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        Algebras of operators on Hilbert space, $C^*-$algebras, von Neumann algebras, non-commutative geometry

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        votes
        0answers
        79 views

        Anzai flow in noncommutative geometry

        Consider Anzai flows (cf. Anzai: Ergodic Skew Product Transformations on the Torus, Osaka Math. J. 3 (1951), 83-99) on the two dimensional torus $T^2$. I would like to know if there exists some ...
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        votes
        0answers
        110 views

        Cartan MASA and group measure space construction

        Let $N$ be a ${\rm II}_1$ factor. A maximal abelian self-adjoint subalgebra (MASA) is a $*$-subalgebra $A \subset N$ such that $A' \cap N = A$. It is called Cartan if $\mathcal{N}(A)''=N$, with the ...
        6
        votes
        0answers
        145 views

        A characterisation of certain $C^*$-algebras

        I was wondering if there is a characterisation for $C^*$-algebras (unital) for which the bidual does not have any central atoms. It is not sufficient for example to demand that the $C^*$-algebra does ...
        2
        votes
        1answer
        110 views

        Strongly Continuous Group Actions on the $ C^{\ast} $-Algebra of Compact Operators on a Hilbert Space

        Let $ \mathcal{H} $ be a not-necessarily-separable Hilbert space. Let $ G $ be a locally compact Hausdorff group. It is easy to see that if $ U: G \to \mathbb{U}(\mathcal{H}) $ is a norm-continuous ...
        0
        votes
        1answer
        45 views

        Examples of isomorphic W* algebra with non-homeomorphic weak topology

        Due to the uniqueness of the predual, a W* algebra, when realized as a von Neumann algebra in any way, always has a unique, well-defined ultraweak (or $\sigma$-weak) topology. The same can be said ...
        3
        votes
        2answers
        138 views

        Commutative direct summands of C*-algebras

        I have a question about commutative direct summands of $C$*-algebras. Let $A$ be a $C$*-algebra (with unit) and suppose that its bidual $A^{**}$ has a commutative direct summand, that is, $A^{**}=B\...
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        vote
        0answers
        36 views

        A 2- cocycle $\tau$ which is not cyclic but it still satisfies the stability of $\tau(e,e,e)$ for idempotent $e$

        I learned the following statement from page $20$ of the book Noncommutative Geometry by Alain Connes: Let $\tau$ be a $2$-cyclic cocyle on a $C^*$ algebra. Then for every smooth curve $e(t)$ of ...
        1
        vote
        1answer
        134 views

        When is $\inf_{n\geq0}x^n\neq0$?

        Let $M$ be a von Neumann algebra acting on a Hilbert space $H$. Let $x$ be a positive element of $M$ with $\|x\|=1$. So, $(x^n)_{n\in\mathbb N}$ is a decreasing sequence of positive elements and $y:=\...
        0
        votes
        0answers
        89 views

        Unitary element of the group algebra

        Let $G$ be a torsion free group. Are unitary elements of $\mathbb CG$ studied? By unitary element I mean an element $\alpha$ in $\mathbb CG$, such that $\alpha^*\alpha=1$? Do the triviality of unitary ...
        3
        votes
        0answers
        84 views

        Does this element belong to $\mathbb CG$?

        Let $G$ be a torsion-free group. Let $\alpha$ be a symmetric element of $\mathbb CG$, i.e. $\alpha^*=\alpha$, with $\|\alpha\|_1=\sum|\alpha(g)|<1$, so $\beta:=\sum_{n\ge 0}(-1)^n\alpha^n$ is an ...
        6
        votes
        1answer
        171 views

        Direct proof of “Nuclear implies $C_{red}^*(G) \cong C^*(G)$”

        It is well-known that for a discrete group $G$ the following statements are equivalent: $C_{red}^*(G)$ nuclear $C_{red}^*(G) \cong C^*(G)$ canonically i.e. there exists an *-isomorphism between the ...
        24
        votes
        3answers
        615 views

        What does it mean for a category to admit direct integrals?

        Given an infinite countable group $G$, the category of unitary representations of $G$ admits direct integrals. Namely, given a measure space $(X,\mu)$ and a measurable family of unitary $G$-reps $(H_x)...
        3
        votes
        0answers
        50 views

        To what extend can a von Neumann algebra be determined by its projection lattice structure?

        Let $ M, N $ be von Neumann algebras, $ P $ (resp. $Q$) the projection lattice of $M$ (resp. $N$). Any isomorphism $ \varphi : M \to N $ on the level of involutive algebras induces an isomorphism $ \...
        3
        votes
        1answer
        76 views

        Sequence in *-algebra with different limits for two C*-norms?

        The following question looks simple, but the answer is not obvious for me: Let $S$ be a $*$-algebra and $\left\Vert \cdot \right\Vert _{1}$, $\left\Vert \cdot \right\Vert _{2}$ $C^*$-norms on $S$ ...
        4
        votes
        1answer
        120 views

        Behaviour of direct limit with matrices

        I am trying to understand direct limits in the category of $C^*$-algebras by self reading. My last question was also related to direct limits. Here is another of my doubts: Let $(A_n,f_n)$ be a ...

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