Journal of Linear and Topological Algebra
Volume:11 Issue: 2, Spring 2022

‎Topological complexity which plays an important role in motion planning problem can be generalized to homotopic distance $mathrm{D}$ as introduced in cite{MVML}‎. ‎In this paper‎, ‎we study the homotopic distance and mention that it can be realized as a pseudometric on $mathrm{Map}(X,Y)$‎. ‎Moreover we study the topology induced by the pseudometric $mathrm{D}$‎. ‎In particular‎, ‎we consider the space $mathrm{Map}(S^1,S^1)$ and use the non-compactness of it to talk about the non-compactness of $mathrm{Map}(X,Y)$‎.

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‎In this paper we construct a $2$-positive map from $ma_4(Complex)$ to $ma_5(Complex)$ and state the conditions under which the map is positive and completely positive (copositivity of positive)‎. ‎The construction allows us to create a decomposable map, where the Choi matrix of complete positivity is equal to the Choi matrix of complete copositivity‎.

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‎Let $mathcal{A}$ and $mathcal{U}$ be Banach algebras‎, ‎$theta$ be a nonzero character on $mathcal{A}$ and let ${mathcal{A}}times_{theta}{mathcal{U}}$ be the corresponding Lau product Banach algebra‎. ‎In this paper we investigate derivations and multipliers of ${mathcal{A}}times_{theta}{mathcal{U}}$ and study the automatic continuity of these maps‎. ‎We also study continuity of the derivations for some special cases of $mathcal{U}$ and the Banach $({{mathcal{A}}times_{theta}mathcal{U}})$-bimodule ${mathcal{X}}$ and establish various results in this respect‎. ‎Some of the results are devoted to find conditions under which one can represent a derivation on ${{mathcal{A}}times_{theta}mathcal{U}}$ as sum of two derivations in such a way that one of them is continuous‎. ‎Some examples are also given‎.

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‎This is a survey of a variety of equivariant (co)homology theories for operator algebras‎. ‎We briefly discuss a background on equivariant Hochschild cohomology‎. ‎We discuss a notion of equivariant $ L^2 $-cohomology and equivariant $ L^2 $-Betti numbers for subalgebras of a von Neumann algebra‎. ‎For graded $C^*$-algebras (with grading over a group) we elaborate on a notion of graded $ L^2 $-cohomology and its relation to equivariant $L^2$-cohomology‎.

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Geometric algebra provides intuitive and easy description of geometric entities (encoded by blades) along with different operations and orthogonal transformations. Grassmann's Exterior and Hamilton's quaternions lead to the existence of Clifford (Geometric) algebra. Clifford or geometric product has its significant role in whole domain of Clifford algebra, while as contraction (anti outer product or analogous to dot product) is grade reduction operation. The other operations can be derived from the former one. The paper explores elucidation of Clifford algebra and Clifford product with some salient features and applications.