M. Weigt

Bio: M. Weigt is an academic researcher from Nelson Mandela Metropolitan University. The author has contributed to research in topics: Tensor product of algebras & Tensor product of Hilbert spaces. The author has an hindex of 2, co-authored 2 publications receiving 13 citations.

Tensor products of unbounded operator algebras

Abstract: The term $GW^*$-algebra means a generalized $W^*$-algebra and corresponds to an unbounded generalization of a standard von Neumann algebra. It was introduced by the second named author in 1978 for developing the Tomita-Takesaki theory in algebras of unbounded operators. In this note we consider tensor products of unbounded operator algebras resulting in a $GW^*$-algebra. Existence and uniqueness of the $GW^*$-tensor product is encountered, while ``properly $W^*$-infinite" $GW^*$-algebras are introduced and their structure is investigated.

Quantum functional analysis : non-coordinate approach

Abstract: This book contains a systematic presentation of quantum functional analysis, a mathematical subject also known as operator space theory. Created in the 1980s, it nowadays is one of the most prominent areas of functional analysis, both as a field of active research and as a source of numerous important applications. The approach taken in this book differs significantly from the standard approach used in studying operator space theory. Instead of viewing ""quantized coefficients"" as matrices in a fixed basis, in this book they are interpreted as finite rank operators in a fixed Hilbert space. This allows the author to replace matrix computations with algebraic techniques of module theory and tensor products, thus achieving a more invariant approach to the subject. The book can be used by graduate students and research mathematicians interested in functional analysis and related areas of mathematics and mathematical physics. Prerequisites include standard courses in abstract algebra and functional analysis.|This book contains a systematic presentation of quantum functional analysis, a mathematical subject also known as operator space theory. Created in the 1980s, it nowadays is one of the most prominent areas of functional analysis, both as a field of active research and as a source of numerous important applications. The approach taken in this book differs significantly from the standard approach used in studying operator space theory. Instead of viewing ""quantized coefficients"" as matrices in a fixed basis, in this book they are interpreted as finite rank operators in a fixed Hilbert space. This allows the author to replace matrix computations with algebraic techniques of module theory and tensor products, thus achieving a more invariant approach to the subject. The book can be used by graduate students and research mathematicians interested in functional analysis and related areas of mathematics and mathematical physics. Prerequisites include standard courses in abstract algebra and functional analysis.

Derivations of Frechet nuclear GB*-algebras

Abstract: It is an open question whether every derivation of a Fréchet GB$^{\\ast }$-algebra $A[{\\it\\tau}]$ is continuous. We give an affirmative answer for the case where $A[{\\it\\tau}]$ is a smooth Fréchet nuclear GB$^{\\ast }$-algebra. Motivated by this result, we give examples of smooth Fréchet nuclear GB$^{\\ast }$-algebras which are not pro-C$^{\\ast }$-algebras.

Crossed products of algebras of unbounded operators

Abstract: Consider a closed $O^*$--algebra $\mathcal{M}$ on a dense linear subspace $\mathcal{D}$ of a Hilbert space $\mathcal{H}$, a locally compact group $G$ with left invariant Haar measure $ds$ and an action $\alpha$ of $G$ on $\mathcal{M}$. Under some natural conditions, the $O^*$--crossed product $\mathcal{M}\underset{\alpha}{\overset{O^*}{\rtimes}}G$ of $\mathcal{M}$ and the $GW^*$--crossed product $\mathcal{M}\underset{\alpha}{\overset{GW^*}{\rtimes}}G$ are introduced. When $G$ is also abelian, the dual action $\widehat{\alpha}$ of the dual group $\widehat{G}$ on $\mathcal{M}\underset{\alpha}{\overset{O^*}{\rtimes}}G$ and on $\mathcal{M}\underset{\alpha}{\overset{GW^*}{\rtimes}}G$ is defined, which makes it possible to study the crossed products $(\mathcal{M}\underset{\alpha}{\overset{O^*}{\rtimes}}G)\underset{\widehat{\alpha}}{\overset{O^*}{\rtimes}}\widehat{G}$ and $\mathcal{M}\underset{\alpha}{\overset{GW^*}{\rtimes}}G)\underset{\widehat{\alpha}}{\overset{GW^*}{\rtimes}}\widehat{G}$. In case of modular actions, these constructions are used to obtain results on duality of type $\mathrm{II}$--like and type $\mathrm{III}$--like $GW^*$--algebras.