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6 votes
1 answer
170 views

Do projections in an $AW^\ast$-algebra form an orthomodular lattice?

I’m currently studying orthomodular lattices arising out of operator algebras. One of the most standard examples is the projection lattice of a von Neumann algebra - if $M$ acts on a Hilbert space $H$,...
David Gao's user avatar
  • 2,830
2 votes
0 answers
147 views

About normal states in abstract von Neumann algebras

In the book "Fundamental of the theory of operator algebras" (KAdisong and Ringrose, Vol 2) we have the Corollary 7.1.16 but this was state only for concrete von Neumann algebras (because ...
Gabriel Palau's user avatar
2 votes
0 answers
158 views

Question about the ergodic mean

This is a repost from this MathStackExchange question, where unfortunately I was not able to resolve this question. I've read a thesis where there is an example on ergodic mean, where however there is ...
MBlrd's user avatar
  • 33
4 votes
0 answers
169 views

Drinfeld center of a tensor category

Firstly, apologies for the imprecise language, I'm a physicist trying to understand anyonic excitations from the lens of category theory. If I have a category (say $\operatorname{Rep}(\mathbb{Z}_2)$) ...
pyroscepter's user avatar
1 vote
1 answer
119 views

Complemented submodules of a Hilbert C*-module

Let $A$ be a $C^*$-algebra and $E$ be a (right) Hilbert $C^*$-module over $A$. Assume $F$ is a closed submodule of $E$ such that $F^\perp := \{x \in E: \langle x, F\rangle=0\}$ is orthogonally ...
Andromeda's user avatar
  • 175
0 votes
1 answer
154 views

Why is $q(f,g) = (f-g,0)$ not adjointable?

Let $A= C([0,1])$ and $J= \{f \in A: f(0) = 0\}$. Consider the Hilbert $C^*$-module $E:= A \oplus J$ (with the obvious right $A$-action and inner product). I want to prove that $$q: E \to E: (f,g) \...
Andromeda's user avatar
  • 175
5 votes
1 answer
341 views

Reference request for Deterministic $\subset$ Random $\subset$ Quantum

I hope this post is on topic as a reference request. I have seen somewhere the idea of (and saw it written just like this): $$\text{Deterministic }\subset\text{ Random }\subset\text{ Quantum }.$$ I am ...
JP McCarthy's user avatar
  • 1,037
-1 votes
1 answer
210 views

A commuting pair of isometries

Let $H$ be a Hilbert space and $B(H)$ be the space of all bounded operators on $H$. The Wold decomposition says that: an operator $x$ in $B(H)$ is an isometry if and only if $x=x_u\oplus x_s$ where $...
ABB's user avatar
  • 4,058
5 votes
1 answer
388 views

hereditary C*-subalgebra of a non-elementary simple C*-algebra

A is said to be elementary if A is isomorphic to some $K(H)$ or $M_n$. A C*-subalgebra $B$ is said to be hereditary if for every $0≤a≤b∈B$ we have $a∈B$. I wanted to know that is this statement true? ...
Peg Leg Jonathan's user avatar
4 votes
1 answer
110 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 ...
Dave Shulman's user avatar
6 votes
1 answer
119 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}...
Dave Shulman's user avatar
6 votes
2 answers
297 views

Finite-dimensional Hilbert $C^*$-modules

Does there exist a classification, or characterization, of finite-dimensional Hilbert $C^*$-modules? More generally, does there exist a characterization of countable direct sums of finite-dimensional ...
Max Schattman's user avatar
6 votes
1 answer
788 views

A spectral description of Fredholm operators

Let $L:H \to H$ be a bounded operator on a Hilbert space $H$, with finite dimensional kernel, and whose adjoint also has finite dimensional kernel. Is it true that $L$ is Fredholm if and only if its ...
Max Schattman's user avatar
2 votes
0 answers
116 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 (...
Max Schattman's user avatar
8 votes
1 answer
572 views

Comparing the definitions of $K$-theory and $K$-homology for $C^*$-algebras

In Higson and Roe's Analytic K-homology, for a unital $C*$-algebra $A$, the definitions of K-theory and K-homology have quite a similar flavor. Roughly, the group $K_0(A)$ is given by the ...
Dave Shulman's user avatar
7 votes
1 answer
200 views

If $\ $ $yx_n\to 0 $ for all $y$ in a C$^*$-algebra, Is it true that $x_n$ is weakly convergent to $0$?

$A$ is a C$^*\! $-algebra and $(x_n)_{n\in \mathbb{N}} \subseteq A $. If $\ $ $yx_n\to 0 $ for all $y\in A$, Is it true that $x_n$ is weakly convergent to $0$ ? For unitals this is trivial. ...
Darman's user avatar
  • 327
4 votes
1 answer
357 views

Extending maps from dense $*$-algebras of $C^*$-algebras

Given $\cal{A},\cal{B}$ two dense $*$-algebras of two $C^*$-algebras $A$ and $B$ respectively, together with a $*$-algebra homomorphism $f:\cal{A} \to \cal{B}$, is it clear that $f$ extends to a ...
Max Schattman's user avatar
4 votes
1 answer
157 views

Geometric Motivation for Hilbert $C^*$-Bimodules

I'm trying to get an understanding of Hilbert $C^*$-bi-modules from a geometric point of view. As is well-known, we have that i) Commutative unital $C^*$-algebras correspond to compact Hausdorff ...
Ago Szekeres's user avatar
1 vote
1 answer
143 views

strong convergence in Hilbert c* module

For a monotone sequence of projections in Hilbert c* module, do we have similar conclusion as in Hilbert space (such as strong convergence)? If not, what could be said in that situation?
Fiona's user avatar
  • 11
4 votes
1 answer
384 views

A Hilbert-space completion of a Hilbert $ C^{*} $-module over a separable $ C^{*} $-algebra

Let $ B $ be a separable $ C^{*} $-algebra and $ \mathcal{E} $ a Hilbert $ B $-module. We know that $ B $ has a faithful state $ \phi $. Using $ \phi $, we can construct a $ \mathbb{C} $-valued pre-...
Transcendental's user avatar