues near -1, lead to very non-Hermitian output for other basic methods of calculating matrix logarithms. Altering the output of these algorithms to force an Hermitian output creates accuracy issues which are avoided in the considered algorithm. A modification is introduced to deal properly with the $J$-skew symmetric unitary matrices. Applications to numerical studies of topological insulators in two symmetry classes are discussed.

math.NA↗
arXiv2012

Factorization of Matrices of Quaternions

We review known factorization results in quaternion matrices. Specifically, we derive the Jordan canonical form, polar decomposition, singular value decomposition, the QR factorization. We prove there is a Schur factorization for commuting matrices, and from this derive the spectral theorem. We do not consider algorithms, but do point to some of the numerical literature. Rather than work directly with matrices of quaternions, we work with complex matrices with a specific symmetry based on the dual operation. We discuss related results regarding complex matrices that are self-dual or symmetric, but perhaps not Hermitian.

math.OA↗
arXiv2011

The Point in Weak Semiprojectivity and AANR Compacta

We initiate the study of pointed approximative absolute neighborhood retracts. Our motivation is to generate examples of C*-algebras that behave in unexpected ways with respect to weak semiprojectivity. We consider both weak semiprojectivity (WSP) and weak semiprojectivity with respect to the class of unital C*-algebras (WSP1). For a non-unital C*-algebra, these are different properties. One example shows a C*-algebra can fail to be WSP while its unitization is WSP. Another example shows WSP1 is not closed under direct sums.

math.OA↗
arXiv2011

Noncommutative Semialgebraic Sets in Nilpotent Variables

We solve the lifting problem in C^*-algebras for many sets of relations that include the relations x_j^{N_j} = 0 on each variable. The remaining relations must be of the form \| p(x_1,...,x_n) \| \leq C for C a positive constant and p a noncommutative *-polynomial that is in some sense homogeneous. For example, we prove liftability for the set of relations x^3=0, y^4=0, z^5=0, xx^*+yy^*+zz^* \leq 1. Thus we find more noncommutative semialgebraic sets that have the topology of noncommutative absolute retracts.

math.OA↗
arXiv2010

Noncommutative Semialgebraic sets and Associated Lifting Problems

We solve a class of lifting problems involving approximate polynomial relations (soft polynomial relations). Various associated C*-algebras are therefore projective. The technical lemma we need is a new manifestation of Akemann and Pedersen's discovery of the norm adjusting power of quasi-central approximate units. A projective C*-algebra is the analog of an absolute retract. Thus we can say that various noncommutative semialgebraic sets turn out to be absolute retracts. In particular we show a noncommutative absolute retract results from the intersection of the approximate locus of a homogeneous polynomial with the noncommutative unit ball. By unit ball we are referring the C*-algebra of the universal row contraction. We show projectivity of alternative noncommutative unit balls. Sufficiently many C*-algebras are now known to be projective that we are able to show that the cone over any separable C*-algebra is the inductive limit of C*-algebras that are projective.

math.OA↗
arXiv2010

C*-Algebra Relations

We investigate relations on elements in C*-algebras, including *-polynomial relations, order relations and all relations that correspond to universal C*-algebras. We call these C*-relations and define them axiomatically. Within these are the compact C*-relations, which are those that determine universal C*-algebras, and we introduce the more flexible concept of a closed C*-relation. In the case of a finite set of generators, we show that closed C*-relations correspond to the zero-sets of elements in a free sigma -C*-algebra. This provides a solid link between two of the previous theories on relations in C*-algebras. Applications to lifting problems are briefly considered in the last section.

math.OA↗
arXiv2009

Weakly Projective C*-Algebras

The noncommutative analog of an approximative absolute retract (AAR) is introduced, a weakly projective C*-algebra. This property sits between being residually finite dimensional and projectivity. Examples and closure properties are considered.

math.OA↗
arXiv2009

From Matrix to Operator Inequalities

We generalize Loewner's method for proving that matrix monotone functions are operator monotone. The relation x \leq y on bounded operators is our model for a definition for C*-relations of being residually finite dimensional. Our main result is a meta-theorem about theorems involving relations on bounded operators. If we can show there are residually finite dimensional relations involved, and verify a technical condition, then such a theorem will follow from its restriction to matrices. Applications are shown regarding norms of exponentials, the norms of commutators and "positive" noncommutative *-polynomials.

math.OA↗
arXiv2008

A Projective C*-Algebra Related to K-Theory

The C*-algebra qC is the smallest of the C*-algebras qA introduced by Cuntz in the context of KK-theory. An important property of qC is the natural isomorphism of K0 of D with classes of homomorphism from qC to matrix algebras over D. Our main result concerns the exponential (boundary) map from K0 of a quotient B to K1 of an ideal I. We show if a K0 element is realized as a homomorphism from qC to B then its boundary is realized as a unitary in the unitization of I. The picture we obtain of the exponential map is based on a projective C*-algebra P that is universal for a set of relations slightly weaker than the relations that define qC. A new, shorter proof of the semiprojectivity of qC is described. Smoothing questions related the relations for qC are addressed.

math.OA↗
arXiv2007

Renyi Dimension and Gaussian Filtering II

We consider convolving a Gaussian of a varying scale epsilon against a Borel measure mu on Euclidean delta-dimensional space. The Lq norm of the result is differentiable in epsilon. We calculate this derivative and show how the upper order of its growth relates to its lower Renyi dimension. We assume q is strictly between 1 and infty and that mu is finite with compact support. Consider choosing a sequence epsilon_n of scales for the Gaussians. The differences between the usual Lq norms at adjacent scales can be made to grow more slowly than any positive power of n by setting the epsilon_n by a power rule. The correct exponent in the power rule is determined by the lower Renyi dimension. We calculate and find bounds on the derivative of the Gaussian kernel versions of the correlation integral. We show that a Gaussian Kernel version of the Renyi entropy sum in continuous.

math.FA↗
arXiv2007

Renyi Dimension and Gaussian Filtering

Consider the partition function S(ε) associated in theory of Renyi dimension to a finite Borel measure μon Euclidean d-space. This partion function S(ε) is the sum of the q-th powers of the measure applied to a partition of d-space into d-cubes of width ε. We further Guerin's investigation of the relation between this partition function and the Lebesgue Lp norm (Lq norm) of the convolution of μagainst an approximate identity of Gaussians. We prove a Lipschitz-type esimate on the partition function. This bound on the partition function leads to results regarding the computation of Renyi dimension. It also shows that the partion function is of O-regular variation. We find situtations where one can or cannot replace the partition function by a discrete version. We discover that the slopes of the least-square best fit linear approximations to the partion function cannot always be used to calculate upper and lower Renyi dimension.

math.FA↗