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      89 views

      On examples of action of C-star simple group on von Neumann algebra

      Can there exist a faithful action of a $C^{*}$-simple group $G$ on a von Neumann algebra $(M,\varphi)$ equipped with faithful normal state $\varphi$ such that action preserves the state $\varphi$ and ...
      0
      votes
      1answer
      170 views

      A certain class of representations

      Let $g$ be a non-identity element in a torsion-free amenable group, does there exist a finite-dimensional unitary representation $\pi$ with $\pi(g)\neq 1$? (The word "finite-dimensional" was ...
      6
      votes
      0answers
      51 views

      Analogue of Friedrichs extension for Hilbert $C^*$-modules

      Suppose one has a densely defined symmetric operator $T:\mathcal{M}\rightarrow\mathcal{M}$, where $\mathcal{M}$ is a Hilbert $A$-module for a $C^*$-algebra $A$. Suppose that $T$ is non-negative, so ...
      4
      votes
      1answer
      74 views

      Graded adjointable operators on a graded Hilbert space

      Given a graded Hilbert space $\mathbf{H} = \bigoplus_{k \in \mathbb{N}} \mathbf{H}_k$, one might extend the notion of adjoint to a "graded adjoint" defined as follows: $L \in B(\mathbf{H})$ is said to ...
      2
      votes
      1answer
      91 views

      Extending $C^*$-norms from $*$-subalgebras

      Let $A$ be a unital $*$-algebra, and $B$ a unital $*$-subalgebra of $A$. In addition, assume that there exists a $B$-$B$-sub-bimodule $C \subset A$, such that $$ A \simeq B \oplus C, $$ where $\...
      2
      votes
      1answer
      125 views

      Does the square root of a finite propagation operator have finite propagation?

      Let $X$ be a non-compact manifold and let $C_0(X)$ act on $L^2(X)$ by pointwise multiplication. We say $T\in\mathcal{B}(L^2(X))$ has finite propagation if there exists an $r>0$ such that: for all ...
      6
      votes
      1answer
      77 views

      Tensoring adjointable maps on Hilbert modules

      Given a right Hilbert $A$-module $E$, and a right Hilbert $B$-module $F$, together with non-degenerate $*$-homomorphism $\phi:A \to \mathcal{L}_B(F)$, we can form the tensor product $$ E \otimes_{\phi}...
      2
      votes
      1answer
      75 views

      Twisted canonical commutation relations

      I am dealing with universal C*-algebra generated by $x,y$ with the following relations: $xy = qyx$, $x^{*}y = qyx^{*}$, $y^{*}x = qxy^{*}$, $x^{*}x = q^2xx^{*} - (1-q^2)yy^{*}$, $y^{*}y = q^2yy^{*} - (...
      1
      vote
      0answers
      133 views

      A criterion for abelian $C^*$ algebra [closed]

      Let $A$ be a unital $C^*$ algebra such that for any two positive elements $x$, $y$ in $A$, whenever $x\leq y$ we have that $x^2\leq y^2$. Prove that $A$ is abelian.
      5
      votes
      1answer
      207 views

      Variations on Kaplansky Density

      Let $A$ be a $C^*$-algebra and $\pi:A\rightarrow B(H)$ a faithful $*$-representation, so $M=\pi(A)''$ is a von Neumann algebra and $A\rightarrow M$ is an inclusion. von Neumann's Bicommutant Theorem ...
      3
      votes
      1answer
      234 views

      Norm of two operators on $l^2(\mathbb{Z}_{2}*\mathbb{Z}_{2})$ different

      In my research I encounered the following (very concrete) question: Consider the (discrete) group $G:=\mathbb{Z}_{2}*\mathbb{Z}_{2}$. Let $s\text{, }t\in G$ be the generating elements and define for $\...
      0
      votes
      0answers
      47 views

      On cyclicity of a module

      Let $A$ be a $\text{ von Neumann algebra }$, $\mathcal{H}$ is a cyclic $A$ module, $G$ be a finite group acting on $A$, is $\mathcal{H}$ cyclic module over fixed point subalgebra of the action? ...
      2
      votes
      0answers
      147 views

      A Banach or $C^*$ algebraic analogy of a classical fact in real analysis

      Let $A$ be a commutative unital Banach algebra.The maximal ideal space of $A$ is denoted by $\hat A$. Assume that $D:A \to A$ is a derivation. Fix an element $a\in A$. Assume that for every $\phi\in \...
      2
      votes
      0answers
      59 views

      Closable operators on Hilbert modules

      For $T:{\frak{Dom}}(T) \to H$ a densely defined operator, with $H$ a (separable) Hilbert space, we know that $T$ is closable if its adjoint $T^*$ has dense domain in $H$. Does this extend to the (...
      2
      votes
      1answer
      78 views

      Complete positivity with infinite dimensional auxillary spaces

      The usual definition of complete positivity (e.g. Stinespring (1955), or Holevo's Statistical Structure of Quantum Theory) is that a linear map between (sub $C^*$ algebras of) the bounded operators on ...

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